reduced complexity signal detection and channel estimation

Reduced Complexity Signal Detectionand Channel Estimation for Iterative

MIMO-OFDM Systems

Licai Fang

This thesis is presented for the degree of Doctor of PhilosophySchool of Electrical, Electronic and Computer Engineering

May 2016

Abstract

Multi-Input Multi-Output (MIMO) is a key technology in broadband wireless commu-nications, and it has been used in WiMax, LTE and WiFi (802.11n/ac). As OrthogonalFrequency Division Multiplexing (OFDM) can transform a frequency selective fadingchannel into a set of parallel frequency flat fading channels and thus greatly reducethe complexity of equalization, MIMO is typically combined with OFDM in practicalapplications. For a MIMO-OFDM system, the channel estimation and signal detectionalgorithms based on linear-minimum-mean-square-error (LMMSE) are often employedbecause of their good performance. But conventional algorithms typically require amatrix inversion with cubic level complexity, which is a major obstacle for practicalimplementation.

To reduce the complexity, in this thesis, we focused on algorithms design by reducingthe number of costly operations and the cost of each operation. Due to the law of largenumbers, the matrix to be inverted, in both the LMMSE channel estimation of an OFDMsystem and the uplink signal detection of a massive MIMO system (i.e., both the numberof transmit and receive antennas are large), approaches a diagonally dominant matrix.By exploiting this special structure, the Neumann series expansion was employed toreduce the complexity of matrix inversion from cubic to quadratic level. At the sametime, we found that in a massive MIMO-OFDM system there are strong correlationsbetween the matrix inversions in uplink LMMSE detection of adjacent subcarriers.Similar correlations were also found between different iterations of an LMMSE detectorin a turbo MIMO-OFDM system. By exploiting the correlations between adjacentsubcarriers or different iterations, interpolation based methods can effectively reduce thenumber of costly operations.

Specifically, in this thesis, an LMMSE detection algorithm for turbo-MIMO systems,which exploits the correlation of matrix inversion between different iterations, wasproposed to reduce the complexity of non-first iterations from O(N3

t ) to O(N2t ) where

Nt is the number of transmit antenna. Then a Partial Gaussian method was proposed tobe employed for spatially correlated channels, and a branch-and-bound algorithm wasproposed to reduce the complexity of the Partial Gaussian algorithm. For LMMSE chan-

iv

nel estimation of OFDM systems, a low complexity algorithm based on Neumann seriesexpansion was investigated. This proposed algorithm can achieve mean-square error(MSE) performance close to the optimal LMMSE estimator but with only O(N logL)complexity where N is the number of subcarriers and L is the number of time domainchannel coefficients taps. With the aid of turbo processing, we also proposed a data-aidedchannel estimator which can track time-varying channels caused by terminals movement(up to 100 km/hour) with very low pilot overhead.

We also investigated medium-sized massive MIMO systems. A low cost LMMSEdetection algorithm based on Neumann series expansion for uplink applications wasproposed. Compared to alternative algorithms, the algorithm can significantly reducethe total detection complexity to O(KNtNr) where Nr is the number of receive antennaand K (typically K < 3) is the number of Neumann series expansion. The computationsaving comes from the fact that proposed algorithm can not only avoid computing matrixinversion but also replace matrix-matrix multiplications with matrix-vector multiplica-tions.

List of Publications

[1] L. Fang, and D. Huang. Neumann Series Expansion Based LMMSE ChannelEstimation for OFDM Systems. IEEE Communications Letters, vol. 20, no. 4, pp.748-751, April 2016. (Chapter 4)

[2] L. Fang, L. Xu, and D. Huang. Low complexity iterative MMSE-PIC detec-tion for medium-size massive MIMO. IEEE Wireless Communications Letters,5(1):108–111, Feb 2016. (Chapter 5)

[3] Licai Fang, Lu Xu, Qinghua Guo, Defeng Huang, and S. Nordholm. A lowcomplexity iterative soft-decision feedback MMSE-PIC detection algorithm formassive MIMO. In 2015 IEEE International Conference on Acoustics, Speech andSignal Processing (ICASSP), pages 2939–2943, 2015. (Chapter 2)

[4] Licai Fang, Lu Xu, Qinghua Guo, D.D. Huang, and S. Nordholm. A hybrid iterativeMIMO detection algorithm: Partial Gaussian approach with integer programming.In 2014 IEEE/CIC International Conference on Communications in China (ICCC),pages 463–468, 2014. (Chapter 3)

Acknowledgements

First, I would like to thank my supervisors Prof. David (Defeng) Huang andDr. Qinghua Guo for their support, for giving me the opportunity to pursue my Ph.D.Without their directions, enlightenments and encouragements, this thesis would havebeen impossible.

Then I would like to thank the colleagues in the Signal Processing Wireless Commu-nication Laboratory (SPWCL) research group at the University of Western Australia,namely, Dr. Lu Xu, Dr. Jindan Yang, Dr. Hang Li and Dr. T.-U. I. Khandoker. Theirinsightful academic discussion is invaluable to my research.

Most importantly, my sincere thanks go to my wife Dr. Wei Hou and our families.Their consistent supports are the main driving force for me to finish this thesis duringmy 40s.

Table of contents

List of Publications v

List of figures xiii

List of tables xv

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 MIMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Turbo Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.3 OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Turbo MIMO-OFDM System . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 LDPC Encoder and Decoder . . . . . . . . . . . . . . . . . . . 81.2.3 Soft Mapper and Soft Demapper . . . . . . . . . . . . . . . . . 111.2.4 Signal Detection . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.5 Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . . 19

1.3 Motivations and Contributions . . . . . . . . . . . . . . . . . . . . . . 221.3.1 Signal Detection . . . . . . . . . . . . . . . . . . . . . . . . . 221.3.2 Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . . 24

1.4 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2 A Low Complexity Soft-Decision Feedback MMSE-PIC Detection Algo-rithm 272.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2 Gaussian Model Based MMSE Detection Algorithm . . . . . . . . . . 292.3 Complexity Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3.1 Low Complexity Matrix Inversion . . . . . . . . . . . . . . . . 302.3.2 A Heuristic Approach to Solve the Stability Problem . . . . . . 32

x Table of contents

2.3.3 Computational Complexity Comparison . . . . . . . . . . . . . 32

2.3.4 Iterative Method to Improve First-pass Performance . . . . . . 33

2.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4.2 BER Performance . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 MIMO Detection Algorithm: Partial Gaussian Approach with Integer Pro-gramming 393.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 Partial Gaussian Approach with Integer Programming . . . . . . . . . . 42

3.3.1 PGA Detection Algorithm . . . . . . . . . . . . . . . . . . . . 42

3.3.2 Simplified Marginalization Calculation . . . . . . . . . . . . . 42

3.3.3 Resolving QIP with the Branch-and-Bound algorithm . . . . . . 45


3.4.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4.2 BER Performance . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4.3 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4 A Low Cost LMMSE Channel Estimator for OFDM Systems 514.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3 LMMSE Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . 53

4.4 Newmann Series Expansion Based Channel Estimation . . . . . . . . . 54

4.4.1 Neumann Series Expansion . . . . . . . . . . . . . . . . . . . 54

4.4.2 Computational Complexity Comparison . . . . . . . . . . . . . 56


4.5.1 Mean-Square Error (MSE) Performance for Time-Invariant Chan-nels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.5.2 Bit Error Rate (BER) Performance for Iterative Systems . . . . 58

4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.6.1 The Power Delay Profile (PDP) . . . . . . . . . . . . . . . . . 60

4.6.2 The Assumption of Quasi-static Channel . . . . . . . . . . . . 60

4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Table of contents xi

5 Low Complexity Iterative MMSE-PIC Detection for Medium-Size MassiveMIMO 635.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.3 MMSE Detection Based on Neumann Series Expansion . . . . . . . . . 66

5.3.1 Neumann Series Expansion . . . . . . . . . . . . . . . . . . . 685.3.2 Computational Complexity Comparison . . . . . . . . . . . . . 685.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6 A Novel Interpolation Algorithm for Massive MIMO OFDM System Detec-tion 736.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.2 System Model and Soft-output MMSE Detector . . . . . . . . . . . . . 746.3 MMSE Detection Based on Interpolation . . . . . . . . . . . . . . . . . 76

6.3.1 Correlation of Matrix Inversion for Massive MIMO-OFDMSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.3.2 Interpolation Based Matrix Inversion . . . . . . . . . . . . . . 796.3.3 Computational Complexity Comparison . . . . . . . . . . . . . 80

6.4 BER Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

7 Summary and Future Work 857.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7.2.1 Channel estimation for MIMO-OFDM systems . . . . . . . . . 867.2.2 Channel estimation for Massive MIMO . . . . . . . . . . . . . 867.2.3 Uplink Signal Detection for Massive MIMO-OFDM . . . . . . 87

Appendix A Proof of the Equality of Algorithm 1 and Algorithm 2 89

Bibliography 93

List of figures

1.1 An Iterative MIMO-OFDM Communication System . . . . . . . . . . 51.2 QC-LDPC Base Parity Check Matrix . . . . . . . . . . . . . . . . . . . 81.3 4-PAM Constellation Diagram . . . . . . . . . . . . . . . . . . . . . . 14

2.1 Iterative Detection and Decoding of a MIMO Communication System . 292.2 Iterative Soft-in Soft-Out MMSE Detector . . . . . . . . . . . . . . . . 332.3 BER Performance Comparison Between Exact Implementation and

Proposed Approximation for a 16×16 MIMO System. . . . . . . . . . 342.4 BER Performance Comparison Between Different Number of Self-

iterations for 32×32 MIMO. . . . . . . . . . . . . . . . . . . . . . . . 352.5 BER Performance Comparison Between Different Number of Self-

iterations for 16×16 MIMO. . . . . . . . . . . . . . . . . . . . . . . . 372.6 BER Performance Comparison Between Different Number of Self-

iterations for 4×4 MIMO. . . . . . . . . . . . . . . . . . . . . . . . . 38

3.1 Iterative Detection and Decoding of a MIMO Communication System . 403.2 An example of the proposed branch and bound algorithm where d is the

tree level, lb means low bound, ub means upper bound and m∗ is thevector that minimizes f (m). Because the first heuristic solution happensto be the final solution, there are only 6 nodes visited. . . . . . . . . . . 46

3.3 BER performances of 16-QAM 40×40 MIMO with correlation factorρ = 0.5 and ρ = 0.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4 BER performance comparison between PGA-Exact and PGA-IP under16-QAM 40×40 MIMO correlated channel (ρ = 0.4) . . . . . . . . . . 50

4.1 MSE performance with different L at SNR of 14dB . . . . . . . . . . . 554.2 MSE performance for the 10-tap COST259_RAx channel . . . . . . . . 574.3 BER performance for 10-tap COST259_RAx Channel at speed of 100

km/hour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

xiv List of figures

4.4 MSE Under Channel No.1-6 . . . . . . . . . . . . . . . . . . . . . . . 62

5.1 BER performance comparison for exact MMSE, proposed and SORbased [1] with MIMO size of K ×M = 16×128 . . . . . . . . . . . . . 70

6.1 Correlations of Ch(d) and Cg(d) of adjacent subcarriers with N = 64,Nt = 20, different ρ and different subcarrier distance d. . . . . . . . . . 77

6.2 Correlations of Ch(d) and Cg(d) of adjacent subcarriers (with differentd) under different channel models with N = 64, Nt = 20 and ρ = 8. . . 79

6.3 Complexity comparison with ρ = 8, N = 128 and I = 5 . . . . . . . . . 816.4 BER performance comparison for exact MMSE, Matched filter, Pro-

posed Vpn with exact Hn and Proposed Vp

n with interpolated Hn forNt ×Nr = 16×128 MIMO. . . . . . . . . . . . . . . . . . . . . . . . . 83

List of tables

2.1 Computational Complexity Comparison . . . . . . . . . . . . . . . . . 33

3.1 Average CPU run time (s) comparison between MMSE_PIC, PGA_IPand PGA_Exact for detecting 2000bits with 3 iterations under 40×40MIMO with 16-QAM on a X86 Linux PC . . . . . . . . . . . . . . . . 49

4.1 Simulated Channel Models [2] . . . . . . . . . . . . . . . . . . . . . . 61

5.1 Computational Complexity Comparison . . . . . . . . . . . . . . . . . 69

6.1 Simulated Channel Models [2] . . . . . . . . . . . . . . . . . . . . . . 786.2 Computational Complexity Comparison . . . . . . . . . . . . . . . . . 80

Chapter 1

Introduction

1.1 Background

1.1.1 MIMO

In late 1980’s, the multiple-input multiple-output antenna (MIMO) systems was pro-

posed for wireless communications. By using multiple antennas at both transmitter

and receiver side, MIMO can create multiple parallel channels using the same radio

spectrum [3] [4]. MIMO techniques can improve communications performance by either

increasing reliability or maximizing throughput. In order to increase reliability, some

form of space-time coding (STC) is typically employed to combat multipath scatting

by creating spatial diversity [5]. While for improving throughput, spatial multiplexing

techniques [6] [7] are employed to exploit multipath scatting. It was shown that the

achievable transmitting rate of MIMO systems scales as min(Nt ,Nr)log(1+SNR) and

the link outage scales as SNRNtNr [8] where Nt and Nr are the numbers of transmitter

and receiver antennas, respectively.

MU-MIMO

For the cellular systems, the conventional MIMO technology has some limitations

because the terminals can not employ many antennas due to the cost, power and size

2 Introduction

constraint. Another issue of conventional MIMO is the propagation limitations; in case

of LOS (line-of-sight) propagation, channel rank loss or antenna correlation, the spatial

multiplexing gain in conventional MIMO will be severely degraded [9]. To achieve the

gain of multiple access capacity and overcome above two issues, the multi-user MIMO

(MU-MIMO) scheme had been proposed and researched in recent years. By treating

every user’s terminal as a virtual MIMO antenna, the MIMO spatial multiplexing gain

can be preserved. Although the individual users will not experience increased throughput

by MU-MIMO, but the overall system performance will improve dramatically. So many

state-of-the-art wireless communication standards have adopted MU-MIMO, like 3GPP

long-term evolution advanced (3GPP LTE-A)(Release 10) [10], IEEE 802.16m (WiMAX

Profile 2.0) [11] and Wifi (802.11ac) [12].

Massive MIMO

With the maturing of MU-MIMO, by making the number of antennas much larger at Base

Station side, comes the concept of massive MIMO, which is characterized with hundreds

of antennas at Base Station and can serve tens of terminals simultaneously. Massive

MIMO can reap all the benefits of conventional MIMO and MU-MIMO in a much

greater scale [13] [14]. Firstly, high energy efficiency can be obtained by focusing the

energy with extreme sharpness into small regions in space. Specifically, by appropriately

shaping the signals sent out by the antennas, all radio wave fronts collectively emitted by

all antennas interfere constructively at the intended terminals, but destructively almost

everywhere else. [15] illustrated that the energy focus effect by comparing M = 10

transmit antennas (M-element Uniform Linear Array (ULA)) and M = 100 antennas. It

shows that when the number of transmit antenna at the transmitter is 100, by applying

spatial precoding, the field strength can be focused to a point rather than in a certain

direction as done in conventional MIMO or MU-MIMO. This energy focus property can

greatly reduce the interference between spatially separated users and reduce the total

radiated signal power, thereby the Base Station can benefit from this property to greatly

reduce the total output RF power. At the same time, based on information theory [15],

1.1 Background 3

massive MIMO can increase the spectral efficiency 10+ times from the aggressive spatial

multiplexing.

Besides the above base station scenario where the communication is multipoint-

to-point for uplink or point-to-multipoint for downlink, there is also point-to-point

applications like the back-haul connections between base stations. For this kind of

configuration, a large number of antennas can be used both at transmit and receive base

stations.

It is also worth noting that when the number of receive antennas at the base station is

large and much larger than the total number of transmit antennas in user terminals, a

simple detection algorithm such as a matched filter can achieve very good performance,

as with the assumption of i.i.d. entries for channel matrix H, the channel vectors become

orthogonal to each other and HHH converges to a scaled identity matrix. But from

practical implementation point of view, medium size antenna arrays are also of interest.

1.1.2 Turbo Principle

Nearly at the same time as the emerging of MIMO technology, the invention of turbo

codes and iterative decoding [16] paved the way for achieving system performance

close to the Shannon limit. By exchanging information between several decoding

units iteratively, the system performance was shown to be close to optimal decoding,

but with feasible complexity. Then the “turbo principle” [16] was used to improve

performance of other tasks in the wireless receiver, e.g., equalization [17] [18] [19],

channel estimation [20], multi-user detection [21] and MIMO detection [22] [23] [24].

For a coded communication system, as the complexity of the optimal receiver is

exponential in the length of the data transmitted, most practical receivers include two

separate blocks: signal detection and channel decoding. The signal detectors have been

designed to process the received observations to account for the effects of the channel

and to estimate the transmitted channel symbols that best fit the observed data. Then the

soft information (in the form of Log-Likelihood Ratio (LLR)) is passed to the channel

decoder for decoding.

4 Introduction

Applying the “turbo principle” to this kind of receiver, comes the iterative detection

and decoding (IDD) system. In IDD, a soft-input and soft-out detector is required which

can accept soft information from the decoder and output soft information to the decoder.

In general, only extrinsic information can be exchanged between the detector and the

decoder [25].

“Turbo principle” can also be applied to the task of channel estimation. In order to

track channel variation caused by movement of terminals, data-aided scheme is often

employed. For slow fading channel with preamble-type pilots, the channel coefficients

copied from last symbol can be improved by exploiting the soft or hard information

feedback from the decoder as the virtual pilot [26]. Similarly, for superimposed-type

pilot, it is common to perform iterative channel estimation and decoding by exploiting

data fed back from the channel decoder [20].

1.1.3 OFDM

Most modern wireless communication systems are broadband systems which have high

data rates. As a result, the symbol rate is much higher than the channel coherence

bandwidth and thus the channel is frequency selective. The major issue about frequency

selective fading is the inter-symbol interference (ISI), which is caused by the fact that the

symbol period is shorter than the delay spread. To combat with ISI, one way is to employ

equalization with single carrier. As the computational complexity of equalization is

quite high, another popular technique for coping with frequency-selective fading effects

is using orthogonal frequency division multiplexing (OFDM).

The idea behind OFDM is to split a broadband signal that experiences frequency-

selective fading into multiple narrow sub-bands (subcarrier) so that each subcarrier

experiences flat fading. Because the bandwidths of the sub-bands is less than the

coherence bandwidth of the channel, each sub-stream is far less vulnerable to the ISI

than the original input stream. At the same time, although each OFDM subcarrier

is narrowband , the bandwidth of the OFDM symbol is greater than the coherence

bandwidth of a frequency selective channel. To mitigate the effects of the ISI between

1.2 Turbo MIMO-OFDM System 5

OFDM symbols, guard intervals are inserted between OFDM symbols so that time

dispersion of current OFDM symbol will not interfere with subsequent OFDM symbols.

In practice, an OFDM symbol is obtained by taking the inverse discrete Fourier trans-

form (IDFT) of a block of modulation symbols at the transmitter. Then at the receiver

the forward discrete Fourier transform (DFT) is performed to restore the modulated

symbols. As both the IDFT and DFT can be implemented using fast Fourier transform

(FFT) algorithms, OFDM is considered as a low cost technique.

1.2 Turbo MIMO-OFDM System

Fig. 1.1 An Iterative MIMO-OFDM Communication System

The research interest of this thesis is to reduce the complexity of iterative MIMO-

OFDM systems which combine all major benefits of above three key technologies. Fig.

1.1 is a block diagram of the iterative MIMO-OFDM system. At the transmit side, a

convolution code encoder or LDPC code encoder is employed for channel encoder. Then

the serial encoded bits sequence is split to Ns parallel sub-streams. Each sub-stream will

be scrambled by an interleaver and followed by constellation mapper to map a chunk of

bits to a constellation symbol. Then all the sub-streams data pass the pre-coding block

to map Ns sub-streams to Nt transmit chains. After spatial mapping, each transmit chain

has OFDM modulation applied to it by processing it through an IFFT block that converts

6 Introduction

a block of modulated constellation points to a time domain block of symbols followed

by adding the cyclic prefix (CP). The resulting baseband sequence of symbols in each

chain are then passed to the analog and RF blocks before being applied to a transmit

antenna.

At the receive side, after the CP of the data received on every receive antenna is

removed, FFT will be performed to generate the frequency domain symbols. Then

the channel estimator estimates the frequency domain channel coefficients based on

the received pilot data. With the frequency domain channel coefficients, the MIMO

detection is performed on every subcarrier. The detected data is then de-mapped to

soft information (typically in LLR format) and sent to the channel decoder. In an IDD

system, the decoded bits (or soft information) will be sent back (after re-mapping them

to symbols) to the channel estimator and/or the symbol detector to purify the results of

last iteration.

1.2.1 Channel

The nature of the wireless environment results in the transmitted signal experiencing var-

ious forms of corruption including noise and fading. The background noise and thermal

noise of the channel are the major contributors of noise which is commonly modelled

as additive white Gaussian noise (AWGN). Fading, which is the variation of the signal

amplitude over time and frequency, may either be due to multipath propagation, referred

to as multi-path fading, or to shadowing from obstacles that affect the propagation of a

radio wave, referred to as shadow fading.

The fading phenomenon can be broadly classified into two different types: large-

scale fading and small-scale fading. The large-scale fading is characterized by average

path loss and shadowing. On the other hand, small-scale fading refers to the result of

multipath propagation. In a wireless environment, the transmitted signal may be scattered

into multiple paths as a result of reflection and refraction off environmental obstacles and

atmospheric effects. An attenuated version of the transmitted signal propagates through

each path and arrives at the receiver at different times. Consequently, the received signal


is distorted by one symbol interfering with subsequent symbols, which is commonly

referred as inter symbol interference (ISI).

Characteristics of a multipath fading channel are often specified by a power delay

profile (PDP). Using a PDP, different signal paths are characterized by their relative delay

(τi) and average power (P(τi)). Then the RMS delay spread στ can be calculated by the

square root of the second central moment of PDP as στ =√

τ2 − τ2 where the mean

excess delay τ is given by the first moment of PDP as τ = ∑k τkP(τk)∑k P(τk)

and τ2 =∑k τ2

k P(τk)

∑k P(τk).

In general, the coherence bandwidth, denoted as Bc, is inversely-proportional to the

RMS delay spread, that is, Bc ≈ 1στ

.

Fading Due to Time Dispersion

Due to time dispersion, a transmit signal may undergo fading over a frequency domain

either in a selective or non-selective manner, which is referred to as frequency-selective

fading and frequency-flat fading. For the given channel frequency response, frequency

selectivity is generally governed by signal bandwidth. When the signal bandwidth (Bs ∝

1/Ts, Ts is the symbol period) is narrow compared with the coherence bandwidth (Bc)

of the channel, the signal experiences flat fading; otherwise, it experiences frequency-

selective fading.

Fading Due to Frequency Dispersion

Variation in the time domain is closely related to movement of the transmitter or receiver,

which incurs a spread in the frequency domain, known as a Doppler shift. The maximum

Doppler shift can be calculated by fm = vmax fC/c0 where vmax is the maximum velocity

between the receiver antenna and the transmitter antenna, fC is the frequency of carrier

and c0 is the speed of electromagnetic wave. Depending on the extent of the Doppler

spread, the received signal undergoes fast or slow fading. When the coherence time

Tc ≈ 1fm

is smaller than the symbol period Ts (Ts > Tc), a channel impulse response

quickly varies within the symbol period. Under this condition, the transmit signal is

subject to fast fading.

8 Introduction

1.2.2 LDPC Encoder and Decoder

Low-density parity-check (LDPC) codes are linear block codes which can provide near-

capacity performance. They were proposed by Gallager in his dissertation [27] in 1960.

Then in 1981 Tanner generalized LDPC codes and introduced a graphical representation

of LDPC codes in [28]. In mid-1990’s Mackay, Luby and others [29] [30] [31] also

independently discovered the advantages of spare parity-check matrices. The most

obvious character of LDPC codes is that the parity-check matrix has a low density of 1’s

for binary LDPC codes. For a LDPC code with (n− k)×n parity-check matrix H, if the

number of 1’s in each column wc equals to the number of 1’s in each row wr, this code

is called regular LDPC code and otherwise called irregular LDPC code with the code

rate of k/n.

A special subclass of LDPC codes, called Quasi-Cyclic LDPC (QC-LDPC) codes

has received much attention because of their superb error correction performance [27].

QC-LDPC codes is characterized that a cyclic shift of one codeword results in another

codeword and due to this regular structure their encoding is proved to be linear with

code length. As QC-LDPC has near capacity performance and can be decoded by

low-complexity iterative decoding algorithm, it has been adopted by many industrial

standards like IEEE 802.11n, IEEE 802.11ac and IEEE 802.16e, as an error correction

code [32] [12] [33].

LDPC Encoder Algorithm

Fig. 1.2 QC-LDPC Base Parity Check Matrix


The base parity check matrix of rate 5/6 length-1944 QC-LDPC codes (employed

in IEEE 802.11n/ac standards) is defined in Fig. 1.2. The digits indicate the cyclic

shift values of identity sub-matrices. The ’-’ indicates a zero matrix and the sub-matrix

size Z is defined as 81. The base parity check matrix can be partitioned into the two

sub-matrices as shown in Fig. 1.2. Let H = [H1 H2] be the partitioned base parity

check matrix, where H1 is an (n− k)× k matrix, and H2 is an (n− k)× (n− k) matrix.

Let c = [m p] be a codeword block, where m and p denote information and parity bit

sequences, respectively. From the property that the correct codeword satisfies the parity

check equation, the parity bit sequence p can be derived as follows,

HcT = H1mT +H2pT = 0, (1.1)

pT = H−12 H1mT . (1.2)

From (1.2), it is clear that this encoding requires to compute an inverse of matrix with

size of (n−k)× (n−k) and the direct computation has big computational complexity of

O((n−k)3). But when we check the structure of H2 carefully, we can see that this matrix

has a very regular structure which can be exploited for low complexity implementation.

It can be seen from Fig. 1.2 that H2 contains either identity submatrix (with some

shift factor) or zero submatrix. More importantly, two of the three sub-matrices of the

first columns have the same value and every other column contains two same value.

Therefore, if we let H1mT = [λ0,λ1, ...,λn−k−1]T and p = [p(0),p(1), ...,p(n− k−1)],

the first subvector of p0 can be easily obtained with

p(0)T =n−k−1

∑i=0

λi (1.3)

Then the remainder of the parity bits can be obtained by forward substitution. This

algorithm leads to linear complexity solution for QC-LDPC encoding. Actually, many

10 Introduction

efforts now focus on how to improve encoding throughput and reduce implementation

complexity at the same time [34] [35].

LDPC Decoder Algorithm

Based on the Tanner graph representation of LDPC codes, the iterative massage passing

algorithm (MPA) is typically exploited to do the decoding. Tanner graph is a kind

of bipartite graph whose nodes can be separated into two types, and edges may only

connect two nodes of different types. These two nodes in Tanner graph are the variable

nodes (v-node) and the check nodes (c-node). The Tanner graph is drawn based on

the following rule: check node j is connected to variable node i whenever element h ji

of parity check matrix H is a 1. So, it is easy to know that there are m = n− k check

nodes for check equations and n variable nodes for code bits. The task of LDPC decoder

is to compute the a posteriori probability (APP) for a bit in the transmit codeword

c = [c0,c1, ...,cn−1] equals 1 given the received word y = [y0,y1, ...,yn] in LLR:

L(ci) = log(

Pr(ci = 0|y)Pr(ci = 1|y)

). (1.4)

When drawing a Tanner graph, typically we put the c-nodes above the v-nodes. Then

the message passing from a v-node i to a c-node j is noted as m↑i j. This extrinsic

information message is the probability of Pr(ci = b | input message), b ∈ {0,1} which

comes from channel input and all its neighbours excluding the c-node itself. In the

reverse direction, the message passing from a c-node to a v-node m↓ ji is the probability

of Pr(check equation f j is satisfied | input message). Now we introduce the following

notations [36]:

• Vj=v-nodes connected to c-node f j

• Ci=c-nodes connected to v-node ci

• Mv(i) = messages from all v-nodes except node ci

• Mc( j) = messages from all c-nodes except node f j


• Pi = Pr(ci = 1 | yi)

• Si = event that the check equations involving ci are satisfied

• qi j(b) = Pr(ci = b | Si,yi,Mc( j)

), where b ∈ {0,1}. For LLR format, m↑i j =

log[qi j(0)]/qi j(1)]

• r ji(b) = Pr(check equation f j is satisfied | ci = b,Mv(i)

), where b ∈ {0,1}. For

LLR format, m↓ ji = log[r ji(0)/r ji(1)

]Then, the MPA can be summarized as follows,

• Step 1: Initialization: For every v-node, initialize pi = Pr(ci = 1|yi), then qi j(0) =

1− pi and qi j(1) = pi for each hi j = 1. Under AWGN channel, pi = 1/(1+

exp(2yi/σ2)).

• Step 2: For each c-node, update r ji by r ji(0) = 12 +

12 ∏

i′∈V j\i(1− 2qi′ j(1)) and

r ji(1) = 1− r ji(0).

• Step 3: Update qi j by qi j(0)=Ki j(1−Pi) ∏j′∈Ci\ j

r j′i(0), qi j(1)=Ki jPi ∏j′∈Ci\ j

r j′i(1)

and Ki j is selected to ensure that qi j(1)+qi j(0) = 1.

• Step 4: Update Qi by Qi(0) = Ki(1−Pi) ∏j∈Ci

r ji(0) and Qi(1) = KiPi ∏j∈Ci

r ji(1)

and Ki j is selected to ensure that Qi(1)+Qi(0) = 1.

• Step 5: Hard decision: For i = 0,1, ...,n− 1, if Qi(1) > Qi(0) then ci = 1; else

ci = 0.

• Step 6: If cHT = 0 or reaching the maximum iteration number, stop; else, go to

Step 2.

1.2.3 Soft Mapper and Soft Demapper

Soft Mapper

The function of a soft mapper module is to calculate the symbol mean and variance

from the extrinsic LLRs of code bits coming from the Soft-Input Soft-Output (SISO)

12 Introduction

decoder [37]. The soft mapper calculates {mn,vn} based on extrinsic LLR L(cn) using

the following equations:

mn = E(xn) =2Q

∑i=1

αi p(xn = αi) (1.5)

vn =Cov(xn,xn) =2Q

∑i=1

|αi|2 p(xn = αi)−m2n (1.6)

where each constellation symbol αi corresponds to a binary vector si = [si,1,si,2, ...,si,Q]T ,

and the symbol’s probability p(xn = αi) can be calculated as:

p(xn = αi) =Q

∏j=1

p(cn, j = si, j) (1.7)

while p(cn, j = si, j) is the probability of a code bit, which is normally represented by

LLR:

L j = lnp(cn, j = 0)p(cn, j = 1)

= lnp(cn, j = 0)

1− p(cn, j = 0). (1.8)

With (1.5) - (1.7), the computational complexity is O(Q2Q). When high order mod-

ulation is exploited, the computational complexity is high and the low complexity

algorithms can be found in [38] and [39].

Soft Demapper

In a coded system, the soft output from the equalizer (or detector) typically can greatly

improve the system BER performance compared to hard output. The symbol output

from equalizer (or detector) should be demapped to bit information in LLR format

which is the input requirement from most of the channel decoders like the Turbo code

or the LDPC code. When the soft output symbols are assumed as Gaussian distributed,

they can be described by their mean vector m and auto-covariance diagonal matrix V.

The task of the demapper is to compute the LLR for each code bit cn,q, which can be


expressed as [18]

L(cn,q) = lnP(cn,q = 0|y)P(cn,q = 1|y)

= ln

∑xn∈A 0

q

P(xn|y)

∑xn∈A 1

q

P(xn|y)(1.9)

where A 0q (A 1

q ) denotes the subset of all αi ∈A corresponding to a binary subsequence

with the qth bit given by 0 (1). When IDD is adopted, only extrinsic information will be

passed to the channel decoder. The extrinsic LLR [17]

Le(cn,q) = L(cn,q)−La(cn,q)

= ln

∑xn∈A 0

q

P(y|xn)P(xn)

∑xn∈A 1

q

P(y|xn)P(xn)−La(cn,q)

(1.10)

will be the input to the decoder, where La(cn,q) is the output extrinsic LLR of the decoder

in the last iteration and P(xn) can be calculated from La(cn,q). The probability of the data

symbol xn being the constellation point αi is given by P(xn = αi) ∝ exp(− |αi−me

n|2ve

n

).

After some manipulation, we can get

Le(cn,q) = ln

∑αi∈A 0

q

exp(− |αi−me

n|2ve

n

)∏

q′ =qP(cn,q′ = si,q′)

∑αi∈A 1

q

exp(− |αi−me

n|2ve

n

)∏

q′ =qP(cn,q′ = si,q′)

(1.11)

Directly computing LLR in (1.11) needs exhaustively search every constellation point

which results high computational complexity if high order constellation is employed.

To reduce this complexity, quite a lot of works can be referred although they are

implemented in different background [38] [40]. The basic idea of these methods are

using the regularity of constellation points and employing the approximation used by

max_log_map algorithm in [41] to change the exhaustively search to a piecewise linear

combination. If we ignore the a priori information which has been found with little

14 Introduction

performance penalty and apply this approximation, (1.11) can be represented as

Le(cn,q)≈ ln

∑αi∈A 0

q

exp(− |αi−me

n|2ve

n

)∑

αi∈A 1q

exp(− |αi−me

n|2ve

n

)≈ 1

ven

maxαi∈A 0

q

(−|αi −men|2)−

1ve

nmax

αi∈A 1q

(−|αi −men|2)

=1ve

n

[min

αi∈A 1q

(|αi −men|2)− min

αi∈A 0q

(|αi −men|2)

](1.12)

Fig. 1.3 is a 4-PAM constellation diagram and the men is located in the × point. It is

0010 0111

5

3

5

15

1�

5

3✁

Fig. 1.3 4-PAM Constellation Diagram

easy to see that the results of the two min operation in (1.12) are the white constellation

point and the black one, and thus (1.12) can be easily calculated as follows:

Le(cn,0) =

1√5ve

n(4me

n −8) (men ≥ 0)

1√5ve

n(−4me

n −8) (men < 0)

(1.13)

and

Le(cn,1) =

1√5ve

n(8me

n −8) (men ≥ 2)

1√5ve

n(4me

n) (|men|< 2)

1√5ve

n(8me

n +8) (men ≤−2).

(1.14)


1.2.4 Signal Detection

After the data symbols are transmitted over a MIMO channel and corrupted by AWGN,

the receiver receives superimposed and noised version of these symbols. The data

detection block is responsible for recovering those corrupted data symbols based on

certain estimation criterion. At the receiver side, in order to improve performance,

iterative detection and decoding can be employed based on the “turbo principle”. From

the iterative receiver diagram Fig. 1.1, it can be seen that the SISO decoder and the SISO

detector iteratively exchange soft extrinsic information between them.

The following is a brief review of conventional detection methods. If there is no

a-priori information available, the ML (Maximum Likelihood) method can be employed

while the MAP (Maximum a-Posteriori ) method can be employed if the a-priori in-

formation is available. But, both ML and MAP based methods suffer from the huge

computational complexity which is exponential in the number of transmit antenna Nt

and modulation constellation size Q. In order to reduce complexity, linear methods such

as zero forcing (ZF) or Minimum Mean Square Error (MMSE) can be employed. In

the family of non-linear detection algorithms, Sphere Decoding (SD) based search algo-

rithms have been deeply studied [42] [43]. Basically, SD algorithms have exponential

average complexity [3], and most importantly the complexity depends on channel status

and received SNR. In order to make the complexity deterministic, Fixed-Complexity

Sphere Decoder (FCSD) has been proposed with medium complexity and near ML per-

formance [43] [44]. Another non-linear detection is called Partial Gaussian method [45].

This algorithm has low and fixed computational complexity and near MAP performance

by using an adjustable parameter M. The basic idea behind this method is taking M

important symbols as discrete symbols but others as continuous. The continuous symbols

can be assumed to be Gaussian distributed which makes the whole computational com-

plexity very low. The last type of detection algorithm is based on factor graph [46] [47]

[17] [48] [49] [50] [51] [52].

In this thesis, we will focus on MIMO spatial multiplexing technique which can

transmit data at a higher speed than the system employing spatial diversity. Consider

16 Introduction

a MIMO-OFDM system with spatial multiplexing technique in Fig. 1.1 which has Nt

antennas at the transmit side, Nr antennas at the receiver side and N subcarriers. The

cyclic prefixes (CP) are inserted before the IFFT of x(n) to ensure the orthogonality

among the subcarriers and prevent inter-symbol interference (ISI) between consecutive

OFDM symbols. Considering a quasi-static channel which is constant during one OFDM

symbol, this OFDM system can be described as a set of parallel frequency flat additive

white Gaussian noise (AWGN) channels. Then the channel H can be denoted by a matrix

sized Nr ×Nt with its (i, j)th entry hi j denoting the channel gain between the ith transmit

antenna and the jth receive antenna where j ∈ [1,2, ...,Nr] and i ∈ [1,2, ...,Nt ]. So, for

every subcarrier, a length-Nr observation vector y at the receive side can be written as

y = Hx+w (1.15)

where w denotes a length-Nr circularly symmetric additive white Gaussian noise (AWGN)

vector with zero-mean and covariance of σ2I. It is worth noting that there are totally N

such equations in a MIMO-OFDM system.

Conventional Detection Algorithms

Linear signal detection algorithms like ZF and MMSE treat all other transmitted signals

as interferences and minimize or nullify these interferences when detecting the desired

signals. Specifically, according to the system model of (1.15), the ZF detection algorithm

can be described as:

xZF = (HHH)−1HHy (1.16)

while MMSE algorithm can be listed as

xMMSE = (HHH+σ2I)−1HHy (1.17)

where x is the detected transmit symbols. The noise enhancement effect of the above

two algorithms is significant when the condition number of the channel matrix is large


(the minimum singular value is very small) [53] while the effect of noise enhancement

in MMSE algorithm is less critical than that in ZF algorithm.

In order to improve performance, Maximum likelihood (ML) is often employed

which calculates the Euclidean distance between the received signal vector and the

product of all possible transmitted signal vectors with the given channel H and finds the

one with the minimum distance. Mathematically, the ML algorithm can be described as:

xML = arg minx∈A Nt

(||y−Hx||2). (1.18)

It is obvious that the complexity of ML algorithm is exponential in Nt which is too

complex for a practical implementation, but its performance is much better than afore-

mentioned ZF and MMSE algorithms, especially for small-size MIMO. But for large

MIMO system, linear detection algorithms such as MMSE-PIC can have near optimal

performance [54]. To reduce the computational complexity of ML algorithm, search

based algorithms like Sphere Decoding (SD) can be exploited. After applying QL

decomposition to H (H = QL, QT Q = I and L is lower triangular), the problem (1.18)

can be visualized as a decision tree with Nt layers [55] as follows:

min{x1,x2,...,xNt }

f1(x1)+ f2(x1,x2)+ ...+ fNt (x1,x2, ...,xNt ) (1.19)

where fk(x1,x2, ...,xk) = (yk − ∑kl=1 Lk,lxl)

2 and y = Qy. The basic idea under SD

algorithm is to use efficient tree traversal algorithms to eliminate the number of nodes

visited and thus reduce the total complexity.

Soft-In Soft-Out Detection Algorithms for Turbo MIMO-OFDM Systems

The more reliable feedback from the decoder is a good information source to perform

interference cancellation. A lot of multi-user detection algorithms can be applied to

MIMO detection like the minimum mean square error parallel interference cancellation

(MMSE-PIC) algorithms [56] [57]. These algorithm involves a matrix inversion when

detecting every symbol. To reduce the complexity, an iterative method to implement

18 Introduction

the MMSE filter was proposed in [58]. Then [59] presented a method which needs pre-

computing one matrix inversion only and then detects every symbol with low complexity

incremental calculations. In 2011, [60] proposed a well optimized version of MMSE-PIC

with only one matrix inversion for detecting a block of data and implemented it in ASIC

which has been widely cited as the state-of-the-art MIMO detection implementation

benchmark. This algorithm is listed in Algorithm 1:

Algorithm 1 MMSE-PIC MIMO Detection Algorithm

Input: y,H, La

Output: Le ◃ extrinsic LLR value for every bit1: Compute the Gram matrix G = HHH and the matched filter output yMF = HHy.2: Compute the a priori soft-symbols m and variances V with (1.5) and (1.6).3: Perform PIC based on yMF according to yMF

i = HH yi = yMF −∑ j, j =i g jm j, j =1, ...,Nt where g j denotes the jth column of G.

4: Compute the matrix inversion of A−1 = (GV+σ2INt )−1.

5: Compute the MMSE filter outputs as µi = aHi gi and xi = aH

i yi, i = 1, ...,Nt , whereaH

i is the ith row of A−1.6: Compute the extrinsic variance and extrinsic mean by7: ve

i = 1/µi −18: me

n = xi/µi9: Compute LLRs Le(ci,q) with (1.12), i = 1, ...,Nt , q = 1, ...,Q.

Also in 2011, [17] proposed a generic method to implement a Soft-Input Soft-Output

(SISO) detector, where the a posteriori distribution of a multivariate Gaussian vector

was calculated first, followed by the calculation of the extrinsic information of each

individual variable. The calculation of multiple variables together naturally enables

sharing of computational units, thereby reducing system complexity. This algorithm is

described in Algorithm 2;

After applying this algorithm to MIMO detection, we found that although [17]

and [60] have very different formulae, they actually can generate the same extrinsic

mean and variance and thus the same soft-output to the channel decoder. The proof is

given in Appendix A.


Algorithm 2 Gaussian model based MMSE detectionInput: y,H, La

Output: Le ◃ extrinsic LLR value for every bit1: Compute the Gram matrix G = HHH.2: Compute the a priori soft-symbols m and variances V with (1.5) and (1.6).3: Calculate the a posteriori mean mp and variance Vp by4: Vp = (V−1 + 1

2σ2 G)−1

5: mp = m+ 12σ2 Vp(HHy−Gm).

6: Calculate the extrinsic mean men and variance ve

n by7: ve

n = ( 1vp

n− 1

vn)−1

8: men = ve

n(mp

nvp

n− mn

vn).

9: Compute the LLRs Le(ci,q) with (1.12), i = 1, ...,Nt , q = 1, ...,Q.

1.2.5 Channel Estimation

In OFDM systems, a long enough cyclic prefixes (CP) insertion before the IFFT can

ensure the orthogonality among the subcarriers and prevent inter-symbol interference

(ISI) between consecutive OFDM symbols. Considering a quasi-static channel which

is constant during one OFDM symbol, this OFDM channel can be described as a set

of parallel additive white Gaussian noise (AWGN) channels. The orthogonality allows

each subcarrier component of the received signal to be expressed as the product of the

transmitted signal and channel frequency response at the subcarrier. Then the channel

can be estimated by using a preamble or pilot symbols known to both transmitter and

receiver for pilot subcarriers, then various interpolation techniques can be applied to

estimate the channel response of the subcarriers between pilot subcarriers. Depending on

the arrangement of pilots, four different types of pilot structures are typically employed.

• 1: Block Type: OFDM pilot symbols at all subcarriers are transmitted periodically.

Typically, a time domain interpolation is performed to get the whole channel

information. It is suitable for frequency-selective slow fading channels .

• 2: Comb Type: Every OFDM symbol has pilot tones at the periodically-located

subcarriers. It is suitable for fast-fading channels.

20 Introduction

• 3: Lattice Type: As a combination of block type and comb type, pilot tones are

inserted along both the time and frequency axes with given periods.

• 4: Superimposed Pilot: Low power of training (pilots) signal is added to the data

signal at the transmitter. The data-aided scheme, where the signal from the detector

or the channel decoder, is typically exploited to do interference cancellation for

the channel estimation.

Channel Estimation for OFDM System

After dropping the CP and performing FFT, the received frequency domain signal for

OFDM symbol n is given by

y(n) = X(n)η(n)+w(n) (1.20)

where y(n) denotes a length-N observation vector, X(n) ≡ diag{x(n)} denotes an

N × N diagonal matrix with x(n) (data transmitted in nth OFDM symbol, x(n) =

[x1,x2, · · · ,xN ]T ) on its diagonal, η(n) is the frequency domain channel coefficients and

w(n) denotes a length-N circularly symmetric AWGN vector with PDF C N (w;0,σ2I).

For notation simplicity, from now on we omit the time index n.

Pilot based channel estimation When training symbols are available the least-

square (LS ) and minimum-mean-square-error (MMSE) techniques are widely used for

channel estimation.

• HLS = X−1y, LS channel estimation.

• HMMSE = NFPFHXH(NXFPFHXH + σ2I)−1y, MMSE channel estimation,

where P is the channel power profile, F is the DFT matrix with the (k, l)th element

given by (F)k,l =√

Ne− j 2πklN with j =

√−1.

Although LS channel estimation has very low complexity, it suffers from noise

enhancement issue. In order to improve the performance of OFDM channel estimation,


the DFT based channel estimation algorithm can be employed. Specifically, after taking

IDFT of the estimated frequency domain channel coefficients, we get the time domain

channel coefficients with length N. But the actual time domain coefficients only have

the length of L and typically L < N. By assigning the coefficients to zero for those with

index larger than L and transforming them back to frequency domain, we get the channel

estimation with better performance.

The MMSE channel estimation algorithm is much robust from noise enhancement but

the matrix inversion requires O(N3) complexity. To reduce the cubic complexity, there

are many algorithms have been proposed such as [61] [62] and [63] using windowed

discrete Fourier transform (WDFT) methods and [64] using Dual-Diagonal LMMS

algorithm.

Channel Estimation for MIMO-OFDM System

Classical channel estimation techniques for OFDM cannot be used in MIMO-OFDM

system directly, since the received signal is a superposition of signals transmitted from

different antennas for each OFDM subcarrier. The Expectation-Maximization (EM)

algorithm can convert a multiple-input channel estimation problem into a number of

single-input channel estimation problems [65].

MIMO-OFDM System Model In Fig. 1.1, the received signal on the mRth receive

antenna at time n after performing a DFT can be expressed as:

y(n)mR = X(n)FhmR(n)+w (1.21)

where y(n)mR = [ymR,1,ymR,2, ...,ymR,N ], X = [X1,X2, ...,XNT ] are the transmitted sym-

bols, XmT includes the symbols transmitted over N subcarriers from the mT th trans-

mit antenna on its diagonal, F = INT

⊗F and F is the truncated DFT matrix, with

[F]u,s = 1√N

e− j2πus/N , and u = 0, ...,N −1,s = 0, ...,L−1, hmR= [hT

1,mR, ...,hT

NT ,mR]T is

the time domain channel vector, with hmT ,mR= [hmT ,mR,0, ...,hmT ,mR,l, ...,hmT ,mR,L−1].

22 Introduction

LS for MIMO-OFDM The LS channel estimate for (1.21) is expressed as

hmR(n) = (FHXH(n)X(n)F)−1FHXH(n)y(n)mR (1.22)

Obviously, the matrix to be inverted is with the size of NT L×NT L and involves the

complexity of O(N3T L3).

1.3 Motivations and Contributions

1.3.1 Signal Detection

In massive MIMO applications [54], as the number of transmit antennas Nt is very large,

many of the conventional MIMO detection algorithms like Sphere Decoding (SD) [42]

have prohibitive complexity. As a result, new algorithms were proposed to reduce the

complexity [66]-[67]. In [66] and [68], two local neighborhood search methods known

as likelihood ascent search (LAS) and reactive tabu search (RTS) were presented. Both

can achieve near-optimal performance for BPSK or QPSK modulations but perform

poorly with high-order quadrature amplitude modulation (QAM). To further improve the

performance for high-order QAM, layered tabu search (LTS) was presented in [67] but

with much higher complexity. Interestingly, when turbo-processing is employed, recent

research shows that for massive MIMO and under well conditioned channels, the linear

detection method such as iterative minimum mean-squared error with soft interference

cancellation can achieve near optimal performance [54]. Together with the iterative

detection and decoding (IDD) technology, linear detection algorithm like the minimum

mean square error parallel interference cancellation (MMSE-PIC) algorithm [56] [57] is

attractive because of its low complexity and good bit error rate (BER) performance. To

reduce the burden of performing matrix inversion for detecting every symbol in MMSE-

PIC algorithm, some reduced complexity algorithms have been proposed [58] [59] and

implemented in ASIC [60] [69] which require only one matrix inversion to detect one

block of receive data.

1.3 Motivations and Contributions 23

For iterative MIMO detection application, the matrix inversion has to be computed

for every iteration because the a priori variance is different for every iteration. As

this matrix inversion varies only according to this a priori variance between different

iterations, it is possible that the second and the subsequent iterations can exploit the

matrix inversion result of the first pass thereby reducing the total complexity. Chapter 2

will focus on this topic.

With more and more antennas are employed in modern communication systems, the

physical limitation forces the system designer to reduce the space between different

antennas and thus leads to correlated channels. The spatial correlation between antennas

should be taken into account when performing signal detection. We found that for a turbo

massive MIMO system, the MMSE-PIC performs poorly under correlated channels and

the Partial Gaussian Algorithm (PGA) in [45] can handle correlation channel effectively.

But due to the marginalization of M discrete symbols in PGA is exponential in MQ (Q

is the number of bits in a symbol), it is obvious that with larger M (say M ≥ 3) PGA

algorithm will have high computational complexity. So in Chapter 3, we proposed an

approximation method and a search algorithm to reduce this complexity. Extensive

simulation shows that the approximation only causes marginal performance loss and the

proposed branch-and-bound algorithm has roughly 5% of the exact PGA algorithm’s

complexity.

Although the matched filter algorithm is optimal and with low complexity when

a large number of antennas are employed by the base station, for practical medium-

size massive MIMO, more complex algorithms have to be used for good performance.

Together with the IDD technology, MMSE-PIC algorithm [60] is attractive for detection

of medium-size massive MIMO signals. But the algorithm proposed in [60] still needs

cubic level complexity when detecting a block of data. To reduce this complexity,

[70] and [71] employ Neumann series expansion to avoid matrix inversion involved in

MMSE filter calculation. Then in [72] the authors proposed to use similar method to

perform 3GPP-LTE uplink signal detection and proved the convergence of the Neumann

series expansion.These works can all successfully avoid computing matrix inversion

24 Introduction

directly, and reduce complexity from O(N3t ) to O(N2

t ) where Nt is the total number of

antennas of end terminals. But they all need the pre-computed Gram matrix as an input.

The Gram matrix computation involves complexity of N2t Nr/2 which is much higher

than the matrix inversion of N3t /2 in massive MIMO uplink detection where Nr ≫ Nt

and Nr is the number of antennas in base station. This means that they cannot reduce the

total detection complexity significantly. This motives us to study the method that how

to reduce the total complexity. In Chapter 5, we proposed a novel detection algorithm

which can avoid both matrix inversion and matrix-matrix multiplication. The proposed

algorithm has the complexity of O(KNtNr) where K is the terms number of Neumann

series expansion (typically k ≤ 5).

Then we consider a medium-size massive MIMO-OFDM system. As the matched

filter detection algorithm cannot achieve good enough bit error rate (BER) performance,

the MMSE-PIC based Soft-Input Soft-Output (SISO) detector is often used for signal

detection of every data subcarrier. But because the number of tones N is typically large

and the MMSE-PIC algorithm involves cubic level complexity from a matrix inversion

and Gram calculation, the tone by tone (per subcarrier) detection methods still incur very

high computational complexity. Although there are works which can perform matrix

inversion using interpolation method, they were all designed for small-size MIMO and

cannot be easily extended to massive MIMO applications. In Chapter 6, we will exploit

the strong correlation between the MMSE matrix inversions of adjacent subcarriers and

propose a linear interpolation method to compute the matrix inversions thus significantly

reducing the number of matrix inversion required. Extensive simulations show that

the proposed algorithm can reduce the complexity to the matched filter level but with

significantly better BER performance than it.

1.3.2 Channel Estimation

Accurate channel estimation is an essential requirement for high performance signal

detection at the receiver. In an OFDM system, the frequency selective channel is of-

ten assumed time invariant within one OFDM symbol and the frequency correlation

1.3 Motivations and Contributions 25

among different subcarriers is often exploited to reduce the computational complexity

of the channel estimator. If the channel power delay profile (PDP) is available, the

linear-minimum-mean-square-error (LMMSE) estimation is typically employed with

the aid of pilot signals (and/or data fed back from the detector or the channel decoder).

However, directly implementing such an estimator typically involves a matrix inversion

with cubic complexity of channel length. To reduce the cubic complexity, windowed

discrete Fourier transform (WDFT) methods were proposed in [61] [62] [63] to achieve

a complexity of O(N logN) (N is the number of subcarriers) but with significant per-

formance loss. In [73], using the law of large numbers, an approximation to the matrix

inversion was proposed to reduce the complexity to O(N logN). However, this incurs a

mean-square error (MSE) floor in the high signal-to-noise ratio (SNR) region. In [64], a

Dual-Diagonal LMMSE channel estimation for OFDM systems was proposed and the

corresponding MSE was analyzed. With this method, the channel estimation can be

achieved with complexity of O(N logN) and the MSE performance is close to the exact

LMMSE algorithm from low to medium SNR. But for high SNR, both the simulation

results and MSE analysis showed that there is still some performance loss. Recently,

basis expansion model (BEM) algorithms based on discrete prolate spheroidal (DPS)

sequences have attracted much interest as they need no channel statistics but the knowl-

edge of the maximum delay spread and the maximum Doppler spread. Assuming that the

CIR is invariant within one OFDM symbol, a low complexity Linear MMSE estimation

of time-frequency variant channels for MIMO-OFDM systems was proposed in [74]

by replacing a two-dimensional Slepian-basis expansion with two serially concatenated

one-dimensional Slepian-basis expansions. Then in [75], the time variant CIR within

one OFDM symbol was taken into account and algorithms with complexity of O(N2)

were proposed. These DPS based algorithms were well designed for fast-fading environ-

ment. For block fading channels, [76] proposed several DPS based algorithms with low

complexity.

In this thesis, by employing the fact that the matrix to be inverted in MMSE channel

estimation is diagonally dominant1, we proposed to use a K terms Neumann series

26 Introduction

expansion in Chapter 4 to approximate the inversion. In this way, the matrix inversion

can be implemented with Fast Fourier Transform (FFT) or Inverse Fast Fourier Transform

(IFFT) operations with L inputs or L outputs, thus has the complexity of O(N logL)

where L is the number of time domain channel taps. It is worth noting that the proposed

channel estimation algorithm has close MSE performance as the exact implementation

from low to high SNR. In this chapter, we also found that with the knowledge of the

number of channel taps (i.e. L) and SNR, an uniform distributed PDP can be used to

replace the exact PDP with marginal performance loss, which is desirable because the

exact PDP is typically difficult to obtain.

1.4 Notations

The notations used in this thesis are as follows. Lower and upper case letters denote

scalars. Bold lower and upper case letters represent column vectors and matrices,

respectively. As customary, given a matrix Q we will let Qi j denote its entry in ith row

and jth column, and vi is used to present the ith element of a vector v. We use ∝ to

denote equality of functions up to a scale factor. The superscriptions “T ” and “H” denote

the transpose and conjugate transpose, respectively. Let IN denote an N ×N identity

matrix, E[·] the expectation operation and tr{·} the trace operation. The function of

diag{a} returns a diagonal matrix with vector a being the main diagonal and {M}diag

returns M with the off-diagonal elements of M set to be zero. The probability density

function (PDF) of a continuous random variable and the probability mass function of a

discrete random variable are represented by p(·) and P(·), respectively.

1A square matrix A is called diagonally dominant if |Aii| ≥ ∑ j =i |Ai j| for all i, where Ai j denotes theentry in the ith row and jth column.

Chapter 2

A Low Complexity Soft-Decision

Feedback MMSE-PIC Detection

Algorithm

In [17], a generic method to implement a Soft-Input Soft-Output (SISO) detector was

proposed, where the a posteriori distribution of a multivariate Gaussian vector was

calculated first, followed by the calculation of the extrinsic information of each individual

variable. The calculation of multiple variables together naturally enables sharing of

computational units, thereby reducing system complexity. So in this chapter, we firstly

employ [17] to implement the MMSE-PIC in MIMO applications, which can reduce

system complexity as the matrix to be inverted is a Hermitian positive definite (HPD)

matrix with size Nt ×Nt (Nr and Nt are the number of receive antennas and transmit

antennas, respectively). A HPD matrix enables us to use the more computational efficient

matrix inversion method.

In order to reduce the complexity of the second and subsequent iterations, we derive

a new method to calculate the matrix inversion by a linear combination of two matrices

which have been computed in the first pass (from the detector to the decoder). With this

method, we can reduce the complexity of matrix inversion from O(N3t ) to O(N2

t ) along

with small performance penalty. Compared to other matrix inversion approximation

28 A Low Complexity Soft-Decision Feedback MMSE-PIC Detection Algorithm

methods, the proposed method does not rely on any special requirement of the random

channel matrix.

The power of turbo processing comes from the more and more reliable a priori infor-

mation from the decoder, but for the first pass, there is no such information available.

At the same time, as the employed iterations between the decoder and the detector will

inevitably reduce the throughput and increase the system latency, for high speed appli-

cations it is difficult to perform IDD when they run at the highest throughput [60] [77].

Considering this, it is desirable to improve the first pass performance. So, we propose a

self-iteration method, which feeds back the detector’s soft decision output directly to

its a priori input, to improve the performance of the detector. By employing a low cost

approximation of matrix inversion, the method of self-iteration is attractive due to the

fact that with only a slight increase of complexity, a performance gain of 1dB to 2dB

can be achieved. It is worth noting that this self-iteration method is also applicable to

non-turbo systems to improve system performance.

The remainder of this chapter is organized as follows. Section 2.1 describes the

turbo-MIMO system model. Then the Gaussian model based MMSE detection algorithm

is detailed in Section 2.2. In Section 2.3, we introduce a proposal of how to reduce the

complexity of matrix inversion and the self-iteration method to improve the first pass

BER performance. Simulation results are shown in Section 2.4.

2.1 System Model

As shown in Fig. 2.1, we consider a single carrier coded MIMO system with Nr receive

antennas and Nt transmit antennas. The received signal at the receiver is as follows

y = Hx+w (2.1)

where y denotes a length-Nr observation vector, H denotes an Nr ×Nt MIMO system

transfer matrix, w denotes a length-Nr circularly symmetric additive white Gaussian

noise (AWGN) vector with PDF C N (w;0,2σ2I), and x = [x1,x2, · · · ,xNt ]T is mapped

2.2 Gaussian Model Based MMSE Detection Algorithm 29

SISO Decoder

✝n

MIM

O D

ete

cto

r

Interleaver

De-interleaver

-

-

y

La

Le

Channel Encoder

an MIMO Modulator

Inter-leaver

...

bncn

Fig. 2.1 Iterative Detection and Decoding of a MIMO Communication System

from an interleaved code sequence c, i.e., each xn ∈ A = {α1,α2, · · · ,α2Q}(|A |= 2Q)

corresponds to a length-Q subsequence of c denoted by cn = [cn,1,cn,2, · · ·,cn,Q]T .

The task of the detector is to compute the log-likelihood ratio (LLR) for each code

bit cn,q, which can be expressed as [19]


= ln

∑xn∈A 0

q

P(xn|y)

∑xn∈A 1

q

P(xn|y)(2.2)

where A 0q (A 1


with the qth bit given by 0 (1). The extrinsic LLR [17]

Le(cn,q) = L(cn,q)−La(cn,q)

= ln

∑xn∈A 0

q

P(y|xn)P(xn)

∑xn∈A 1

q

P(y|xn)P(xn)−La(cn,q)

(2.3)

will be the input to the decoder, where La(cn,q) is the output extrinsic LLR of the decoder

in the last iteration and P(xn) can be calculated from La(cn,q).

2.2 Gaussian Model Based MMSE Detection Algorithm

Let G = HHH and y = HHy, the linear MMSE detection algorithm in [17] is shown in

Algorithm 3 . Due to the use of the interleaver, different bits of a symbol can be assumed


to be independent, and thus P(xn = αi) = ∏Qj=1 p(cn, j = si, j) where p(cn, j = si, j) is

calculated from the a priori LLR of Laj with the LLR definition of La

j = ln p(cn, j=0)p(cn, j=1) .

Algorithm 3 Gaussian model based MMSE detectionInput: y,G, La

Output: Le ◃ extrinsic LLR value for every bit1: Calculate a priori mean m and variance V2: mn = ∑

αi∈AαiP(xn = αi) ◃ m = [m1,m2, · · · ,mNt ]

T

3: vn = ∑αi∈A

|αi −mn|2 P(xn = αi) ◃ V = diag[v1,v2, · · · ,vNt ]

4: Calculate a posteriori mean mp and variance Vp

5: m = Gm6: Vp = (V−1 + 1

2σ2 G)−1

7: mp = m+ 12σ2 Vp(y− m)

8: Calculate extrinsic mean men and variance ve

n9: ve

n = ( 1vp

n− 1

vn)−1 ◃ vp

n is the nth diagonal element of Vp

10: men = ve

n(mp

nvp

n− mn

vn) ◃ mp

n is the nth element of mp

11: Calculate extrinsic LLR Le

12: Le(cn,q) = ln∑

αi∈A 0q

exp(− |αi−me

n|2ven

)∏

q′ =q

P(cn,q′=s

i,q′ )

∑

αi∈A 1q

exp(− |αi−me

n|2ven

)∏

q′ =q

P(cn,q′=s

i,q′ )

It is worth noting that the LLR calculation in Line 12 can be further simplified by

exploiting the constellation regularity after applying the log_max approximation and

ignoring the a priori terms like [38].

2.3 Complexity Reduction

2.3.1 Low Complexity Matrix Inversion

It can be seen that the matrix inversion in Line 6 of Algorithm 3 contributes the major

complexity of N3t /2. If Nr and Nt are large enough (e.g. greater than 200 [78]), matrix G

tends to be an identity matrix from random matrix theory, which makes the computational

complexity of this matrix inversion trivial. On the other hand, if Nr is much bigger

than Nt (like Nr/Nt > 8 [71]), matrix G becomes diagonal dominant, then the 2-term

2.3 Complexity Reduction 31

Neumann series can be employed to approximate this matrix inversion with complexity

of O(N2t ). We aim to find a more generic method which does not depend on any special

requirement for the size of this random matrix H. As [18], by averaging the diagonal

elements of V, we have V = kI where k = ∑n vnNt

. So, Line 6 of Algorithm 3 can be

rewritten as

Vp = (kI+1

2σ2 G)−1 (2.4)

where k = 1/k = 1/(∑n vn/Nt). For the first pass, there is no a priori information

available, thus we assume m to be a zero vector and V to be the identity matrix I. So, we

change (2.4) to

Vp =((I+

12σ2 G)+(k−1)I

)−1

= (A+(k−1)I)−1(2.5)

where A = I+ 12σ2 G. Thus, we can represent (2.5) as a function of k as Vp = f (k). By

using the approximation of f (k) = f (1)+ f′(1)(k−1) and the derivative of a matrix in-

verse dM−1

dk′=−M−1 dM

dk′M−1, we have a direct formula to compute this matrix inversion

as

Vp = A−1 − (k−1)A−1A−1. (2.6)

We can pre-compute E1 = A−1 and E2 = A−1A−1 and save them in memory. Then the

matrix inversion can be calculated by linear combination of these two fixed matrices as

Vp = E1 − (k−1)E2. (2.7)

Using this method, we reduce the complexity of matrix inversion from O(N3t ) to O(N2

t ).

It is worth noting that [59] also proposed an approximation method which incrementally

calculates the second and subsequent pass matrix inversion based on a pre-computed

exact matrix inversion result, but the method is only applicable to constant envelope

constellations. And in [79], a singular value decomposition (SVD) based matrix inversion

method was proposed, but this method needs linear combination of Nt pre-computed

matrices and thus has higher computational complexity than the proposed method.


2.3.2 A Heuristic Approach to Solve the Stability Problem

As the approximation of f (x) = f (1)+ f′(1)(k−1) has an error term of O((k−1)2), to

achieve a high accuracy (k−1) must be small enough (|k−1|< 1). But unfortunately

this constraint cannot always be met because when the a priori information becomes

more and more reliable, vn will be less than 0.5, leading to a unstable BER performance.

Heuristically, we propose to revise k as k = 1/(∑n vn/Nt + 0.5), thus Line 6 of Algo-

rithm 3 is replaced with the following: Hereafter, we refer this updated algorithm as

1: k = 1/(∑n vn/Nt +0.5)2: Vp = E1 − (k−1)E2

Algorithm 3.

2.3.3 Computational Complexity Comparison

In [60], a well optimized MMSE-PIC algorithm, which employs only one matrix inver-

sion to detect a length-Nr received data block for every iteration, has been proposed and

implemented in ASIC and now it has been widely cited as a MMSE-PIC implementation

benchmark. The core part of this algorithm is listed in Algorithm 4 which is equiva-

lent to Line 4 to Line 10 of Algorithm 31. From Algorithm 4, it is easy to see that

the computational complexity of Line 1 is N2t +N3

t as the matrix to be inverted is not

Hermitian. By contrast, the complexity of the matrix inversion in Algorithm 3 is N3t /2

by using LDL decomposition and modified backwards substitution [80]. As HHH is a

Hermitian matrix, we assume that this matrix multiplication has a complexity of NrN2t /2.

We summarize the complexity of above mentioned algorithms in Table 2.1. From

this table, Algorithm 3 and Algorithm 4 have the same pre-computing complexity.

But for every pass Algorithm 3 has only half of the complexity of Algorithm 4. At the

same time, compared to Algorithm 4, the proposed Algorithm 3 has great computation

saving for the second and subsequent pass processing while maintaining the same level

of pre-computing complexity.1Please see Appendix I for the proof of this equivalent.

2.3 Complexity Reduction 33

SISO

MMSE Detector

SIS

O D

ec

od

er

n

Interleaver

De-

interleaver-

-

y

La

Le

Fig. 2.2 Iterative Soft-in Soft-Out MMSE Detector

Algorithm 4 Core Part of MMSE-PIC Algorithm in [60]

1: A−1 = (GV+2σ2I)−1 ◃ One matrix inversion per iteration2: for n = 1 to Nt do3: yn = y− ∑

j, j =ng jm j ◃ g j is jth column of G

4: µn = aHn gn ◃ an is the nth row of A−1

5: xn = aHn yn

6: men = xn/µn ◃ extrinsic mean

7: ven = 1/µn − vn ◃ extrinsic variance

8: end for

Table 2.1 Computational Complexity ComparisonPre-computing Every Pass

Algorithm 4 12NrN2

t +NrNt 4N2t +N3

tAlgorithm 3 1

2NrN2t +NrNt 2N2

t +12N3

tAlgorithm 3 1

2NrN2t +NrNt +N3

t 4N2t

2.3.4 Iterative Method to Improve First-pass Performance

By employing SISO MMSE’s soft decision output as its a priori input (see Fig. 2.2),

the SISO MIMO detector itself can run in an iterative manner and we call it iterative

MMSE detection algorithm (I-MMSE). Compared to conventional MMSE turbo receiver,

a 1dB to 2dB performance gain can be obtained. Actually, the fact that self-iteration

can improve system performance had been observed in other literatures [72] [81] where

only one self-iteration has been reported. By contrast, the simulations show that there is

performance gain up to four iterations. More importantly, after employing the proposed


0 5 10 15 20 25 30

10−4

10−3

10−2

10−1

SNR per receive antenna (dB)

BE

R

Iter0Exact Iter2Approx Iter2Exact Iter4Approx Iter4

4−QAM 16−QAM 64−QAM

Fig. 2.3 BER Performance Comparison Between Exact Implementation and ProposedApproximation for a 16×16 MIMO System.

low complexity matrix inversion, I-MMSE seems more attractive because of its much

lower complexity cost of 4N2t for the second and subsequent pass calculation.

2.4 Simulation Results

2.4.1 Simulation Setup

We consider a Rayleigh slow fading random channel so H does not change over a

codeword. The elements of H are independent and identically Gaussian distributed with

zero mean and variance 1. During simulation, we assume perfect channel information

is available in the detection module. A rate-1/2, regular (3,6) low-density parity-check

(LDPC) code with codeword length of 2000 bits is employed as the channel code and the

maximum number of iterations of the decoder is 25. The square quadrature amplitude

2.4 Simulation Results 35

modulations (2Q-QAM) with Gray mapping are used. For each signal-to-noise (SNR)

value, we run at least 100000 codewords in the Monte Carlo simulations. We set the

scaling factor of output LLR to 0.7 [82]. In the simulations, there are clipping both

in soft-output part and soft-input part of the detector. The soft-in clipping threshold2

for the a priori LLR is ±2, and soft-output module constrains the output LLR range to

[−50,50].

2.4.2 BER Performance

0 5 10 15 20 25 30

10−4

10−3

10−2

10−1


BE

R

MMSE−PIC Iter0I−MMSE (1) Iter0I−MMSE (2) Iter0I−MMSE (4) Iter0MMSE−PIC Iter1MMSE−PIC Iter2I−MMSE (1) Iter2I−MMSE (2) Iter2I−MMSE (4) Iter2

4−QAM 64−QAM16−QAM

Fig. 2.4 BER Performance Comparison Between Different Number of Self-iterations for32×32 MIMO.

Fig. 2.3 shows the performance comparison between exact implementation (Algorithm

3) and the proposed approximation (Algorithm 3) of a 16× 16 MIMO system with

4-QAM, 16-QAM and 64-QAM signaling. The legend of Iter=0 stands for the the first2This clipping threshold can also help resolve the numerical stability issue of Line 10 of Algorithm 3

when the a priori variance vn is close to zero.


pass without the a priori information. The legend of Iter=2 stands for performance

after running two outer loops (between the decoder and the detector). It is clear that

this approximation has nearly no performance loss for 4-QAM signaling, but has small

performance loss for 16-QAM and 64-QAM compared to the exact one. For IDD sys-

tems employing I-MMSE algorithm, there exist two iterative loops. The self-iteration

of detector is the inner loop. The outer loop from the decoder to the detector is the

same as that in a typical turbo system. Through extensive simulation we have found

that the self-iteration method has only marginal performance gain beyond the first outer

pass, thus we only perform the inner loop for the first outer pass. Then we compare

performances between I-MMSE (using Algorithm 3 together with low complexity inner

loop) and the conventional MMSE-PIC (using Algorithm 3) for MIMO systems with

different sizes and various modulation signaling. In Fig. 2.4, the number in the bracket

denotes the number of self-iterations and IterX denotes X outer iterations. It is clear

that the proposed method can significantly improve the first pass system performance

(1dB to 2dB at BER of 10−4). It can also be seen that with more than two self-iterations

there is still performance gain although after four self-iterations the performance gain is

marginal. The simulations show that similar performance gain can also be obtained in

other sized MIMO systems like 16×16 in Fig. 2.5 and 4×4 in Fig. 2.6.

2.5 Conclusion

In this chapter, we firstly employed a low complexity Gaussian model based MMSE

algorithm to perform the MMSE-PIC detection. This algorithm can detect a length-

Nr received data block with only one Hermitian matrix inversion, and the matrix to

be inverted has the size of Nt ×Nt which is especially preferable for massive MIMO

uplink applications where Nt << Nr. Then we proposed a generic method to reduce the

computational complexity of the matrix inversion from O(N3t ) to O(N2

t ) without the

dependence on the size of the random channel matrix. At last, a self-iteration method

2.5 Conclusion 37

0 5 10 15 20 25 30

10−4

10−3

10−2

10−1


BE

R


4−QAM 16−QAM 64−QAM


was proposed to improve a turbo receiver’s first pass performance by 1dB to 2dB with

only a small complexity increase.


0 5 10 15 20 25 30 35

10−4

10−3

10−2

10−1


BE

R


64−QAM16−QAM

4−QAM


Chapter 3

MIMO Detection Algorithm: Partial

Gaussian Approach with Integer

Programming

3.1 Introduction

As mentioned in Chapter 1, the minimum mean square error parallel interference can-

cellation (MMSE-PIC) algorithm can achieve near optimal performance for massive

MIMO and under well conditioned channels. But high density antennas deployment

may reduce the space between different antennas and thus leads to correlated channel.

The MMSE-PIC algorithm performs poorly under correlated channels. The spatial

correlation between antennas should be taken into account when performing signal

detection.

Recently, we have proposed a Partial Gaussian Approach (PGA) which is very effec-

tive in turbo equalization [45]. The basic idea behind this method is taking M important

symbols as discrete symbols but others as continuous and the continuous symbols can be

assumed to be Gaussian distributed which makes the whole computational complexity

low.

40 MIMO Detection Algorithm: Partial Gaussian Approach with Integer Programming

Channel

Encoder

an MIMO

Modulator

SISO

Decodern

MIM

O D

ete

cto

r

Interleaver

Interle

aver

De-interleaver

...

...

--

bn cn

y

La

Le

Channel

Encoder

MIMO

Modulator

Interle

aver SISO

DecoderInterleaver

De-interleaver-

-

...

Fig. 3.1 Iterative Detection and Decoding of a MIMO Communication System

In this chapter, we investigate the application of PGA for detection of massive MIMO

systems. Simulation results show that under correlated channel PGA has impressive

performance than MMSE-PIC (e.g. under heavily correlated 40×40 MIMO with 16-

QAM signaling, a 5 dB gain can be observed). Due to the marginalization of M discrete

symbols in PGA is exponential in MQ (Q is the number of bits in a symbol), we find that

with larger M (say M ≥ 3) PGA algorithm will have high computational complexity.

In order to reduce the complexity, firstly we apply the “max-log" algorithm and ap-

proximate the APP calculation with the minimization of a quadratic function with integer

variables, thereby reformulating the marginalization problem into a quadratic integer

programming (QIP) problem. Then we implement the depth-first branch-and-bound

algorithm to solve this QIP problem. Simulation results show that the approximation

only causes marginal performance penalty and the proposed branch-and-bound algorithm

has roughly 5% of the exact PGA algorithm’s complexity.


iterative detection system model. Then the Partial Gaussian Approach and the proposed

complexity reduction algorithm are presented in Section 3.3. Simulation results are

shown in Section 3.4.

3.2 System Model

As shown in Fig. 3.1, we consider the uplink of a multiuser MIMO system with Nr

receive antennas at the Base Station and Nt users each with one transmit antenna. The

3.2 System Model 41

received signal in the coded system is represented as

y = Hx+w (3.1)

where y denotes a length-Nr observation vector, H denotes an Nr ×Nt MIMO system

transfer matrix, w denotes a length-Nr circularly symmetric additive white Gaussian

noise (AWGN) vector with PDF N (w;0,σ2I), and x = [x1,x2, · · · ,xNt ]T is mapped

from an interleaved code sequence c, i.e., each xn ∈ A = {α1,α2, · · · ,α2Q}(|A |= 2Q)

corresponds to a length-Q subsequence of c denoted by cn = [cn,1,cn,2, · · ·,cn,Q]T .

The task of the detector is to compute the log-likelihood ratios (LLR) for each code

bit cn,q, which can be expressed as [37]


= ln∑xn∈A 0

qP(xn|y)

∑xn∈A 1q

P(xn|y)(3.2)

where A 0q (A 1


with the qth bit given by 0 (1). The extrinsic LLR [17]

Le(cn,q) = L(cn,q)−La(cn,q) (3.3)

will be input to the decoder, where La(cn,q) is the output extrinsic LLR of the decoder in

the last iteration. The key task of the detector is to compute the a posteriori probability

(APP) P(xn|y) for each symbol xn. According to Bayes’ rule, we have

P(xn|y) = ∑x\xn

P(x|y) ∝ ∑x\xn

P(x)P(y|x) (3.4)

where the length-(Nt −1) vector x\xn consists of the elements of x except xn. Given x, y

is Gaussian distributed, i.e., p(y|x) = N (y;Hx,σ2I) (3.4) can then be rewritten as

P(xn|y) ∝ ∑x\xn

P(x)exp[− (y−Hx)H(y−Hx)

σ2

](3.5)


3.3 Partial Gaussian Approach with Integer Program-

ming

3.3.1 PGA Detection Algorithm

We summarize the PGA detection algorithm [45] in Algorithm 5. After marginalizing

out the contribution from the continuous symbols, the approximate APP of xn in (3.5)

can be represented as the marginalization over M(M ≪ Nt) discrete received symbols as

follows:

P(xn|y) ∝ ∑xD\xn

P(xD)exp[−(xD − z)HZ(xD − z)] (3.6)

where the operation of ∑xD\xncan be performed by enumerating all M data points, and

z and Z are defined in Line 10 and Line 11 in Algorithm 5. It is easy to see that the

total complexity of this marginalization (Line 12) is O(2Q(M+1)), and when M is large

(e.g. larger than 3), it is still too complex for hardware implementation for high order

signaling like 64-QAM.

3.3.2 Simplified Marginalization Calculation

For separable complex symbol constellations such as squared quadrature amplitude

modulation (QAM), the constellation can be separated into two real-valued PAM signals.

Thus we can get a real-valued system model as:

yr = Hrxr +wr (3.7)

where yr = [ℜ(yT ),ℑ(yT )]T , xr = [ℜ(xT ),ℑ(xT )]T , wr = [ℜ(wT ),ℑ(wT )]T and

Hr =

ℜ(H) −ℑ(H)

ℑ(H) ℜ(H)

.

In above equations, ℜ(·) and ℑ(·) represent the real part and the imaginary part

of a complex number, respectively. This way, we can reduce the number of bits in a

modulated symbol to Q = Q2 , thus reduce the complexity.

3.3 Partial Gaussian Approach with Integer Programming 43

As in [60], by ignoring P(xDr ) in (3.6), P(xn|yr) can be approximated by

P(xn|yr) ∝ ∑xD

r \xn

exp[−(xDr − zr)

T Zr(xDr − zr)]. (3.8)

Then, after applying the approximation of ln(ea + eb) ≈ max(a,b) , (3.8) can be

Algorithm 5 Partial Gaussian Approach

Input: y,H,M, La

Output: Le ◃ extrinsic LLR value for every bit1: Calculate a priori mean m and variance V by2: mi = ∑α∈A αP(xi = α)3: Vi = ∑α∈A |α −mi|2 P(xi = α)4: Calculate Vector c and Matrix C by5: C = V−1 + 1

σ2 HHH6: c = C−1[V−1m+ 1

σ2 HHy]7: for n = 1 to Nt do8: Set matrix S by selecting most important M symbols based on HHH [45]9: Calculate Vector z and Matrix Z by

10: Z = (SCST )−1 − (VD)−1

11: z = Z−1[(SCST )−1Sc− (VD)−1mD]12: Calculate P(xn|y) using (3.6) for all xn13: for q = 1 to Q do14: Calculate LLR L(cn,q) using (3.2)15: Calculate Le(cn,q) with (3.3)16: end for17: end for

changed to

P(xn|yr) ∝ exp(−min[(xDr − zr)

T Zr(xDr − zr)]). (3.9)

In Line 12 of Algorithm 5, there are 2Q APP P(xn|yr) should be calculated and each xn

has an enumerated value from the set ˆA = {(2Q −1), · · · ,3,1,−1,−3, · · · ,−(2Q −1)}.

Without loss of generality, we assume that the first variable xD0 of xD

r is the variable of

interest (i.e., xn in (3.8)), thus the scope of the search in the minimizing operation in

(3.9) covers M variables (elements xD1 to xD

M of xDr ).

As each element of xDr (say xD

i ) belongs to the set ˆA , let mi =12(2

Q −1− xi), then

mi ∈ [0,1, · · · ,2Q−1] is the index of the constellation point corresponding to xDi . With a


little bit of algebra, (3.9) can be reformulated as a quadratic function (after ignoring the

constant factor) as follows:

P(xn|yr) ∝ exp(−min(mT Zm+LT m+C)) (3.10)

where m = [m1, · · · ,mM]T , Z is obtained by deleting the first row and first column of

Algorithm 6 QIP with Branch-and-Bound

Input: Function f (m) = mT Zm+LT m+COutput: Integer vector r minimizing f (m)

1: for d = 1 to M do2: Get Zd by deleting Z’s first d rows and first d columns;3: Calculate the inverse matrices Z−1

d ;4: end for5: m∗ =−1

2(Z−1L); ◃ Minimizing f (m)

6: Set lb = f (m∗); ub = f (r∗); d = 0;7: Rounding m∗ to an initial feasible solution r∗;8: while d ≥ 0 do9: if 0 < d < M then

10: Compute L and C using (3.14);11: Compute m∗ =−1

2(Z−1d L);

12: Set lb = frd(m∗);

13: end if14: if d = M then ◃ Accessing leaf node15: Set lv = f (rM);16: if ub ≥ lv then17: Set r = rM; ub = lv;18: end if19: Set lb = ub;20: end if21: if lb < ub then ◃ Branch on md+122: Set d = d +1; rd = ⌊m∗

1⌉;23: else ◃ Always holds if d = M24: Set d = d −1; ◃ Prune current node25: if d > 0 then ◃ Enumerate next node26: Assign rd based on [83];27: end if28: end if29: end while

3.3 Partial Gaussian Approach with Integer Programming 45

Zr, while C and each element of L can be obtained by:

L j−1 =−M

∑i=0

Z j,iki +2Z0, jxD0 , j ∈ [1, · · · ,M]

C =14

M

∑i=0

M

∑j=0

k jZ j,iki +L0xD0 + xD

0 Z0,0xD0

(3.11)

where ki = 2Q −1− zi with zi representing the ith element of zr and Z j,i is the ( j, i)th

element of matrix Zr.

3.3.3 Resolving QIP with the Branch-and-Bound algorithm

An effective algorithm to handle integer programming is the branch-and-bound algo-

rithm [84]. In our massive MIMO detection application, as the box constraint is typically

not big, e.g. [0,1,2,3] for 16-QAM and [0,1, · · · ,7] for 64-QAM, we select the branching

strategy that consists of fixing a single variable to an integer value in the box constraint

each time. After fixing d variables, we get a reduced function (with M−d variables) as

frd(x) : RM−d → R := f (r1, · · · ,rd,x1, · · · ,xM−d) (3.12)

where {r1, · · · ,rd} (denoted by rd) are the values of those fixed variables. This reduced

function still has a quadratic form of

frd(x) = xT Zdx+ LT x+C (3.13)

where the matrix Zd is obtained from Z by deleting the first d rows and the first d

columns, while C and the elements of the vector L can be calculated by:

C =C+d

∑i=1

Liri +d

∑i=1

d

∑j=1

Zi, jrir j,

L j−d = L j +2d

∑i=1

Zi, jri, j ∈ [d +1, · · · ,M]

(3.14)


with Zi, j representing the (i, j)th element of Z.

We adopt the well-known Schnorr-Euchner method [83] as enumeration rule and set

the continuous minimum f (x∗) of the reduced function (3.13) as the lower bound. The

initial upper bound is a heuristics based feasible solution, which is a mapping of every

element of m∗ (m∗ =−12(Z

−1L))

to the closest integer in the set of [0,1, · · · ,2Q −1].

Then the upper bound will be tightened after visiting a leaf node with smaller target

function value (Step 4 in Fig. 3.2). When the lower bound is above the upper bound, the

current node will be pruned (Step 5 and 6 in Fig. 3.2) and the next enumeration value

based on the enumeration rule will be attempted. The proposed algorithm is presented in

Algorithm 6.

Fig. 3.2 An example of the proposed branch and bound algorithm where d is the treelevel, lb means low bound, ub means upper bound and m∗ is the vector that minimizesf (m). Because the first heuristic solution happens to be the final solution, there are only6 nodes visited.



3.4.1 Simulation Setup

We use the following correlated complex channel model [85]

H = (RRx)1/2Hw(RT x)

1/2 (3.15)

where RRx and RT x are covariance matrices representing the receive antenna correlation

and transmit antenna correlation, (·)1/2 denote a square root matrix and the elements

of Hw is independent and identically Gaussian distributed with zero mean and variance

one. The channel H is considered as a Rayleigh slow fading channel which means that

H does not change over a codeword. For multiuser uplink application scenario, it is

reasonable to assume that different users are located far away which causes nearly no

transmit side correlation, i.e. RT X = I. We assume

RRx =

1 ρ ρ4 · · · ρ(Nr−1)2

ρ 1 ρ · · · ...

ρ4 ρ 1 . . . ρ4

... . . . . . . . . . ρ

ρ(Nr−1)2 · · · ρ4 ρ 1

(3.16)

where ρ ∈ [0,1] is the fading correlation between two adjacent receive antenna elements

and it is approximated by:

ρ(d)≈ exp(−23 ·△2 ·d2) (3.17)

where △ is the angular spread and d is the distance in wavelengths between the antenna

elements. A rate-1/2, regular (3,6) LDPC code with codeword length 2000 bits is

employed as the channel code. The maximum iteration number of the decoder is 25. The

square 2Q-QAM modulation with Gray mapping is used. During simulation, we assume


10 15 20 25 30 35 4010

−4

10−3

10−2

10−1

SNR Per Antenna (dB)

BE

R

MMSE−SIC Iter1PGA−IP Iter1MMSE−SIC Iter3PGA−IP Iter3MMSE−SIC Iter5PGA−IP Iter5

ρ=0.5 ρ=0.8

Fig. 3.3 BER performances of 16-QAM 40×40 MIMO with correlation factor ρ = 0.5and ρ = 0.8.

perfect channel information is available in the detection module. For each signal-to-noise

(SNR) value, we run at least 20000 frames in the Monte Carlo simulation.

3.4.2 BER Performance

Firstly we evaluate the performance of PGA under correlated channels and the channel

correlation factor ρ with 0.5 and 0.8 are chosen to represent the lightly correlated channel

and the heavily correlated channel, respectively. From Fig. 3.3, it is clear that when

the correlation becomes larger, the performance gap between MMSE-PIC and PGA

becomes bigger. When ρ = 0.5, PGA-IP outperforms MMSE-PIC by 0.7 dB, while

the performance gain can reach 2 dB when ρ = 0.8. It is worth noting that above

simulation only considered the receive side correlation, if we take the transmit side

3.5 Conclusion 49

Table 3.1 Average CPU run time (s) comparison between MMSE_PIC, PGA_IP andPGA_Exact for detecting 2000bits with 3 iterations under 40×40 MIMO with 16-QAMon a X86 Linux PC

SNR (dB) 16 17 18 19 20 21 22PGA_EXACT 11.7639 11.7474 11.7556 11.7683 11.7826 11.7798 11.8237

PGA_IP 0.6972 0.6943 0.6874 0.6821 0.6786 0.6779 0.6816MMSE_PIC 0.1158 0.1152 0.1123 0.1083 0.1029 0.0999 0.0983

spatial correlation into account (such as multiple antennas are employed in a single

terminal), PGA-IP has much bigger gain over MMSE-PIC (e.g. 5 dB has been observed

if RT x = RRx and ρ = 0.7). Then, in order to validate the effectiveness of our proposed

PGA-IP algorithm, we compare the BER performance between the PGA-IP and the

exact implementation of PGA. Form Fig. 3.4, it is easily seen that the PGA-IP only

incurs marginal performance loss at the first iteration.

3.4.3 Complexity

PGA in Algorithm 5 is a fixed complexity algorithm, but PGA-IP has a variable com-

plexity because the branch-and-bound algorithm is a kind of data-driven tree search

algorithm. With simulation, we found that the average number of nodes visited is about

8 for every xn of 64-QAM with M = 3 which is much lower than the full search of

23×3 = 512 nodes. From Table 3.1, it is clear that the computational complexity of the

proposed PGA-IP algorithm is much lower than the exact PGA algorithm. We also list

the run time of MMSE-PIC as a reference. We can see that the complexity of PGA-IP

algorithm is only several times higher than the reduced complexity MMSE-PIC which

adopts [17] for iterative MMSE and [38] to reduce the complexity of LLR calculation.

3.5 Conclusion

In this chapter, we have presented the PGA-IP detection algorithm to handle correlated

massive MIMO channel and the simulation results show that it can outperform MMSE-

PIC about 1.5 dB. In order to reduce complexity, a novel algorithm based on Integer


11 12 13 14 15 16 17 18 19 20 21

10−4

10−3

10−2

10−1

SNR Per Antenna (dB)

BE

R

PGA−Exact Iter1PGA−IP Iter1PGA−Exact Iter2PGA−IP Iter2

Fig. 3.4 BER performance comparison between PGA-Exact and PGA-IP under 16-QAM40×40 MIMO correlated channel (ρ = 0.4)

Programming has been proposed. By computational complexity simulation, we could

see under massive MIMO scenario, PGA-IP detection has nearly the similar complexity

level as the MMSE-PIC algorithm but with better performance.

Chapter 4

A Low Cost LMMSE Channel

Estimator for OFDM Systems

4.1 Introduction

In this chapter, we focus on how to reduce the complexity of the traditional LMMSE

channel estimator for slow fading channels in an OFDM system. The proposed algorithm

can be used in data-aided channel estimation in an iterative system where the estimated

data or decoded data are employed as the virtual pilot. A preamble-type of pilots

based frame structure is employed to provide the initial LMMSE channel estimation.

For the channel estimation, we firstly reformulate the conventional LMMSE channel

estimation to a form that has small sized L×L matrix inversion where L is the number

of non-zero time domain channel taps. Then by exploiting the fact that this small sized

matrix is diagonally dominant due to the law of large numbers, we propose to use

a K terms Neumann series expansion to approximate its inversion. In this way, the

LMMSE estimator can be achieved by a cascade of matrix vector products, which can

be implemented with Fast Fourier Transform (FFT) or Inverse Fast Fourier Transform

(IFFT) operations with L inputs or L outputs, thus has the complexity of O(N logL).

Simulation results show that with small K (K ≤ 2), the performance of the proposed

52 A Low Cost LMMSE Channel Estimator for OFDM Systems

approximation is close to the exact LMMSE implementation from low to high SNR for

both pilot-aided channel estimation and data-aided channel estimation.


OFDM system model. Then the conventional LMMSE channel estimation algorithm is

presented in Section 4.3. In Section 4.4, we propose to use Neumann series expansion to

perform LMMSE channel estimation with low complexity. Simulation results are shown

in Section 4.5 and Section 4.6 concludes this chapter.

4.2 System Model

We consider a coded OFDM system with N subcarriers. The data vector of the nth

OFDM symbol x(n) = [x1,x2, · · · ,xN ]T , which is mapped from an interleaved code

sequence c, i.e., each xi ∈ A = {α1,α2, · · · ,α2Q}(|A |= 2Q) corresponds to a length-Q

subsequence of c denoted by ci = [ci,1,ci,2, · · ·,ci,Q]T , is stacked into one OFDM symbol.

The cyclic prefixes (CP) are inserted before the IFFT of x(n) to ensure the orthogonality

among the subcarriers and prevent inter-symbol interference (ISI) between consecutive

OFDM symbols. Considering a quasi-static channel which is constant during one OFDM

symbol, this OFDM system can be described as a set of parallel additive white Gaussian

noise (AWGN) channels. After dropping the CP and performing FFT, the received

frequency domain signal for OFDM symbol n is given by

y(n) = X(n)η(n)+w(n) (4.1)

where y(n) denotes a length-N observation vector, X(n)≡ diag{x(n)} denotes an N ×

N diagonal matrix with x(n) on its diagonal, η(n) is the frequency domain channel

coefficients and w(n) denotes a length-N circularly symmetric AWGN vector with PDF

C N (w;0,σ2I). For notation simplicity, from now on we omit the time index n.

4.3 LMMSE Channel Estimation 53

The time domain channel coefficients h = [h1,h2, ...,hL]T is related to the frequency

domain channel coefficients η with

η = FLh (4.2)

where FL is a truncated DFT matrix (sized N ×L) with the (k, l)-th element given by

FL(k, l) = exp(− j 2πklN )/

√N with j =

√−1. We assume that the power delay profile

(PDP) of a multipath channel is known1, which can be exploited by the channel es-

timator. The channel coefficients [hi] have zero mean and the covariance E[hhH ] =

diag{p1, ..., pL} ≡ P is regarded as the PDP, where pi is the average power of the i-th

delay path.

4.3 LMMSE Channel Estimation

From (4.1), the LMMSE estimation of frequency domain channel coefficients η can be

computed by [87]

η = CηyC−1yy y (4.3)

where Cηy and Cyy are the covariance matrix of η and y, and the auto-covariance matrix

of y, respectively. Based on the definition of the covariance matrix of two vectors a and

b

Cab = E[(a−E[a])(b−E[b])H], (4.4)

we have

Cηy = CηηXH = NFLPFHL XH (4.5)

and

Cyy = XCηηXH +σ2IN = NXFLPFH

L XH +σ2IN (4.6)

From (4.3), (4.5) and (4.6), it can be seen that directly computing frequency domain

channel coefficients needs an N ×N matrix inversion with O(N3) complexity.

1PDP can also be estimated with low complexity of O(L2)), see [86]


Considering that the number of non-zero time domain channel delay taps L can be

much less than the number of subcarriers N, using matrix inversion lemma, (4.3) can be

reformulated to

η = NFLPFHL XH(NXFLPFH

L XH +σ2IN)

−1y

= NFL√

P(N√

PH

FHL XHXFL

√P+σ

2IL)−1√P

HFH

L XHy.(4.7)

As P is a positive definite diagonal real-valued matrix, it is easy to get that√

PH=

√P = diag{√p1,

√p2, · · · ,

√pL}. As a result, (4.7) involves a matrix inversion (sized

L×L) and FFT (IFFT) with computational complexity of O(N logL+L3). At the same

time, by the law of large numbers, it is easy to see that matrix FHL XHXFL is diagonally

dominant, which can be exploited to enable Neumann series expansion to approximate

the matrix inversion, thereby reducing the complexity.

Algorithm 7 LMMSE Channel Estimation for OFDMInput: y,P,X ◃ X is the pilot or feedback dataOutput: η ◃ Frequency domain channel coefficients

1: D = NPtr{XHX}+σ2IL2: v =

√PFH

L XHy3: v0 = D−1v ◃ Initialize Neumann series expansion4: s0 = v05: for i = 1 to K do6: vi = vi−1 −ND−1√PHFH

L XHXFL√

Pvi−1 −σ2D−1vi−17: si = si−1 +vi8: end for9: η = NFL

√PsK

4.4 Newmann Series Expansion Based Channel Estima-

tion

4.4.1 Neumann Series Expansion

Neumann series expansion [88] can be employed to approximate matrix inversion with

the summation of a series of matrix multiplications. For a diagonal dominant matrix M,

4.4 Newmann Series Expansion Based Channel Estimation 55

let D = {M}diag. A K terms Neumann series expansion of M can be written as

M−1 ≈K

∑i=0

(I−D−1M)iD−1. (4.8)

It is easy to see that the multiplication of M−1 and a vector v can be computed by K

loops as

v0 = D−1v, s0 = v0

for i = 1 to K do

vi = (I−D−1M)vi−1

si = si−1 +vi

end for

(4.9)

with M−1v ≈ sK . With (4.9), only matrix-vector multiplications are required, which can

greatly reduce the computational complexity.

4 6 8 10 12 14 16

10−4

10−3

L

MS

E

DPS LMMSE

Dual−Diagonal

Proposed K=1

Proposed K=2

Proposed K=3

Exact LMMSE

Fig. 4.1 MSE performance with different L at SNR of 14dB


Let M = N√

PHFHL XHXFL

√P+σ2IL and v =

√PHFH

L XHy and plug them into

(4.9), then (4.7) can be computed by Algorithm 7. In this algorithm, line 1 uses the fact

that {FHL NFL}diag = tr{N}IL for any diagonal matrix N.


In Algorithm 7, the matrices D, X, P are all diagonal. So lines 1, 3, 4, and 7 have

trivial complexity. For Line 2 and Line 9, an IFFT (with L outputs) and an FFT

(with L inputs) can be employed, respectively. Then in Line 6, the computation of

N√

PHFHL XHXFL

√Pvi−1 can be implemented by an FFT with L inputs, followed by an

IFFT with L outputs. In summary, the total complexity of the proposed LMMSE channel

estimation for OFDM system is dominated by 2(K +1) FFTs (or IFFTs) with L inputs

or L outputs. Note that the complexity of FFT or IFFT with L inputs or L outputs can be

computed with complexity of O(N logL) [89].

In [64], the authors have proved that the WDFT based technique [61] [62] [63] can

be treated as special cases of their proposed Dual-Diagonal algorithm. So we focus on

Dual-Diagonal LMMSE algorithm for complexity comparison. For the dual-diagonal

LMMSE algorithm, the frequency domain channel coefficients η = FBFHAy, where

both A and B are diagonal matrices, and F is an N×N DFT matrix, therefore the number

of FFT (IFFT) required is two. Based on the fast algorithm in Appendix I of [64], there

are also two FFTs needed to compute the matrix B. As a result, the total number of N

point FFTs (IFFTs) of the Dual-Diagonal LMMSE algorithm [64] is four, which has the

computational complexity of O(N logN).


We consider an OFDM system with N = 128 subcarriers, the carrier frequency is 2.4

GHz and the symbol duration is 0.25 µs. The CP is set to be one-eighth of the number

of subcarriers. The modulation is 64-QAM with Gray mapping. We constrain the total


0 2 4 6 8 10 12 14 16

10−4

10−3

10−2

10−1

SNR (dB)

MS

E

Proposed K=1

Dual−Diagonal LMMSE

Proposed K=2

Proposed K=3

Exact LMMSE

Fig. 4.2 MSE performance for the 10-tap COST259_RAx channel

transmit power to one, and set the noise variance at receive side to σ2, then the average

received signal-to-noise ratio (SNR) is given by 1/σ2.

4.5.1 Mean-Square Error (MSE) Performance for Time-Invariant

Channels

In order to determine the required minimum K under different channel length, we

select a channel model with the PDP given by pi = Γe−0.1(i−1), i ∈ [1,L] where Γ is

a normalization factor (∑i pi = 1). Fig. 4.1 shows the MSE performance of the exact

LMMSE, the Dual-Diagonal LMMSE [64], the DPS based LMMSE [74]-[76]2 and the

2For the application scenario of this chapter, the DPS algorithm is reduced to one dimension. Thecomplexity of the exact DPS is O(N3). In [76], the complexity is reduced to O(IN) based on space-alternating expectation maximization (SAGE), where I is the number of DPS sequences and is in the orderof L.


proposed algorithm with different L at SNR of 14dB. We treat all transmit data as pilot

in order to get the upper bound of the MSE performance.

It is obvious that the proposed algorithm outperforms both the Dual-Diagonal

LMMSE [64] and the DPS based LMMSE [76] even with K = 1. At the same time, for

the proposed algorithm with K = 2, there is small MSE performance loss compared with

the exact LMMSE when L is greater than 8, and with K = 3 the proposed algorithm

has nearly the same performance as the exact LMMSE algorithm for all channel length.

The DPS based LMMSE algorithm has the worst performance as it only requires the

maximum normalized delay spread as the input, but all other algorithms require the

exact channel power profile.

Then we use the 10-tap COST259_RAx [90] channel to compare the MSE per-

formance in Fig. 4.2. It can be seen that the proposed method has nearly the same

performance as the exact LMMSE with K = 3 while there is small performance loss in

high SNR region with K = 2. It is also obvious that even with K = 2 the proposed algo-

rithm outperforms the Dual-Diagonal LMMSE algorithm and the DPS based LMMSE

algorithm from low to high SNR range.

4.5.2 Bit Error Rate (BER) Performance for Iterative Systems

As in Section V.A of [26], we consider an iterative channel estimation scheme, where

the hard decision from the output of the channel decoder is employed as the virtual

pilot. We employ a frame structure that every frame contains 25 OFDM symbols and

the first symbol is the pilot symbol to provide the initial LMMSE channel estimation

for the iterative channel estimation scheme3. With slow fading assumption, the channel

coefficients of the last OFDM symbol are used for the current symbol detection, then

the hard decision fed back from the decoder is mapped to a data symbol, which is

exploited to update the channel estimation. Although we assume that the channel is

static within one OFDM symbol in the design of the channel estimation, the channel

coefficients generated in the simulations changed at every sampling time according to

Jakes model [91] and the received singal was generated using the time-varying channel.


11 12 13 14 15 16 17 18

10−4

10−3

10−2

10−1

SNR (dB)

BE

R

DPS Iter0

DPS Iter1

Dual−Diagonal Iter0

Proposed Iter0

Exact LMMSE Iter0

Dual−Diagonal Iter1

Proposed Iter1

Exact LMMSE Iter1

Fig. 4.3 BER performance for 10-tap COST259_RAx Channel at speed of 100 km/hour

For the Jakes model, the relative speed between the transmitter and the receiver is

assumed to be 100 km/hour. In the simulations, we used the 10-tap COST259_RAx

channel model [90]. A rate-1/2, regular (3,6) low-density parity-check (LDPC) code

with codeword length of 768 bits was also used. For the LDPC decoder, the maximum

number of iterations between the variable nodes and the check nodes was 25.

Fig. 4.3 shows the BER performance of the system with exact LMMSE, Dual-

Diagonal LMMSE [64], DPS based LMMSE [76] and the proposed LMMSE approx-

imation. It is clear that the proposed algorithm with K = 2 nearly has the same BER

performance as the exact LMMSE algorithm and always better than the Dual-Diagonal

method and the DPS based LMMSE algorithm.

3Besides the preamble-type pilot in our example, the proposed algorithm can be easily applied to othertypes of pilots.


4.6 Discussion

4.6.1 The Power Delay Profile (PDP)

For the LMMSE channel estimation algorithm, there is a limitation that it requires the

PDP as a input which is typically difficult to obtain. This means that the PDP exploited

in the proposed algorithm may not be the same as that experienced by the transmit signal.

But fortunately, we have found that with the knowledge of the number of channel taps

(i.e. L) and SNR only, we can get close MSE performance to the exactly known PDP

case by artificially using an uniform distributed PDP for the proposed algorithm.

In order to validate this result under different channels, we have simulated the

channel models shown in Table 4.1 and the MSE performances are shown in Fig. 4.4. In

the simulations, the uniform PDP is defined as pi = 1/L, i ∈ [1, ...,L], the exponential

PDP 1 is defined as pi = Q1e−0.1(i−1), i ∈ [1, ...,L] where Q1 is a normalization factor

(∑i pi = 1), and the exponential PDP 2 is defined as pi = Q2e−0.5(i−1), i ∈ [1, ...,L] where

Q2 is also a normalization factor. Fig. 4.4 shows the MSE performance comparison

between the exact PDP, the uniform PDP, the exponential PDP 1 and the exponential

PDP 2 under different channel models. It is easy to see that the uniform PDP nearly has

the same MSE performance as the exact PDP.

It is worth noting that the above finding was also reported and analysed in [92]

4.6.2 The Assumption of Quasi-static Channel

While we assume that the channel is static within one OFDM symbol in the design of

the channel estimation, the channel coefficients generated in the simulations changed

at every sampling time according to Jakes model. Thus the maximum Doppler spread

normalized by the duration of one OFDM symbol is an important design factor for

our proposed channel tracking scheme to work properly. If this Doppler spread is too

big, there is too big difference between the channel coefficients of two subsequent

OFDM symbols and thus the channel tracking mechanism will fail. The Doppler spread

4.7 Conclusion 61

Table 4.1 Simulated Channel Models [2]No. Channel Model PDP in dB

1COST259_RAx

Rural ara, 10-tap channel3GPP_TR_25.943

−5.2−6.4−8.4−9.3−10−13.1−15.3−18.5−20.4−22.4

2COST207_TU12

Typical urban, 12-tap channel−4−30−2.6−3−5−7−5

−6.5−8.6−11−10

3COST207_HT

THilly terrain, 6-tap channel 0−2−4−7−6−12

4ITU_Vehicular_A

ITU Vehicular A, 6-tap channel 0−1−9−10−15−20

5ITU_Pedestrian_A

4-tap channel 0−9.7−19.2−22.8

6ITU_Pedestrian_B

6-tap channel 0−0.9−4.9−8−7.8−23.9

normalized by the duration of one OFDM symbol can be calculated by:

fdmax =vmax fC

c0Ts

=vmax fC

c0(N +NCP)Tsample

(4.10)

where vmax is the maximum movement speed, c0 = 3.0×108m/s is the speed of light, fC

is the carrier frequency, Ts is the OFDM symbol length, N is the number of subcarriers,

NCP is the length of CP and Tsample is the sample rate. For above simulations, for

vmax = 100 km/h the the Doppler spread normalized by the duration of one OFDM

symbol is 0.008.

4.7 Conclusion

In this chapter, we have proposed a low complexity LMMSE channel estimation algo-

rithm for OFDM systems by approximating the matrix inversion with Neumann series

expansion. This enables the channel estimation to be implemented with L-point input

FFT or L-point output IFFT, which have the complexity of O(N logL). Extensive sim-

ulation results show that under different channel models, the proposed algorithm can


0 5 10 15

10−4

10−3

10−2

10−1

Channel NO.6

Channel NO.6Channel NO.6

0 5 10 15

10−4

10−3

10−2

10−1

Channel NO.5


0 5 10 15

10−4

10−3

10−2

10−1

Channel NO.4


0 5 10 15

10−4

10−3

10−2

10−1

Channel NO.3


0 5 10 15

10−4

10−3

10−2

10−1

Channel NO.2


0 5 10 15

10−4

10−3

10−2

10−1

Channel NO.1


Exact

Uniform

Exp 1

Exp 2

Exact

Uniform

Exp 1

Exp 2

Exact

Uniform

Exp 1

Exp 2

Exact

Uniform

Exp 1

Exp 2

Exact

Uniform

Exp 1

Exp 2

Exact

Uniform

Exp 1

Exp 2

Fig. 4.4 MSE Under Channel No.1-6

achieve good MSE performance from low to high SNR range, and nearly the same

BER performance as the exact LMMSE algorithm. We also found that using a uniform

distributed PDP only incurs marginal performance loss but can relaxing the requirement

from the exact PDP to the maximum number of time domain channel taps.

Chapter 5

Low Complexity Iterative MMSE-PIC

Detection for Medium-Size Massive

MIMO

5.1 Introduction

When the number of receive antennas at the base station becomes large, in particular,

much larger than the number of total transmit antennas in user terminals, a simple

detection algorithm such as a matched filter can achieve very good performance, as

with the assumption of i.i.d. entries for channel matrix H, the channel vectors become

orthogonal to each other and HHH converges to a scaled identity matrix. But for practical

medium-size massive MIMO, matched filter based detection algorithm suffers perfor-

mance loss [72]. Therefore, alternative linear detection algorithms such as the minimum

mean square error parallel interference cancellation (MMSE-PIC) algorithm [60] are

often employed due to their relatively low complexity and good bit error rate (BER) per-

formance. However, the MMSE-PIC still requires complexity of O(K3) for calculating

a matrix inversion and O(K2M) for calculating the Gram matrix, where K is the number

of transmit antennas and M is the number of receive antennas.

64 Low Complexity Iterative MMSE-PIC Detection for Medium-Size Massive MIMO

To reduce the complexity, [70] and [71] employed Neumann series expansion to

approximate the matrix inversion by a matrix polynomial. Then in [72] the authors

proposed to use the same method to perform 3GPP-LTE uplink signal detection and

proved the convergence of the Neumann series expansion. Different from using Neumann

series expansion, in [1] an iterative method based on successive overrelaxation (SOR)

is employed to calculate the product of the inversion of a matrix and a vector, which

can converge to the exact solution. These work can successfully reduce the complexity

of computing matrix inversion from O(K3) to O(K2). But they all require the pre-

computed Gram matrix as an input. In massive MIMO with M ≫ K, the Gram matrix

computation involves computational complexity of O(K2M), which is much higher than

the O(K3) complexity of matrix inversion .

In this chapter, based on the MMSE detection algorithm [17], we exploit Neumann

series expansion to reduce the total complexity of MMSE-PIC for massive MIMO. With

the proposed method, computational complexity is reduced by avoiding direct matrix

inversion and replacing the matrix-matrix multiplication of Gram matrix with matrix-

vector multiplications. Specifically, we propose to employ an L (typically L ≤ 3) terms

Neumann series expansion for calculating the means of data symbols to be detected, and

a first order approximation for calculating the variances and thus reducing the complexity

from O(K2M+K3) to O(LKM) with marginal performance loss when L = 3 for MIMO

size of K×M = 16×128. We also investigate the application of the proposed algorithm

in an iterative detection and decoding (IDD) system, where the symbol detector and the

channel decoder work iteratively. We found that with one iteration between the decoder

and the detector, the proposed approximation algorithm with L = 3 can achieve the same

performance as the exact MMSE-PIC algorithm.


turbo-MIMO system model. Then in Section 5.3, we propose to use Neumann series

expansion to perform MMSE detection without computing the Gram matrix. Simulation

results are shown in Section 5.4 and Section 5.5 concludes this chapter.

5.2 System Model 65

5.2 System Model

Consider a multiuser massive MIMO system with M receive antennas at the base station

and K single-antenna user terminals. Let x = [x1,x2, . . . ,xK]T denote the transmit vector

comprising the symbols transmitted simultaneously by all users in one channel use where

xn ∈ A = {α1,α2, . . . ,α2Q}(|A |= 2Q) denotes transmitted symbol from user n, then

each xn corresponds to a length-Q subsequence of c denoted by cn = [cn,1,cn,2, · · ·,cn,Q]T.

Let H = [h1,h2, . . . ,hK] denote the channel gain matrix, where hn = [h1n,h2n, . . . ,hMn]T

is the channel gain vector from user n to the base station, and h jn denotes the channel

gain from the n-th user to the j-th receive antenna at the base station. Assuming rich

scattering, adequate spatial separation between the base station antenna elements and

perfect user power control, h jn,∀ j are assumed to be i.i.d. complex Gaussian distributed

with zero mean and variance one. Thus a length-M observation vector y at the base

station can be written as

y = Hx+w (5.1)

where w denotes a length-M circularly symmetric additive white Gaussian noise (AWGN)

vector with zero-mean and covariance of σ2I.

The task of the Soft-In Soft-Out (SISO) detector is to compute the extrinsic log-

likelihood ratio (LLR) for each code bit cn,q, which is the input to the decoder and can

be expressed as [17]

Le(cn,q) = ln

∑xn∈A 0

q

P(y|xn)P(xn)

∑xn∈A 1

q

P(y|xn)P(xn)−La(cn,q) (5.2)

where La(cn,q) is the output extrinsic LLR of the decoder, xn ∈ A 0q (A

1q ) represents

constellations whose q-th bit is 0(1) and P(xn) is the a priori probability of xn which

can be calculated from La(cn,q).


5.3 MMSE Detection Based on Neumann Series Expan-

sion

We employ the method proposed in [17] to perform MIMO MMSE detection. With

this algorithm, it is easy to reformulate the matrix to be inverted with the size of K ×K

which is preferable for massive MIMO applications with M ≫ K. The core part of this

algorithm is to compute the a posteriori mean mp and variance Vp of x by

Vp = (V−1 +1

σ2 HHH)−1, (5.3)

mp = m+1

σ2 Vp(HHy−HHHm), (5.4)

where m and V are the a priori mean and variance of x, respectively, and they can be

calculated from the feedback of the decoder1. Then the extrinsic mean men and variance

ven of the n-th element of x (which are used to generate soft-out LLR) can be calculated

by

ven = (

1vp

n− 1

vn)−1, (5.5)

men = ve

n(mp

n

vpn− mn

vn), (5.6)

where vn, vpn are the (n,n)-th elements of matrix V and Vp, respectively, and mn, mp

n are

the n-th elements of vector m and mp, respectively. It is easy to see that (5.3) and (5.4)

require a computational complexity of O(K2M) for calculating HHH and O(K3) for

calculating the matrix inverse.

1At the beginning of the IDD, there is no feedback from the decoder. Assuming that the constellationof the modulation is with zero mean and normalized with unit power and data streams from differenttransmit antennas are statistically independent, we have m be a zero vector and V be the identity matrixIK with size K ×K.

5.3 MMSE Detection Based on Neumann Series Expansion 67

Algorithm 8 Reduced Complexity Neumann Series expansion based MMSE detection

Input: y, H, La

Output: Le ◃ extrinsic LLR value for every bit1: Calculate a priori mean m and variance V from La

2: mn = ∑αi∈A

αiP(xn = αi)

3: vn = ∑αi∈A

|αi −mn|2 P(xn = αi)

4: Calculate a posteriori mean mp

5: D = diag(V−1 + 1σ2 HHH)

6: v0 = D−1(HHy−HHHm)7: s0 = v08: for i = 1 to L do9: vi = vi−1 −D−1(V−1 + 1

σ2 HHH)vi−110: si = si−1 +vi11: end for12: mp = m+ 1

σ2 sL13: Approximate the diagonal elements of Vp

14: vpn = dn ◃ dn is the (n,n)-th element of D−1

15: Calculate extrinsic mean men and variance ve

n16: ve

n = ( 1vp

n− 1

vn)−1

17: men = ve

n(mp

nvp

n− mn

vn)

18: Calculate extrinsic LLR Le

19: Le(cn,q) = ln∑

αi∈A 0q

exp(− |αi−me

n |2ven

)∏

q′ =q

P(cn,q′

=si,q′

)

∑

αi∈A 1q

exp(− |αi−me

n |2ven

)∏

q′ =q

P(cn,q′

=si,q′

)


5.3.1 Neumann Series Expansion

The convergence of Neumann series expansion for detection has been proved in [72]. It

has been shown in [72] that, for large ρ = M/K, the Gram matrix G = HHH tends to be

diagonally dominant, which enables the convergence of the Neumann series expansion.

Let us decompose the regularized Gram matrix A = V−1+ 1σ2 G to A = D+E, where

D is the main diagonal of A. As V is a diagonal matrix, the complexity of computing D

is the same as computing the diagonal elements of G. We can then approximate A−1 in

the Neumann series as

A−1 ≈L

∑i=0

(IK −D−1A

)iD−1

=L

∑i=0

(IK −D−1V−1 − 1

σ2 D−1G)iD−1.

(5.7)

Using A−1 of (5.7) to replace Vp and plugging it into the representation of mp of (5.4),

it can be seen that only matrix-vector multiplications are needed for calculating mp and

the calculation of the Gram matrix G itself is avoided. But we should note that in (5.5)

and (5.6) the diagonal elements of Vp are also required to compute the extrinsic mean

and variance. To reduce the complexity, we propose to use the first order approximation

(L = 0) of (5.7) for computing the diagonal elements of Vp (i.e. Vp ≈ D−1).

From (5.7), it is obvious that the multiplication of A−1 and a vector v can be

computed by L loops. The proposed MIMO MMSE detection algorithm with Neumann

series expansion is summarized in Algorithm 8. We note that when L = 0, the proposed

algorithm coincides with the matched filter detector as mp = 1σ2 D−1HHy (Note that we

assume m is a zero vector at the beginning of IDD).


We focus on the number of real-valued multiplications needed and only count quadratic

or beyond terms. For the real-valued system model, the matrix size of H is 2K×2M, y is

a length-2M vector and m is a length-2K vector. Note that using the symmetric property


of matrix G and Vp can reduce the complexity by a half. Table 5.1 is a summary

of complexity comparison between MMSE, the proposed algorithm, Neumann series

expansion based algorithm in [70] and SOR based algorithm in [1]. In the table, the

term 4K2M corresponds to the computing of Gram matrix G. Note that for SOR based

algorithm in [1], the number of iterations Ls may be smaller than that of Neumann series

expansion.

Table 5.1 Computational Complexity ComparisonAlgorithm Number of multiplicationsExact MMSE [17] 8K2 +4K3 +4(K2 +K)MProposed (16+8L)KMNeumann series based [70] 4K2M+8(L−2)K3

SOR based [1] 4K2M+4LsK2

5.3.3 Discussion

In contrast to [70], [71], and [72], which also use the Neumann series expansion to

approximate matrix inversion, the proposed methods avoid direct matrix inversion and

replace the matrix-matrix multiplication by matrix-vector multiplications, which result

in considerable saving in computations.

The method proposed in [1], after optimizing a parameter by off-line exhaustive

searching, can converge faster than Neumann series expansion. But it requires each

element of matrix G as its input, which means that HHH has to be computed explicitly,

thus it cannot reduce the total complexity significantly.


We consider a Rayleigh block fading random channel where H does not change over a

codeword. During simulations, we assume that perfect channel information is available

in the detection module. A rate-1/2, regular (3,6) low-density parity-check (LDPC) code

with codeword length of 2000 bits is employed as the channel code and the maximum

number of iterations of the decoder is 25. The constellation of 64-QAM with Gray


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

10−4

10−3

10−2

10−1


BE

R

Proposed (L=0)Proposed (L=0) VarProposed (L=1)Proposed (L=1) VarProposed (L=2)Proposed (L=2) VarSOR (L

s=2)

Proposed (L=3)Proposed (L=3) VarExactProposed (L=3) IDD

Fig. 5.1 BER performance comparison for exact MMSE, proposed and SOR based [1]with MIMO size of K ×M = 16×128

mapping is used. We constrain the total transmitter power to one, and set the noise

variance at each receive antenna to σ2. Then the average received signal-to-noise ratio

(SNR) at each receive antenna is given by 1/σ2. For each SNR value, we simulate at

least 100000 codewords. In the simulations, clipping is applied to both the soft-output

and the soft-input of the detector. The soft-in clipping threshold2 for the a priori LLR is

±2, and soft-output module constrains the output LLR range to [−50,50].

Fig. 5.1 shows the BER performance comparison between the exact MMSE detec-

tion [17], the proposed algorithm and the SOR based algorithm [1]. The MIMO size

is K ×M = 16×128. It is easy to see that the performance of the matched filter (with

legend Proposed (L=0)) is poor. At the same time, with a larger L the approximation is

more accurate and when L = 3 the proposed algorithm can approach the performance

2This clipping threshold can also help resolve the numerical stability issue of Line 16 and Line 17 ofAlgorithm 8 when the a priori variance vn is close to zero.

5.5 Conclusion 71

of the exact algorithm within 0.3dB. It can also be seen that an extra IDD iteration

(with legend Proposed (L=3) IDD) achieves slightly better performance than the exact

MMSE-PIC algorithm without IDD.

To evaluate the performance loss caused by the first order approximation of Vp, we

use (5.7) to explicitly compute the matrix inversion and assign the diagonal elements

to vpn (as in [70]) and the performances are shown in Fig. 5.1 with legends ending with

Var. It is obvious that the proposed approximation to variance only leads to a small

performance penalty.

5.5 Conclusion

In this chapter, we have proposed to use Neumann series expansion to reduce the

complexity of the MMSE-PIC algorithm for massive MIMO applications with M ≫

K. Firstly, an L terms Neumann series was employed to avoid computing the matrix

inversion by replacing it with a cascade of matrix-vector multiplications. Then, a first-

order approximation was employed to compute the diagonal elements of the a posteriori

variance matrix for calculating LLR, which helps to avoid computing the Gram matrix

explicitly. Simulation results showed that with a small L the proposed approximation

methods lead to marginal performance loss compared with the exact implementation,

but with considerable complexity saving.

Chapter 6

A Novel Interpolation Algorithm for

Massive MIMO OFDM System

Detection

6.1 Introduction

The aforementioned detection algorithms can be applied to flat-fading channels (e.g.

signal detection at each subcarrier of an OFDM system). In this chapter, we will focus

on the detection of all subcarriers of a massive MIMO-OFDM system. As the number of

subcarriers N in a MIMO-OFDM system is typically large, the receiver will have very

high computational complexity if we apply the aforementioned algorithms for every

subcarrier.

In OFDM systems, as the frequency domain coefficients are typically highly corre-

lated, interpolation is often employed to reduce the complexity. For MMSE detection,

the matrix inversion is the main contributor to the complexity. There are several works

which use the interpolation method to compute the matrix inversion. For example, [93]

and [94] exploited the fact that the adjoint and the determinant of a matrix can be repre-

sented in a polynomial form and thus they can be interpolated. Then based on the results

of [93] and [94], [95] proposed a Gaussian approximation for phase shifted interpola-

74 A Novel Interpolation Algorithm for Massive MIMO OFDM System Detection

tion method. Recently, [96] proposed a Banachiewicz formula based matrix inversion

with low complexity. But all these interpolation based algorithms were designed for

small-size MIMO and cannot be easily extended to massive MIMO applications.

In this chapter, from the asymptotic property of Gram matrix we conjecture that

there might be strong correlation between the (regularized) Gram matrix inversions of

adjacent subcarriers for medium-size massive MIMO, which was verified by simulations.

By exploiting this strong correlation, we proposed a linear interpolation based MMSE

detection algorithm which can significantly reduce the number of matrix inversion

required. Extensive simulations show that with the same level of complexity as the

matched filter, the proposed algorithm only incurs small BER performance loss compared

to the exact MMSE detector.


massive MIMO-OFDM system model and soft output MMSE detection algorithm.

Then in Section 6.3, the strong correlation of the matrix inversion for massive MIMO is

evaluated and a linear interpolation algorithm is proposed to compute the matrix inversion

with low complexity. Simulation results are shown in Section 6.4 and conclusion is

given in Section 6.5.

6.2 System Model and Soft-output MMSE Detector

Considering a coded massive MIMO-OFDM system with Nr receive antennas, Nt trans-

mit antennas and N subcarriers, the multipath channel can be mapped to N flat-fading

channels. For subcarrier n = 1, . . . ,N, the received signal at the base station can be

modelled as

yn = Hnxn +wn (6.1)

where yn denotes a length-Nr observation vector, Hn denotes an Nr ×Nt MIMO system

transfer matrix of subcarrier n, wn denotes a length-Nr circularly symmetric additive

white Gaussian noise (AWGN) vector with zero means and covariance of σ2I, and xn =

[x1,x2, . . . ,xN t]T is the data symbol vector transmitted on subcarrier n which is mapped

6.2 System Model and Soft-output MMSE Detector 75

from an interleaved code sequence c, i.e., each xi ∈ A = {α1,α2, . . . ,α2Q}(|A |= 2Q)

corresponds to a length-Q subsequence of c denoted by ci = [ci,1,ci,2, . . . ,ci,Q]T .

The task of a soft-output detector is to compute the extrinsic log-likelihood ratio

(LLR) for each code bit. We apply the LMMSE algorithm in [17] to the MIMO detection.

As we only concern the conventional MMSE detection, after setting V = INt and m a

zero vector, the a posteriori mean mp and variance Vp of x can be calculated by [17]

Vpn = (INt +

1σ2 HH

n Hn)−1, (6.2)

mpn =

1σ2 Vp

nHHn yn, (6.3)

Then the extrinsic mean men,i and variance ve

n,i of the i-th element of xn can be calculated

by

ven,i = (

1vp

n,i−1)−1, (6.4)

men,i = ve

n,imp

n,i

vpn,i

, (6.5)

where vpn,i is the (i, i)-th element of matrix Vp

n and mpn,i is the i-th element of vector mp

n,

respectively. At last, the LLR can be calculated by:

Le(ci,q) = ln

∑α j∈A 0

q

exp(− |α j−me

n,i|2ve

n

)∑

α j∈A 1q

exp(− |α j−me

n,i|2ve

n,i

) (6.6)

where α j ∈ A 0q (A

1q ) represents constellations whose q-th bit is 0(1). It is worth noting

that (6.6) can be simplified by exploiting the constellation regularity [38]. It is easy to

see that (6.2) and (6.3) require computational complexity of O(N2t Nr) for calculating

HHH and O(N3t ) for calculating the matrix inversion.


6.3 MMSE Detection Based on Interpolation

It is obvious that with large number of subcarriers N, the brute-force tone-by-tone

detection incurs prohibitive complexity. To reduce the complexity, one method is to

exploit the correlation between adjacent subcarriers to compute matrix inversion by

using interpolation. This idea was investigated in several works [93]-[96]. In [93], [94]

and [95], interpolation based matrix inversion algorithms were proposed by employing

the fact that even though the inverse of a polynomial matrix is generally not polynomial,

the adjoint and the determinant is polynomial, which allows efficient inversion of the

individual matrices through interpolation. But as mentioned in [93] and [94], these

methods are only useful for limited matrix size as the adjoint matrix is difficult to obtain

for arbitrary matrix size. In [96], an interpolation based matrix inversion algorithm based

on Banachiewicz formula for the inverse of a partitioned matrix was proposed for 4×4

MIMO. But this algorithm is difficult to be extended to massive MIMO applications.

In this section, by theoretical analysis and simulation results we will show that

the matrix inversion of Gram matrix required by a zero forceing (ZF) detector or a

regularized Gram matrix required by an MMSE detector has strong correlation between

adjacent subcarriers for massive MIMO-OFDM systems. Then we will propose a low

complexity detection algorithm which computes the inversion matrix from the base tones

by linear interpolation.

6.3.1 Correlation of Matrix Inversion for Massive MIMO-OFDM

Systems

It is well known that for massive MIMO where Nt ≪ Nr, the Gram matrix Gn = HHn Hn

becomes a diagonally dominant matrix and approaches a scaled identity matrix when

both Nt and Nr approach infinite. This implies that the matrix inversions G−1n of adjacent

subcarriers have the strongest correlation when Nt and Nr approach infinite. This fact

inspires us to consider that when Nt and Nr are not so big, there should still be some

correlation between G−1n of adjacent subcarriers. In the following, we use simulation

6.3 MMSE Detection Based on Interpolation 77

−10 −5 0 5 100.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

d

Cor

rela

tion

Channel Correlation Ch(d)

Correlation Cg(d) ρ=6




Fig. 6.1 Correlations of Ch(d) and Cg(d) of adjacent subcarriers with N = 64, Nt = 20,different ρ and different subcarrier distance d.

to confirm this conjecture. As in [97], the channel correlation coefficients between

subcarrier n and subcarrier n+d (assuming modulo addition) are defined as

Ch(d) =E[|hH(n+d)h(n)|2

]√E[||hH(n+d)||2

]E[||h(n)||2

] (6.7)

where h(n) = vec(Hn) is the vector obtained by stacking the columns of Hn one on top

of the other, and n ∈ [1,N].


Similarly, the correlation coefficients between G−1n and G−1

n+d can be defined as

Cg(d) =E[|gH(n+d)g(n)|2

]√E[||gH(n+d)||2

]E[||g(n)||2

] . (6.8)

where g(n) = vec(G−1n ).

Fig. 6.1 shows the correlations Ch(d) and Cg(d) under different subcarrier distance d

for different sized MIMO (Nt = 20 and with different ρ =Nr/Nt) with N = 64 subcarriers.

It is obvious that when the distance d becomes larger the correlations of Ch(d) and Cg(d)

decreases accordingly as expected. On the other hand, the correlation Cg(d) between

adjacent tones drops much slower than Ch(d) does. At the same time, the correlation

coefficient dropping rate (according to the increased d) is smaller for massive MIMO

systems with bigger ρ than those with smaller ρ .

Table 6.1 Simulated Channel Models [2]No. Channel Model Delay in ns

1COST207_TU6alt

Alt typical urban 6-tap 0 200 500 1600 2300 5000

2COST259_RAxRural ara, 10-tap

3GPP_TR_25.943

0 42 101 129 149245 312 410 469 528

3COST207_TU12

Typical urban, 12-tap0 200 400 600 800 1200 1400

1800 2400 3000 3200 5000

4COST207_HT

THilly terrain, 6-tap 0 200 400 600 15000 17200

5ITU_Pedestrian_A

ITU Pedestria A, 4-tap 0 110 190 410

6ITU_Vehicular_A

ITU Vehicular A, 6-tap 0 310 710 1090 1730 2510

In order to evaluate the correlations under different channel models, the correlations

of Ch(d) and Cg(d) of adjacent subcarriers are simulated for the channels listed in Table

6.1 and the results are shown in Fig. 6.2.

From this figure, it is clear that for the channels with small L (taps of time domain

coefficients), the correlations of both Ch(d) and Cg(d) between adjacent subcarriers

are large. For channels with the same L (like channel No. 2, No. 3 and No. 4), the


−10 −5 0 5 100.3

0.4

0.5

0.6

0.7

0.8

0.9

1

d

Cor

rela

tion

Ch2−L10 Ch(d)

Ch2−L10 Cg(d)

Ch1−L6 Ch(d)

Ch1−L6 Cg(d)

Ch3−L12 Ch(d)

Ch3−L12 Cg(d)

Ch4−L6 Ch(d)

Ch4−L6 Cg(d)

Ch5−L4 Ch(d)

Ch5−L4 Cg(d)

Ch6−L6 Ch(d)

Ch6−L6 Cg(d)

Fig. 6.2 Correlations of Ch(d) and Cg(d) of adjacent subcarriers (with different d) underdifferent channel models with N = 64, Nt = 20 and ρ = 8.

channel with a large maximum delay spread has small correlation (e.g. channel No.

4). More importantly, the correlation of Cg(d) is much stronger than the correlation of

Ch(d) which is consistent with Fig. 6.1.

For the correlation of Vp between adjacent subcarriers, extensive simulations show

that the correlation performance of Vpn is similar to that of G−1

n .

6.3.2 Interpolation Based Matrix Inversion

We select the subcarriers with index of {1,1+D, ...,1+KD,N} as the base subcarriers

where K = ⌊N−1D ⌋ is the closet integer not great than N−1

D . Then we compute the matrix

inversion Vp exactly for these base subcarriers. For non-base subcarriers with index


inside the range (1,1+KD), the matrix inversion can be computed by linear interpolation

asVp

kD+1+d = (1− dD)Vp

kD+1 +dD

Vp(k+1)D+1,

d ∈ (1,D) and k ∈ (0,K).

(6.9)

For the subcarriers with index between 1+KD and N, the matrix inversion can be

computed by linear interpolation as

VpKD+1+d = (1− d

D1)Vp

KD+1 +d

D1Vp

N ,

d ∈ (1,D1)

(6.10)

where D1 = N −KD−1.


We focus on the number of complex-valued multiplications needed and only count

quadratic or beyond terms. Note that the symmetric property of matrix Gn and Vpn can

be exploited to reduce the complexity by a half. We use the complexity of matched filter

as the benchmark which has the complexity of O(NrNt) for every tone. We choose [17]

for exact implementation for base tones and using interpolation of (6.9) to compute the

matrix inversion for adjacent subcarriers.

Table 6.2 is the summary of complexity comparison between the exact MMSE, the

matched filter, [98] and the proposed algorithm. In the table, the term N2t Nr corresponds

to the computing of Gram matrix G and I is the terms number of Neumann Series

expansion in [98]. To illustrate the complexity difference, Fig. 6.3 is shown for

Table 6.2 Computational Complexity ComparisonAlgorithm Number of MultiplicationsExact MMSE [17] N(N3

t +N2t Nr +4NtNr)

Matched Filter N(NtNr)Low Complexity [98] N[(5+2I)NtNr]

ProposedND(N

3t +N2

t Nr +4NtNr)(N − N

D)(N2t +2NtNr)


8 10 12 14 16 18 200

0.5

1

1.5

2

2.5x 10

7

Number of transmit antenna

Num

ber

of c

ompl

exed

val

ue m

ultip

licat

ions

Exact

Low Complexity [93]

Proposed (D=16)

Proposed (D=32)

Matched filter

Fig. 6.3 Complexity comparison with ρ = 8, N = 128 and I = 5

comparing complexities of the above algorithms for a typical medium-size massive

MIMO where ρ = Nr/Nt is set to 8, N = 128, D = 16 and I = 5. It is obvious that the

proposed algorithm has great computation saving compared to the exact implementation

and has comparable complexity to the matched filter. For example, when Nt = 16,

Nr = 128,N = 64 and D = 16, up to 85% of computation saving can be obtained by the

proposed algorithm compared to the exact implementation. Compared to the matched

filter algorithm, the complexity of the proposed algorithm is only 2.4 times higher.


6.4 BER Performance

Based on the profile of channel No.1 in Table 6.1, we firstly generate the time domain

channel coefficients for every path between transmit antennas and receive antennas.

Then FFTs are performed to get the frequency domain channel coefficients. Based

on these coefficients, the frequency domain channel gain matrix of every subcarrier

Hn can be obtained. A rate-1/2, regular (3,6) low-density parity-check (LDPC) code

with codeword length of 10000 bits is employed as the channel code and the maximum

number of iterations of the decoder is 25. The signal modulation of 64-QAM with Gray

mapping is used. For each signal-to-noise (SNR) value, we simulate at least 10000

codewords. In the simulations, there are clipping in soft-output part of the detector (LLR

is constrained to range [−50,50]).

Fig. 6.4 shows the BER performance comparison between the exact MMSE detec-

tion [17], the matched filter, the proposed algorithm with exact Hn (with D = 16,32)

and proposed algorithm with interpolated Hn for non-base subcarriers. The MIMO size

is Nt ×Nr = 16× 128, and the number of subcarrier is set to 256. It is obvious that

the performance of the matched filter algorithm is poor. When computing (5.4) with

proposed algorithm and exact Hn, the BER performance is nearly the same as the exact

one when D = 16, while there is about 0.4dB SNR performance loss when D = 32.

Interpolated Hn in non-base subcarriers also causes some performance loss, e.g. a 0.2dB

performance loss can be observed when using interpolated Hn and Vpn.

6.5 Conclusion

In this chapter, through theoretical analysis and simulations, we found that the matrix

inversion of the Gram matrix G−1n or the regularized Gram matrix (Gn +σ2I)−1 is

strongly correlated between adjacent subcarriers in a massive MIMO-OFDM system.

Then we proposed to use linear interpolation to compute the matrix inversion at non-base

subcarriers. Simulation results showed that the proposed approximation method leads to

marginal performance loss but with considerable complexity saving.

6.5 Conclusion 83

8 8.5 9 9.5 10 10.5 11 11.5 1210

−4

10−3

10−2

10−1

100

SNR (dB)

BE

R

ExactMatche filter

Interp Vnp + exact H

n (D=16)

InterP Vnp +InterP H

n(D=16)

Interp Vnp + exact H

n (D=32)

Fig. 6.4 BER performance comparison for exact MMSE, Matched filter, Proposed Vpn

with exact Hn and Proposed Vpn with interpolated Hn for Nt ×Nr = 16×128 MIMO.

Chapter 7

Summary and Future Work

7.1 Summary

In this thesis, we designed a low complexity channel estimation algorithm for OFDM

systems and various low complexity MIMO detection algorithms. To exploit the correla-

tion between different iterations, an algorithm has been proposed in Chapter 2 to reduce

the matrix inversion from cubic to quadratic level for the second and subsequent itera-

tions. Chapter 3 presents a low complexity PGA algorithm which can deal with channel

spatial correlation effectively. Then a Neumann series expansion based LMMSE channel

estimation algorithm was proposed in Chapter 4 which can reduce the complexity to

O(N logL) where N is the number of subcarriers and L is the number of time domain

channel coefficient taps. In Chapter 5, a Neumann series expansion based LMMSE

algorithm is proposed for massive MIMO uplink detection. With this algorithm, the

total complexity of detecting a length-Nr block is reduced to O(KNtNr) where K is the

number of term in Neumann series expansion and typically it is less than 5. Considering

that the per-tone uplink detection has prohibitive computational complexity in a mas-

sive MIMO-OFDM system, we propose a novel interpolation algorithm to reduce the

complexity of the matrix inversion required by a ZF or MMSE detector in Chapter 61.

86 Summary and Future Work

Compared to the matched filter which has the lower bound complexity, the proposed

algorithm has comparable complexity but has much better BER performance.

7.2 Future Work

7.2.1 Channel estimation for MIMO-OFDM systems

It is most likely that the proposed method in Chapter 4 can be applied to MIMO-OFDM

system and still keep low complexity. Based on the literature survey, a time domain

SAGE algorithm in [99] [100] can reduce the complexity to O(INNtNrL) , where I is

the number of SAGE iterations which is typically less than 4. But as this algorithm

performs channel estimation tap by tap, the estimation latency is huge. By employing

proposed algorithm in a SAGE framework, it is expected that the total complexity can

be reduced to O(INNtNr logL) and the latency can be dramatically reduced because of

employing FFT operation. Specifically, by implementing the single antenna LS channel

estimation ((5) in [99]) with proposed algorithm in Chapter 4, the matrix inversion can

be replaced by L-point input FFT and L-point output IFFT.

7.2.2 Channel estimation for Massive MIMO

The pilot contamination is a key issue for massive MIMO technology [101] [102] [103].

There are two reference papers are closely related to our work. The first one [104]

employs preamble based pilots and uses iterative method to perform data-aided channel

estimation. As virtually, all the data can be assumed as pilots, the effect of pilot contam-

ination can be greatly eliminated. Then the second one [105] employs superimposed

pilots and also performs channel estimation in an iterative manner. The superimposed

pilots can naturally reduce the pilot overhead as all time and frequency resources can

be employed to transmit data. As the length of pilot is of the same length as the data,

it can effectively combat with pilot contamination. When the authors perform channel

1A manuscript based on the main idea of Chapter 6 is being prepared and is planned to be submitted tothe Communication Letter.

7.2 Future Work 87

estimation in [104], they use the approximation of A−1 = diag{A}−1 to perform trivial

matrix inversion which incurs big performance loss as it neglected all non-diagonal

elements. It is hopeful that by employing Neumann series expansion to perform this

matrix inversion, there should be performance gain with small complexity increase.

Similar technique can also be applied to the superimposed pilots case to further reduce

complexity.

7.2.3 Uplink Signal Detection for Massive MIMO-OFDM

In Chapter 6, we have proposed an interpolation based low complexity algorithm for

massive MIMO-OFDM system. But the algorithm can only be applied to conventional

soft-output LMMSE based detector. It cannot be used directly in an IDD system for the

second and subsequent iterations, because the a priori variance is different for different

iteration. From Chapter 2, we know that the matrix inversion between different iterations

are highly correlated and thus can also be interpolated to get low complexity in IDD

system. As a result, a potential future work could be to combine the methods in both

Chapter 2 and Chapter 6 to form a low complexity IDD massive MIMO-OFDM uplink

detector.

Appendix A

Proof of the Equality of Algorithm 1

and Algorithm 2

By using the matrix inversion lemma, Line 4 and 5 in Algorithm 2 can be rewritten to

Vp = V−VHHV−1z HV (A.1)

mp = m−VHHV−1z (y−Hm) (A.2)

where Vz = HVHH +2σ2I. Using hn to denote the nth column of H, we get

vpn = vn − v2

nhHn V−1

z hn (A.3)

mpn = mn − vnhH

n V−1z (y−Hm). (A.4)

Then using Line 7 and Line 8 in Algorithm 2 we have

men =

hHn V−1

z (y−Hm+mnhn)

hHn V−1

z hn(A.5)

90 Proof of the Equality of Algorithm 1 and Algorithm 2

ven =

1hH

n V−1z hn

− vn. (A.6)

Now we will show that hHn V−1

z is a scale version of hHn V−1

n = hHn(2σ2I+HVHH)−1 in

Algorithm 1 (V is the same as V with Vi,i = 1). Obviously, Vn = Vz+(1−vn)hnhHn . By

using the Sherman-Morrison-Woodbury formula of (A+uvH)−1 = A−1 − A−1uvHA−1

1+vHA−1u

we get

hHn V−1

n = hHn V−1

z −hH

n (1− vn)V−1z hnhH

n V−1z

1+(1− vn)hHn V−1

z hn

= hHn V−1

z −(1− vn)(hH

n V−1z hn)(hH

n V−1z )

1+(1− vn)hHn V−1

z hn.

(A.7)

As hHn V−1

z hn is a scalar, (A.7) can be rewritten to

hHn V−1

n =hH

n V−1z

1+(1− vn)hHn V−1

z hn. (A.8)

By noting 11+(1−vn)hH

n V−1z hn

to kn, we get hHn V−1

n = knhHn V−1

z . Then we can represent men

in Algorithm 1 as

men =

xn

µn=

fHn y

fHn hn

=hH

n V−1n (y−hm+mnhn)

hHn V−1

n hn

. (A.9)

In (A.9) by substituting hHn V−1

n with knhHn V−1

z and cancelling the scalar kn, we get the

same result as (A.5).

Now we will show that ven of Algorithm 1 is the same as that in Algorithm 2. From

Line 7 of Algorithm 1, we get

ven =

1fHn hn

−1 =1

hHn V−1

n hn

−1. (A.10)

After substituting (A.8) to (A.10), we get

ven =

1hH

n V−1z

1+(1−vn)hHn V−1

z hnhn

−1 =1

hHn V−1

z hn− vn. (A.11)

91

This is exactly the same as (A.6). Thus, while Algorithm 2 and Algorithm 1 have

different formulae, they actually generate the same extrinsic mean and variance.

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