reduced complexity demodulation of mimo bicm using group detection thesis

8/9/2019 Reduced Complexity Demodulation of MIMO BICM using Group Detection Thesis

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THE IBY AND ALADAR FLEISCHMAN FACULTY OF ENGINEERINGDEPARTMENT OF INTERDISCIPLINARY STUDIES

Reduced Complexity

Demodulationof MIMO Bit-InterleavedCoded Modulation

using IQ Group Detection

Thesis submitted for the degreeMaster of Science in Engineering Sciences

by

Zak Levi

Submitted to the Senate of Tel-Aviv University

February 2006


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THE IBY AND ALADAR FLEISCHMAN FACULTY OF ENGINEERINGDEPARTMENT OF INTERDISCIPLINARY STUDIES

Reduced Complexity

Demodulationof MIMO Bit-InterleavedCoded Modulation

using IQ Group Detection

Thesis submitted for the degreeMaster of Science in Engineering Sciences

by

Zak Levi

Submitted to the Senate of Tel-Aviv University

This research work was carried out at the

Department of Electrical Engineering - Systems,Tel-Aviv University

Under the supervision ofDr. Dan Raphaeli

February 2006


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Abstract

In this thesis we propose a novel reduced complexity technique for the de-

coding of multiple-input multiple-output bit interleaved coded modulation

(MIMO-BICM) using IQ Group Detection (GD). It is well known that the

decoding complexity of the MAP detector for MIMO-BICM increases ex-

ponentially in the product of the number of transmit antennas and num-

ber of bits per modulation symbol, and becomes prohibitive even for simple

schemes. We propose to reduce complexity by partitioning the signal into

disjoint groups at the receiver and then detecting each group using a MAP

detector. Complexity and performance can be traded off by the selection

of the group size. Group separation and partitioning is performed such as

to maximize the Mutual Information between the transmitted and received

signal. For schemes employing independent IQ gray mapping Conventional

MMSE per antenna detection is identified as a special case of group detec-

tion corresponding to a group size of one. It is shown that for moderate

to high SNR Using a group size if two with optimized group partitioning

shows a gain of 1-2[dB] under a fast Rayleigh fading channel, and by 3-4[dB]

under a Quasi static Rayleigh fading channel, with some increase in decod-

ing complexity. It is also shown that higher gains can be achieved using a

larger group size with a further increase in complexity. We further propose

an ad hoc Iterative Group Cancelation scheme using hard decision feedback

i


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to enhance performance.

ii


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1.2 TerminologyAWGN Additive White Gaussian NoiseBICM Bit Interleaved Coded ModulationBE R Bit Error RateCSI Channel State InformationGD Group DetectionGDFE Generalized Decision Feedback EqualizerIC Interference CancelingIS Interference Suppression

MAP Maximum Ap-posterioriMI Mutual InformationMIMO Multiple Input Multiple OutputMMSE Minimum Mean Square ErrorMSE Mean Square ErrorN0 White noise spectral densityNT X Number of antennas at the transmitterNRX Number of antennas as the receiverN g Number of groupsN p Number of partitioning possibilities

SI C Successive Interference CancelationSISO Single Input Single OutputSN R Signal to Noise RatioSV D Singular Value DecompositionT x TransmitterV BLAST Vertical Bell Labs Space-Time AlgorithmRx ReceiverW MF Whitening Matched FilterZF Zero Forcing

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iv

Acknowledgments

I wish to express my appreciation and gratitude to Dr. Dan Raphaeli for his

professional guidance.

I wish to express my appreciation and gratitude to RAMOT for providing

finical support throughout the research

I would like to express my appreciation and gratitude to my parents and to

my girlfreind for standing by me along the entire research.

I would also like to express my gratitude to the numerous cafes in the Tel

Aviv area for their warm hospitality during the writing of these lines . . .


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Contents

1 Introduction and Backgroud 11.1 MIMO-BICM Prior Art . . . . . . . . . . . . . . . . . . . . . 3

1.2 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Outline of Thesis Report . . . . . . . . . . . . . . . . . . . . 7

2 System model and notations 9

2.1 Connection to MIMO OFDM . . . . . . . . . . . . . . . . . . 11

2.2 Review of MIMO-BICM MAP detection . . . . . . . . . . . . 12

3 Group Detection 16

3.1 Group Separation Matrix . . . . . . . . . . . . . . . . . . . . . 18

3.2 Group LLR Computation . . . . . . . . . . . . . . . . . . . . 22

3.3 Simplified LLR Computation for group size 2 . . . . . . . . . . 25

3.3.1 Computing P . . . . . . . . . . . . . . . . . . . . . . . 28

3.3.2 Connection to MSE . . . . . . . . . . . . . . . . . . . . 29

3.3.3 Simple Antenna Partitioning . . . . . . . . . . . . . . . 31

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CONTENTS vi

4 Group Partitioning 33

4.1 Simplified Partitioning for 2 2 scheme . . . . . . . . . . . . . 364.2 Ad hoc Simplified Partitioning . . . . . . . . . . . . . . . . . 39

4.2.1 Simplified Partitioning group size 2 . . . . . . . . . . . 40

4.2.2 Simplified Partitioning group size 4 . . . . . . . . . . . 41

4.3 Mutual Information Rayleigh fading channel . . . . . . . . . 43

5 Iterative Group Interference Cancelation 48

5.1 Group Partitioning For Iterative Group Detection . . . . . . . 50

6 Simulation Results 52

6.1 Fast Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.1.1 Fast Fading 2x2 . . . . . . . . . . . . . . . . . . . . . . 54

6.1.2 Fast Fading 4x4 . . . . . . . . . . . . . . . . . . . . . . 55

6.2 Quasi Static Fading . . . . . . . . . . . . . . . . . . . . . . . . 57

6.2.1 Quasi Static Fading 2x2 . . . . . . . . . . . . . . . . . 57

6.2.2 Quasi Static Fading 4x4 . . . . . . . . . . . . . . . . . 57

6.3 Simulation Summary . . . . . . . . . . . . . . . . . . . . . . . 59

7 Conclusions 64

8 APPENDIX 65

8.1 Simplification ofQ(aG) . . . . . . . . . . . . . . . . . . . . . . 65

8.2 Computing Ree for the 2 2 scheme . . . . . . . . . . . . . . 67


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CONTENTS vii

8.3 Whitening Eigenfilter maximizes the Mutual Information . . . 71

8.3.1 Optimization Problem Simplification . . . . . . . . . . 71

8.3.2 Optimization Problem Solution . . . . . . . . . . . . . 74

Bibliography 78


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List of Figures

2.1 MIMO-BICM NRX x NT X System Model. . . . . . . . . . . . . . 10

3.1 Group Detection Scheme. . . . . . . . . . . . . . . . . . . . . . 17

3.2 Simplified LLR computation for Group size 2. . . . . . . . . . . . 28

4.1 2 2 Capacity Loss Per Antenna Vs Group Detection. . . . . . . 45

4.2 4 4 Capacity Loss Per Antenna Vs Group Detection. . . . . . . 46

4.3 4

4 Capacity Loss GD Optimal Search Vs Simple Search. . . . . 47

5.1 Iterative Group Detection Scheme. . . . . . . . . . . . . . . . . . 50

6.1 Simulator Block Diagram (Initial Stage). . . . . . . . . . . . . . . 53

6.2 2 2 16QAM,64QAM Fast Fading Rayleigh. . . . . . . . . . . . . 56

6.3 4 4 16QAM Fast Fading Rayleigh. . . . . . . . . . . . . . . . . 58

6.4 4

4 16QAM Fast Fading Rayleigh Optimal Search Vs Simple Search. 59

6.5 2 2 16QAM Quasi Static Fading Rayleigh. . . . . . . . . . . . . 60

6.6 4 4 16QAM 4 Groups Simple Search Quasi Static Fading Rayleigh. 62

6.7 4 4 16QAM 2 Groups Simple Search Quasi Static Fading Rayleigh. 63

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List of Tables

4.1 Number of Group Partitioning Possibilities . . . . . . . . . . . . 33

4.2 Group Partitioning Possibilities for 2 2 scenario . . . . . . . . . 37

4.3 Error Covariance Matrices for Group Partitioning of 2 2 scenario 37

6.1 Fast Rayleigh fading MAP,GD,PA BER Comparison, 2 2 . . . . 55

6.2 Fast Rayleigh fading MAP,GD,PA BER Comparison, 4 4 . . . . 57

6.3 Quasi static Rayleigh fading GD,PA PER Comparison, 2 2 . . . 61

6.4 Quasi static Rayleigh fading GD,PA PER Comparison, 4 4 . . . 61

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Chapter 1

Introduction and Backgroud

The increasing demand for spectrally efficient transmission techniques,the

constant demand for higher bandwidth and the pioneering work of Foschini

[7] and Telatar [8] led to a tremendous interest in the transmission through

Multiple Input Multiple Output (MIMO) channels. Information theory pre-

dictions suggested that under a rich scattering channel the channel capacity

would scale linearly with the minimum of the number of transmit and receive

antennas. In MIMO transmission the increased number of spatial dimensions

can be used to improve the probability of error and to increase the trans-

mission rate. Loosely speaking, schemes that exploit the spatial dimensions

to improve the probability of error are said to maximize the diversity gain

while schemes that use the spatial dimensions to increase the transmission

rate are said to maximize the multiplexing gain. Space Time Codes (STC)

and especially Space Time Block Codes (STBC) [9] received a lot of attention

due to their ability to exploit the Diversity Gain in a simple way, however

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CHAPTER 1. INTRODUCTION AND BACKGROUD 2

the coding gain provided by STBC was very limited an non full rate STBC

codes introduce bandwidth expansion. In general STBC could not support

the high spectral efficiencies predicted for MIMO. Spatial Multiplexing (SM)

techniques that transmit different bit streams on each one of transmitting an-

tenna were shown to achieve high spectral efficiencies and thus to exploit the

multiplexing gain. The first SM detection algorithm for MIMO signals was

the Vertical Bell Labs Layered Space-Time (VBLAST) algorithm [7]. The

algorithm worked by detecting the strongest data stream, subtracting that

stream from the received signal and repeating the process for the remaining

data streams. The first V-BLAST system [7] demonstrated unprecedent spec-

tral efficiencies ranging from 20-40bits/s/Hz. It was soon realized that the

V-BLAST algorithm suffered performance degradation through the cancela-

tion process via error propagation. Research for alternate MIMO detectionalgorithms was ignited giving rise to various detection techniques amongst

them were Bit Interleaved Coded Modulation (BICM) based techniques.

BICM is a well known transmission technique widely used in practical

single-input single-output (SISO) systems. In SISO the BICM approach re-

ceived a lot of attention due to its ability to exploit diversity under fading

channels in a simple way [1, 2]. The BICM approach is counter intuitive

at first glance since it suggests that in some channels (specifically Rayleigh

fading) more reliable channel codes can be designed by separating coding


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and modulation. Traditional schemes that combine coding and modulation

typically attempt to maximize the minimum Euclidean distance of the code,

however under fading channels code performance depends strongly on the

minimum Hamming distance of the code. The BICM approach attempts

to maximize the code diversity, this is achieved by bit wise interleaving at

the encoder and by using an appropriate soft-decision bit metric as an input

to the decoder. Optimal soft bit metrics are obtained by Maximum Ap-

Posteriori (MAP) decoding, MAP decoding of BICM transmission involves

the computation of the Log Likelihood Ratio (LLR) for each transmitted

bit. Inspired by such an approach BICM was proposed as a transmission

technique for multi carrier MIMO systems [10, 3]. The LLR computation for

each transmitted bit is performed using a detector. For MIMO channels the

detector complexity is exponential in the product of the number of trans-mit antennas and number of bits per modulation symbol. Even for simple

scenarios this complexity becomes overwhelming.

1.1 MIMO-BICM Prior Art

Various techniques have been suggested to reduce the computational burden

of the MAP detector. Most of them can be classified into either list sphere

detector based techniques, or Interference Suppression and Cancelation based

techniques. The list sphere detector was proposed by [18, 11] and is a mod-


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ification of the sphere decoder. The sphere decoder is an efficient algorithm

for finding the nearest lattice point to a noisy lattice observation. The vector

space spanned by the MIMO channel matrix is regarded as a lattice and the

received signal as its perturbed version. The algorithm searches for the near-

est lattice point in a sphere around the perturbed received lattice point. The

list sphere detector is a modification of the sphere decoder and enables the

production of approximate LLR values for coded bits. The complexity and

performance of the list sphere detector greatly depend on the selection of the

sphere radius and the list size. These depend on the Channel matrix and the

SNR. An iterative scheme using the list sphere detector and a turbo code was

reported to closely approach capacity of the multi antenna fast Rayleigh fad-

ing channel [18]. The complexity of the list sphere detector is generally much

higher then that of decoding techniques employing Interference Suppressionand Cancelation.

The second class of decoding techniques employ Interference Suppression

and Cancelation. The MAP detector is approximated by linear processing of

the MIMO channel outputs followed a per antenna LLR computer. In [13]

the authors proposed a Zero Forcing (ZF) detector followed by a per antenna

LLR computer. The authors made the simplifying assumption of white post

equalization interference. In [14] a Minimum Mean Squared Error (MMSE)

based detector was derived followed by a per antenna LLR computer without


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the assumption of white post equalization interference. It was shown that

such a receiver has the same complexity as the one in [13] but offers superior

performance. The authors in [19] proposed an iterative scheme comprising of

a soft Interference Canceler (IC), an adaptive MMSE detector followed by a

per antenna LLR computer and a soft output decoder. Soft outputs from the

decoder were used to both reconstruct estimates of the channel output and

to adapt the MMSE detector. The reconstructed channel output estimates

were subtracted from the true received signal (IC) and the resulting signal

was detected via the adaptive MMSE detector and the LLR computer. It

was shown that when bit reliability at the output of the soft decoder was

high the adaptive MMSE detector coincided with the Matched Filter. A

reduced complexity approximation to [19] was proposed in [12] where the

soft output decoder was replaced by a hard output Viterbi decoder and thesoft Interference Canceler by a hard one. After the first decoding stage

,due to the lack of soft information from the decoder, correct decisions were

assumed. After the initial decoding stage the MMSE detector was replaced

by a Matched Filter.

1.2 Thesis Contributions

In this thesis we propose a Group Detection (GD) Interference Suppression

based technique. GD was widely studied in the context of Multi User De-


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tection (MUD) in CDMA systems [4]. The idea is to jointly detect a subset

of the transmitted information while treating the rest of the transmission as

noise. Many existing detection techniques can be regarded as Group Detec-

tion based techniques, namely the per antenna detection techniques where

each antenna can be identified as a single group. The authors in [5] used

Group Detection in the context of V-BLAST decoding ( [7]) as a remade for

error propagation. In their work a group of the worst p sub-channels was

jointly detected using ML decoding. A DFE was then used to detect the rest

of the sub-channels. In [6] a GD scheme was proposed as a trade off between

diversity gain and spatial multiplexing gain by partitioning the signal at the

transmitter into groups. Each group was encoded separately and per group

rate adaptation was performed.

In our work group detection was employed only at the receiver side withno special treatment at the transmitter. Unlike [5, 6, 14, 12, 19] where a group

was defined as a collection of antennas/sub-channels, we define a group as a

collection of In Phase and Quadrature (I/Q) components of the transmitted

symbols possibly from different antennas. The smallest group is defined

as a single (I/Q) component of the a transmitted symbol. The GD scheme

consists of group partitioning, group separation and detection. The proposed

GD scheme was derived from an information theoretic point of view. Group

separation was performed by linear detection. Under a Gaussian assumption


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on the transmitted signal, the MMSE detector was identified as a canonical

(information lossless) detector for group detection. A group partitioning

scheme was derived such as to maximize the sum rate. The selection of

the group size allowed us to tradeoff performance with complexity. At one

end when the number of groups was set to one, the entire transmission was

jointly detected and the scheme coincided with full MAP, while at the other

end each dimension was decoded separately. An Iterative group interference

canceling technique using hard outputs from the decoder similar to [12] was

also investigated. Finally, performance was evaluated via simulations using

a rate 1/2 64-state convolutional code with octal generators (133,171) and

random interleaving. The proposed GD scheme was compared to the full

MAP detection scheme and the standard MMSE scheme [14, 12] for both

fast Rayleigh fading and quasi static Rayleigh fading channels. Under suchchannels GD showed gains of up to 4[dB] with respect to conventional per

antenna detection with a some increase in complexity. gains of up to 10[dB]

was obtained with further increase in decoding complexity.

1.3 Outline of Thesis Report

The organization of this thesis report is as follows. In Chapter 2 the system

model is presented along with a review of MIMO-BICM MAP detection.

Chapter 3 introduces the concept of Group Detection and deals with group


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separation and detection. Group partitioning is addressed in Chapter 4. It-

erative Group Interference Cancelation is discussed in Chapter 5. Simulation

results for fast and quasi static Rayleigh fading are presented in Chapter 6,

and Chapter 7 concludes this thesis report.


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Chapter 2

System model and notations

We consider a MIMO-BICM system with NT X transmit and NRX receive an-

tennas as illustrated in Fig. 2.1. The information bit sequence b = [b1,...,bNb]

is encoded into coded bits which are then interleaved by a random interleaver.

The interleaved bits, denoted by c = [c1,...,cNc], are mapped onto an 2m

QAM signal constellation using independent I&Q gray mapping. The block

of Nc/m symbols is split into sub-blocks of length NT X . At each instant n a

sub-block ac(n) = [ac1(n),...,acNTX

(n)]T is transmitted simultaneously by the

NT X antennas. We assume that the transmitted symbols are independent

with a covariance matrix Raa = 2a INTXNTX . The NRX 1 received signal

is denoted by yc(n) = [yc1(n),...,ycNRX

(n)]T and is given by

yc(n) = Hc(n) ac(n) + zc(n) (2.1)

Where Hc(n) is the NRXNT X complex channel matrix [hci,j(n)]i=1..NRX ,j=1..NTXand is assumed to be perfectly known at the receiver (full CSI at the re-

9


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CHAPTER 2. SYSTEM MODEL AND NOTATIONS 10

bi

ciRandom

InterleaverBinarySource

EncoderSymbol

Mapper (Gray)

NTX

a

1a

S/P

TX

H

Detector(Bit LLR)

1y

NRX

y

DeInterleaver Decoder

bi

RX

Figure 2.1: MIMO-BICM NRX x NT X System Model.

ceiver). z(n) is an NRX1 additive white complex Gaussian noise vectorzc(n) = [zc1(n),...,z

cNRX

(n)]T with a covariance matrix ofRzz = 2z INRXNRX .

For simplicity we consider the case where NT X =NRX NT. The extensionto an arbitrary number of transmit and receive antennas satisfying NT X

NRX is strait forward.


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2.1 Connection to MIMO OFDM

Orthogonal Frequency Division Multiplexing (OFDM) is a well known trans-

mission technique widely used for transmission over frequency selective chan-

nels. In general OFDM transforms the Inter Symbol Interference (ISI) chan-

nel into a set of orthogonal sub-channels or sub-carriers and thus greatly

simplifying equalization. This property made OFDM transmission a popular

selection in many modern communication systems.

The system model described in the previous section can be used to de-

scribe a MIMO-OFDM system by interpreting the instant index n as a fre-

quency (subcarrier) index. Each block of Nc/m symbols corresponds to a

single OFDM symbol with NC/(mNT X ) sub carriers. In such a model Hc(n)

corresponds to MIMO channel experienced by the nth subcarrier. For per-

formance evaluation in Chapter 6 we consider two channel models, a fast

Rayleigh fading channel and a quasi static flat Rayleigh fading channel.

These two channel models represent two extreme ends. The fast fading chan-

nel assumes no correlation between sub carriers while the flat fading assumes

full correlation between sub carriers. The fast Rayleigh fading channel can

be used to approximate the channel experienced by a well interleaved OFDM

system operating in strong multi-path environment [3]. Adjacent subcarrier

channel coefficients are in general not independent, however with frequency

interleaving within one OFDM symbol the resulting channel can be approx-


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imated by an independent fast fading channel. Strong multi-path environ-

ments are common in Non Line Of Sight (NLOS) channels. Under a quasi

static flat Rayleigh fading channel all sub-carriers experience the same chan-

nel for the duration of a single OFDM symbol, and the channel changes from

symbol to symbol in an independent fashion. The quasi static flat Rayleigh

fading channel models a channel that is flat in the frequency domain, such

channels are common in narrow band systems. For the reminder of this thesis

we omit the instant index n for clarity of notation.

2.2 Review of MIMO-BICM MAP detection

The decoding scheme is shown in Fig. 2.1,the MAP detector performs soft

de-mapping by computing the conditional Log Likelihood Ratio(LLR) for

each coded bit. The conditional LLR of the kth coded bit is the logarithm

of the ratio of the likelihood that the bit was a one, conditioned by the

received signal and channel state, to the likelihood that the bit was a zero,

conditioned by the received signal and channel state (full CSI at the receiver).

The conditional LLR for the kth coded bit is given by

LLR( ck| yc, Hc) = logPrck = 1| yc, HcPr

ck = 0| yc, Hc (2.2)

For clarity and ease of notation from here on we omit conditioning on the

channel matrix Hc from our notation. Using Bayes rule


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Pr

ck = 1| yc

=Pr

yc ck = 1 Pr {ck = 1}

Pr

yc (2.3)

Since the probability Pr

yc

is not a function of the ck will cancel out

Eq. 2.2. Once more by the use of Bayes rule, the conditional bit probability

is given by

Pr

ck = 1| yc

=

ck11 ,cNck+1

Pr

yc ck = 1, ck11 , cNck+1 Prck11 , cNck+1 ck = 1| Pr {ck = 1}(2.4)

Each block of bits at the output of the channel encoder are in general

statistically dependent. We make the assumption of an ideal interleaver that

scrambles the coded bits in such a way that each block of interleaved bits

c contains bits from different coded blocks rendering the bits in c to be

statistically independent. Thus follows that

Pr

ck = 1| yc

= Pr {ck = 1}

ck11 ,cNck+1

Pr

yc ck = 1, ck11 , cNck+1

i=k

Pr {ci}(2.5)

the LLR the kth coded bit is given by

LLR

ck| yc

= LA (ck) + log

ck11 ,cNck+1

Pryc ck = 1, ck11 , cNck+1 i=k

Pr {ci}

ck11 ,cNck+1

Pr

yc ck = 0, ck11 , cNck+1

i=k

Pr {ci}

(2.6)Define the ap-posteriori LLR of the kth bit as


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LA (ck) = logPr {ck = 1}Pr {ck = 0} (2.7)

and noting that

Pr {ck = i}i{0,1}

=eiLA(i)

1 + eLA(i)(2.8)

Substituting Eq. 2.8 into Eq. 2.6 gives

LLR

ck| yc

= LA (ck) + log

ck11 ,c

Nck+1

Pr

yc ck = 1, ck11 , cNck+1 e

Pi=k

ciLA(ci)

ck11 ,c

Nck+1

Pr

yc ck = 0, ck11 , cNck+1 e

Pi=k

ciLA(ci)

LE( ck|yc)

(2.9)

Define Sk,rl CNT as the set of all complex QAM symbol vectors whosekth bit in the rth symbol is l {0, 1}. Under the assumption of AWGN theLLR the kth coded bit is given by:

the conditional LLR of the kth coded bit is given by

LLR

ck| yc

= LA (ck) + log

acSk,r1 e 12z

ycHcac2+Pi=k

ciLA(ci)

acSk,r0

e 12z

ycHcac2+Pi=k

ciLA(ci)

LE( ck|yc)

(2.10)


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The first term in Eq. 2.10 is the ap-priori likelihood of the bit of interest

while the second term is the extrinsic likelihood of the bit of interest. When

using iterative schemes extrinsic and ap-priori likelihoods are exchanged be-

tween the detector and decoder. For non iterative schemes like the ones

considered in this work, the ap-priori likelihoods of all bit are assumed to be

zero. The resulting likelihood of the kth bit is given by

LLR

ck| yc

= log

acSk,r1

e

12z

ycHcac2

acSk,r0

e

12z

ycHcac2

(2.11)

The complexity of the LLR computation for each bit is 2 mNT1 and is

thus exponential in the product of the M-QAM constellation size and the

number of antennas.


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Chapter 3

Group Detection

Before presenting the group detection scheme let us reformulate the system

model using real signals only. Eq. (2.1) is transformed into

yc

Ryc

I

=

HcR HcIHcI H

cR

acRacI

+

zcRzcI

(3.1)

The subscripts R or I imply taking the Real or Imaginary part of the

Vector or Matrix it is associated with. For clarity of notation for the rest

of this paper all real vectors and matrices derived from the complex ones

described in Sec. 2 will inherit the names of their complex versions without

the superscript c. For example from here on the vector

a = real {ac1

}, ..., real acNT , imag {a

c1

}, ..., imagacNTT

Denote the number of real dimensions as N = 2 NT The transmitted signala is partitioned into Ng disjoint groups of equal size M, where M = N/Ng.

The extension to non equal size groups is trivial. Let = {1, 2, . . . N } be

16


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CHAPTER 3. GROUP DETECTION 17

y

1Ga

LLR

LLR

LLR

P/S Deinterleaver Decoder

b

Group Separation

Group Detection

2Ga

NgG

a

M x 1

M x 1

M x 1

N x 1

N x 1

N x 1

1GW

2GW

NgGW

Figure 3.1: Group Detection Scheme.

the set of indexes of entries in the transmitted vector a. Define the group

partitioning of into disjoint groups Gi such that:Gi ,|Gi| = M andNg

i=1Gi = . For any a N, the group aGi |Gi| is a subgroup ofa that is

made up of ak, k Gi. The channel experienced by the group Gi, namelyHGi , is a sub matrix of H and is made up of the columns of H namely hk

such that k Gi. At the receiver the groups are separated by an N Nseparation matrix denoted as W. The sub-matrix WGi of size M N ,thatfilters out the ith group out of the received vector y, is made up of the rows

of the matrix W, namely wTk such that k Gi. The separate detectionof real and imaginary parts of the transmitted symbols is possible due to

the independent I&Q mapping rule. The separation scheme is depicted in

Fig. 3.1


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3.1 Group Separation Matrix

Given a group partitioning scheme, we propose to optimize the separation

matrix W such as to maximize the sum rate

Ngi=i

I

WGiy; aGi

(3.2)

Denote the output of the separation matrix corresponding to the groupGi by

aGi WGiy (3.3)

Eq. (3.2) is maximized by choosing the group separation matrix WGi such

as to maximize the Mutual Information between each transmitted group and

the output of the separation matrix

MN

WoptGi = arg maxWGi{I(aGi; aGi)} (3.4)

The optimization problem in Eq. (3.4) was solved in two ways each giving

rise to a different solution. The first solution was derived from a matrix

algebraic point of view and turned out to be the well known whitening eigen-

filter, the derivation is believed to be novel and is included in Appendix. (8.3).

The second solution was derived form an information theoretic/estimation


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theoretic point of view similar to [16, 25] and turned out to be the well known

MMSE estimation filter. The latter derivation is simple and the solution is

attractive from a computational point of view and will thus be used in our

group detection scheme. Following is the information theoretic/estimation

theoretic proof. From the data processing inequality [15] follows that

Iy; aGi IWGiy; aGi = IaGi; aGi (3.5)Thus if exists a separation matrix WGi that achieves the equality in

Eq. (3.5) then it clearly maximizes the mutual information in Eq. (3.4) and

in Eq. (3.2). We next prove that, under the Gaussian assumption on the

transmitted signal and channel noise, the sub-matrix of the MMSE estima-

tion matirx ofa from y, corresponding to the group Gi achieves the equality

in Eq. (3.5). The MMSE estimation matrix is denoted by Wmmse and given

by

Wmmse = RayR1yy = H

T

HHT +

2z2a

INN

1(3.6)

The sub-matrix for group Gi is given by

WmmseGi = HTGi

HHT +

2z2a

INN

1(3.7)

The estimation error covariance matrix is given by


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32/91


h ( aG| ammseG ) = h (eG) (3.11)

Consider the mutual information between the sub-vector of ammse cor-

responding to the estimation of group G namely ammseG and the transmitted

group aG

IWmmseG y; aG = I(ammseG ; aG) =h (aG) h (aG| ammseG ) =h (aG) h (eG)

(3.12)

The first equality in Eq. (3.12) follows from the definition of ammseG , the

second equality follows from the definition of the mutual information and the

last equality follows from Eq. (3.11). The mutual information between the

transmitted group G and the received signal y is given by

I

aG; y

= h (aG) h (aG| y) =h (aG) h (eG)

(3.13)

The first equality in Eq. (3.13) follows from the definition of the mu-

tual information and the second equality follows since the covariance of the

MMSE estimator of ammseG from y equals the covariance of the estimation

error namely R aG|y aG|y = ReGeG and that aG| y is Gaussian. Thus fromEq. (3.12) and Eq. (3.13) follows that

I

aG; y

= I(aG; ammseG ) (3.14)


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QED

Changing G with G in the above proves the same for the group G. Thus

we have proved that the selection of the MMSE estimation matrix in Eq. (3.6)

as the group separation matrix maximizes the mutual information between

each one of the groups at the transmitter and the appropriate output of

the separation matrix, and thus maximize the sum rate in Eq. (3.2). For the

remainder of this paper, for clarity of notation, we omit the mmse superscript

form ammse and use a. We also refer to Eq. (3.6) as the MMSE separation

matrix.

3.2 Group LLR Computation

The output of the MMSE separation matrix for group G is given by

aG = WmmseG HGaG + W

mmseG (HGaG + z)

vG

(3.15)

Where HG is the N |G| sub matrix ofH corresponding to group G andHG is the N

G sub matrix ofH corresponding to group G,and GG = .Denote by vG the noise experienced by group G, the covariance matrix of vG

is given by

RvGvG = WmmseG

2a2

HGHTG

+ 2z

2INN

(WmmseG )

T (3.16)


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The conditional LLR for the kth coded bit, where k belongs to the set of

coded bits mapped into one of the symbols belonging to group G is given by

LLR(ck| aG) = log

Pr {ck = 1| aG}Pr {ck = 0| aG}

(3.17)

To compute Eq. (3.17) we need the conditional pdf P r ( aG| aG). FromEq. (3.15) and under the Gaussian assumption on the inter group interfer-

ence [24] follows

aG| aG N

WmmseG HGaG, RvGvG

(3.18)

The fact the the noise term in Eq. (3.15) is colored complicates the eval-

uation of Eq. (3.17). We propose to whiten the noise in Eq. (3.15). The

noise covariance matrix is symmetric positive semi-definite and thus has the

following eigen value decomposition

RvGvG = UGGUTG (3.19)

Where UG is a |G| |G| unitary matrix and G is a |G| |G| diagonal

matrix of the eigen values of RvGvG .

UG (UG)T = I|G||G|

G = diag

G1 , . . . G|G|

(3.20)


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The noise whitening matrix is given by

FG = 12G U

TG (3.21)

The output of the group whitening separation matrix for group G is given

by

aG = FGWmmseG HGaG + vG (3.22)

where

RvGvG = I|G||G|

The conditional LLR is then derived by using Bayes law and the ideal

interleaving assumption. The conditional LLR is given by

LLR(ck| aG) = log

aGS

k,rG,1

e12aGFGWmmseG HGaG2

aGS

k,rG,0

e12aGFGWmmseG HGaG2

(3.23)

Sk,rG,l R|G| is the set of all real |G| dimensional PAM symbol vectors

whose kth bit in the rth symbol is l {0, 1}. The complexity of the LLRcomputation for all groups is Ng2

m2|G| and is exponential in the product of

the group size and the number of bits per real dimension. We are then able

to trade off performance with complexity by the selection of group size.


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3.3 Simplified LLR Computation for group

size 2

For a group size of Ng = 2 it is possible to derive a simplified closed form

approximation for the LLR without computing the noise whitening matrix

in Eq. (3.21). The derivation is in the spirit of [14] and can be done for an

arbitrary group size. The LLR will be derived given a zero forcing separation

matrix. The MMSE structure will then emerge from the derivation. As in [14]

by using the log max approximation [2] the conditional LLR can be expressed

as

LLR

ck| y max

dSk,rG,1

logPr

azf

aG = d maxdSk,rG,0

log P r

azf

aG = d(3.24)Where

azf = H#y =

HTH

1HTy = a + w (3.25)

H# is the ZF separation matrix given by the Moore Penrose pseudo in-

verse of H. The noise covariance matrix is given by

Rww =12

2z

HTH1

(3.26)

Let the group G = {i, j}. Under the Gaussian assumption on the postdetection interference [24] follows that


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P r

azf aG = 1

(2)M|G|exp{1

2Q (aG)} (3.27)

where Q (aG) is given by

Q (aG) =

azf GT

1G

azf G

. (3.28)

To find the mean and covariance in Eq. (3.27) we note that

azf = aiei + ajej +

k /{i,j}

akek + w (3.29)

where ei is the N 1 ith unit vector. The mean and variance are givenby

G

= E

azf aG = aiei + ajej

G = E

azf G

azf G

T aG

= 12

2a

INN VGVTG

+ Rww

(3.30)

where VG =

eiej

. Substituting Eq. (3.30) into Eq. (3.27) gives

Q (aG) = aTzf

1G azf

2aTzf

1G aiei + ajej + a2i eTi 1G ei + a2j eTj 1G ej(3.31)

Substituting Eq. (3.31) and Eq. (3.29) into Eq. (3.24) and noting that

the first term in Eq. (3.31) is not a function of aG and thus cancels out we

arrive at


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LLR

ck| y max

dSk,rG,0

Q (d)

max

dSk,rG,1

Q (d)

(3.32)

Where

Q (aG) = 2aTzf1G

aiei + aj ej

+ a2i eTi

1G ei + a

2j e

Tj

1G ej (3.33)

In Appendix. (8.1) we show that

Q (aG) = ii

ammsei

pii ai

2+ jj

ammsej

pjj aj

2+ ij (a

mmsei pij aj)2

+ij

ammsej pijai2

+ C(3.34)

where

ii =12

2a(1pjj)pii

(1pii)(1pjj)p2ij, ij =

12

2a1

(1pii)(1pjj)p2ij, jj =

12

2a(1pii)pjj

(1pii)(1pjj)p2ij(3.35)

Note that C is not a function of aG and will cancel out in Eq. (3.32).pi,j

is the i,jth element of the matrix P, where P is given by

P = INN + 2z2a HTH11

(3.36)

The simplified computation of the LLR is illustrated in Fig. 3.2


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y

mmseia

m m s eja

x

x

x

1

iip

1

jjp

ijp

+

+

+

+

-

-

-

-

,

,{1/ 0}

k r

i Gd S

+

+

+

+

( )2

( )2

( )2

( )2

x

x

x

x

+

c=max(a,b)

i

j

ijmmse

GW

H

x

mmse

GW

P

mmseW

2

2

z

a

iip

ijp

jjp

SeparationMatrix

Computer

ii

ij

jj

Group Processing

,

,{1/ 0}

k r

j Gd S

Q(d)|ck=1

++ -

LLR(ck)

jd

id x

Compute

Sub-BlockProcessing

Q(d)|ck=0

a

bc

ck=0 ck=0

ck=1 c

k=1

Figure 3.2: Simplified LLR computation for Group size 2.

3.3.1 Computing P

The computation of P can be greatly simplified by exploiting its connection

to MMSEestimation. By using the matrix inversion lemma follows that


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P =

INN +2z2a

HTH

11= I

2a2z

HTH+ INN

1(3.37)

Again from the matrix inversion lemma follows that

2a2z

HTH+ INN

1

= INN HT

HHT +

2z

2aINN

1

H (3.38)

Substituting Eq. (3.38) into Eq. (3.37) gives

INN +

2z2a

HTH

11= HT

HHT +

2z

2aINN

1H (3.39)

substituting Eq. (3.6) into Eq. (3.39)

P = WmmseH (3.40)

Thus we identify P as the effective channel at the output of the MMSE

separation matrix and can thus be computed by multiplying the already

computed Wmmse by the channel matrix H.

3.3.2 Connection to MSE

To gain some insight into Eq. (3.34) we show yet another connection between

P and MMSEestimation theory. From Eq. (3.8) and Eq. (3.40) follows that


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Ree =2a2

(INN P) (3.41)

Ree is the error covariance matrix resulting from MMSE estimation, its

diagonal elements [Ree]i,i are the Mean Square Errors (MSEMMSE) in the

estimation of each element of a.

MSEMMSE,i = [Ree]i,i = E|ai ammsei |2 (3.42)

The unbiased SNR (See [17]) of the ith element ai is given by

SN RMMSE U,i = SN RMMSE,i 1 =2a2

MSEMMSE,i 1 = pii

1pii(3.43)

The bias compensation scaling factor is given by

2a2

2a2

M SEMMSE,i=

1

pii(3.44)

Thusammsei

pii,

ammsejpjj

in Eq. (3.34) are the unbiased MMSEestimates of ai

and aj respectively. We next prove that the best unbiased linear estimate of

ammse

i from aj is pijaj. From Eq. (3.41) follows that

pij = 22a E

(ai ammsei )

aj ammsej

= 22a

E

ammsej (ai ammsei )

22a

E{aiaj} + 22aE{ammsei aj} (3.45)


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The first term in the second equality is zero from the orthogonality prin-

ciple and the second term in the second equality is zero since transmitted

symbols are statistically independent. Thus

pij =2

2aE{ammsei aj} = 22aE

ammsej ai

(3.46)

From linear estimation theory follows that

ammsei (aj) =E{ammsei aj}

E{a2j} aj =2

2aE{ammsei aj} aj = pijaj (3.47)

Since both aj and ammsei are zero mean follows that the best unbiased

linear estimate of ammsei from aj is pijaj . The same proof can be repeated for

ammsej .

Thus we have shown that 1/pii is the MMSE bias compensation factor

from [17] and that the best unbiased linear estimate of ammsei from aj is pijaj.

Thus in Eq. (3.34) we identify four euclidian distance terms.

3.3.3 Simple Antenna Partitioning

Let us consider the complex MMSE estimate of the complex symbol trans-

mitted form the kth antenna, namely ack. It is well known from estimation

theory that the MMSE estimate is biased and that the multiplicative bias

factor is real valued [17]. Thus the real/imaginary part of the MMSE es-

timate, form the kth antenna, namely real{ack}/imag{ack} is not a function


43/91


of the imaginary/real part of the symbol transmitted from the kth antenna.

Thus when the indexes i&j are taken as the Real and Imaginary parts of the

same transmit antenna follows that

E{(ammsei aj )} = 0, E

ammsej ai

= 0 (3.48)

.

From Eq. (3.45)and Eq. (3.48) follows that pij = 0,Eammsei ammsej = 0and that pii = pjj . The detection of the Real and Imaginary parts of a

single antenna can thus be performed separately. Using the above Eq. (3.34)

coincides with the known LLR expression from [14] .

LLR

ck| y SN RMMSE U,i

max

dSk,r{i,j},0

2ij (d) maxdS

k,r{i,j},1

2ij (d)

ij(d) = ammseipii d1 +j ammsejpii d2

(3.49)

Since for per antenna partitioning the Real and Imaginary parts of each

antenna can be detected separately Eq. (3.49) can be further simplified. The

LLR for coded bits ck that are mapped onto ai is reduced to

LLR ck| y SN RMMSE U,i maxd1Sk,ri,0 ammsei

pii

d1

2

max

d1Sk,ri,1 ammsei

pii

d1

2

(3.50)Thus per antenna partitioning is actually equivalent to a per dimension

partitioning and is a group detection scheme for group size one. The LLR

computation complexity is then reduced from N2

2m to N2m2


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Chapter 4

Group Partitioning

The number of ways to partition the transmitted signal into groups is a

function of the transmitted signal size N and the groups size M. For example

when both N and M are powers of 2 the number of partitioning possibilities

is given by Eq. (4.1)

NP =12

N

N/2

12

N/2N/4

. . . 1

2

2MM

=12

log2(N/M) N!M!

log2(N/M)Qi=1

(N2i)!(4.1)

Table 4.2 summarizes the number of partitioning possibilities for several

values of N and M.

Table 4.1: Number of Group Partitioning PossibilitiesScheme M NP

2 2 (N=4) 2 34 4 (N=8) 2 1054 4 (N=8) 4 35

8 8 (N=16) 2 6756758 8 (N=16) 4 2252258 8 (N=16) 8 6435

33


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CHAPTER 4. GROUP PARTITIONING 34

We are faced with the problem of choosing a partitioning scheme from

amongst all the partitioning possibilities. A natural selection would be to

choose the partitioning scheme that minimizes some probability of error mea-

sure, this although very intuitive is very difficult to trace analytically. Instead

we propose to select the partitioning scheme that maximizes the sum rate of

the groups.

By using the chain rule of mutual information the mutual information of

the MIMO channel can be written as Eq. (4.2)

I

y; a

= I

y; aG1

+ I

y; aG2 aG1 + + Iy; aGNg aG1, . . . , aGNg1(4.2)

When using the GD scheme information is not exchanged between groups,

and so the mutual information of Eq. (4.2) cannot in general be realized. The

mutual information (sum rate) given the GD scheme is given by Eq. (4.3)

Ngi=1

I

y; aGi Iy; a (4.3)

The sum rate is simply the sum of mutual information since the groups

are disjoint. The mutual information in Eq. (4.3) can be written as

Ngi=1

I

y; aGi

=

Ngi=1

h

aGi N g

i=1

h

aGi y (4.4)


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Using the fact that R aGi|

y aGi

|y = ReGieGi (Eq. (3.13)) and assuming that

the Inter Group Interference is Gaussian [24] follows that aGi y is also Gaus-

sian and that

N gi=1

I

y; aGi

=

Ngi=1

h

aGi 1

2

Ngi=1

logReGieGi (4.5)

Under the assumption that transmitted symbols are i.i.d the first term in

Eq. (4.5) does not depend on the partitioning scheme, the determinants of the

error covariance matricesReGieGi , i = 1 . . . N g are a function the partitioning

scheme. Given a partitioning scheme {G1, G2, . . . GNg} the error covariancematrix for the ith group ReGieGi is obtained from the covariance matrix Ree

by striking out the kth rows and the kth columns k / Gi. Thus finding thepartitioning scheme that maximizes the mutual information in Eq. (4.5) is

equivalent to finding the partitioning scheme that minimizes the product of

the the determinants of the group error covariance matrices. Thus we need

to solve the following optimization problem

{Gopt1 , Gopt2 , . . . GoptNg} = Argmin

G1, G2, . . . GNgs.t{Gi , |Gi| = M,

Gi Gj = : i, j :}

Ng

i=1ReGieGi

(4.6)

The complexity of the above search quickly becomes overwhelming (Eq. (4.1)).

To reduce the complexity of Eq. (4.6) we turn to the structure of Ree.


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Since Ree is a real covariance matrix it is obviously symmetric positive semi-

definite, however its structure goes deeper due to the symmetry in the real

channel matrix H (Eq. 3.1). This structure can be utilized to greatly sim-

plify Eq. (4.6) for the 2 2 scheme. Intuition form the simplified expressionsfor the 2 2 scheme will then lead us to develop simple suboptimal ad hocapproximations to (Eq. (4.6).

4.1 Simplified Partitioning for 2 2 schemeFor the 22 antenna scenario the MMSE error covariance matrix (Eq. (3.8))is given by:

Ree =12

2a

I4x4 HT

HHT +

2z

2aI4x4

1

H

(4.7)

In Appendix. (8.2) we prove that

Ree =2

2a

1

ad b2 c2

d bb a

0 cc 0

0 cc 0

d bb a

(4.8)

Where hi denotes the ith column of H and the scalars ,a,b,c,d are

given by

= 2z

2aa = 1 + hT1 h1 d = 1 + h

T2 h2

b = hT1 h2 c = hT1 h4

(4.9)


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For simplicity of notation we drop the scalar multiplying the matrix in

Eq. (4.8) since it will not affect the minimization of Eq. (4.6). For the 2 2scenario there are 3 ways to partition the transmitted signal into groups of

size 2,namely

Table 4.2: Group Partitioning Possibilities for 2 2 scenarioScheme G1 G2

1 {1, 2} {3, 4}2

{1, 3

} {2, 4

}3 {1, 4} {2, 3}

The group error covariance matrices corresponding to each partitioning

possibility are given in Table. (4.3).

Table 4.3: Error Covariance Matrices for Group Partitioning of 2 2 scenarioScheme ReG1eG1 ReG2eG2

1 d b

b a d b

b a

2

d 00 d

a 00 a

3

d cc a

a cc d

Denote the product of the determinants of the group error covariance

matrices of the ith partitioning scheme by Di, since all the error covariance

matrices are positive semi-definite their determinants are nonnegative.


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D1= (ad b2)2 = (ad)2 b2 (2ad 1) 0

D2= (ad)2 0

D3= (ad c2)2 = (ad)2 c2 (2ad 1) 0

(4.10)

Substituting Eq. (4.10) into Eq. (4.6) we obtain

Scheme =Arg mini

(Di) (4.11)

From Eq. (4.10) and Eq. (4.11) follows that partitioning scheme 2 will

be chosen only in cases where 2ad < 1 or when both b and c equal zero. If

both b and c equal zero then all group selections result in the same mutual

information and any one of the three partitioning schemes can be selected.

To see what happens when either b or c do not equal zero we shall substitute

a and d from Eq. (4.9) and so

2ad = 2+2

hT1 h1 + hT2 h2 +

2

hT1 h1

hT2 h2 (4.12)

The second term in the sum of Eq. (4.12) is non negative, denote it by

x2. Thus Eq. (4.12) can be written as

2ad = 2 + x2 2ad 2 > 1 (4.13)

Thus the second partitioning scheme will never be better than schemes

1 and 3. This is not surprising since in Sec 3.3.3 we made the observation


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4.2.1 Simplified Partitioning group size 2

The algorithm is a greedy one that partitions the transmitted signal into

groups of size 2 such as to maximize the correlation at the output of the

channel. The algorithm stats off with a candidate list consisting of all the

transmitted elements. At each stage, the algorithm finds the two maximally

correlated elements from the candidate list, groups them together and then

erases them from the candidate list.

Simplified partitioning algorithm for a group size of 2

1) n = 1, n = {(i, j) : i < j}2) hi,j =

hTi hj

, (i, j) n

3) [in, jn] = arg max(i,j)n (hi,j)

4) Gn = {in, jn}5) n = {(k, jn), (in, k), (jn, k), (k, in) : (k, jn), (in, k), (jn, k), (k, in) n}6) n+1 = n \ n7) if (+ + n) Ng goto 3 else end

Since matrix HTH is a byproduct from the computation of Wmmse (see

Eq (A-7)) it needs not to be recomputed in stage 2. The above algorithm is

very simple and its complexity is that of finding the maximum entry from


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a list. At each one of the Ng stages of the algorithm the list size decreases

drastically. This is a tremendous reduction compared the combinatorial com-

plexity in Eq. (3.15). In Sec (4.3) we show that under a Gaussian alphabet

and Rayleigh channel assumptions, for NT = 4 the loss of the simplified par-

titioning algorithm with respect to optimal partitioning increases with the

SNR. When transmitting 16bit/ChannelUse the loss is about 0.3[dB] and

when transmitting 30bit/ChannelUse the loss is about 0.45[dB]

4.2.2 Simplified Partitioning group size 4

The algorithm is a greedy one that partitions the transmitted signal into

groups of size 4 namely

Gn = {in, jn, kn, ln}

such as to maximize the following heuristical correlation measure

[in, jn] = arg maxi,jhTi hj

kn = arg maxkhTinhk + hTjnhk

ln = arg maxlhTinhl

+hTjnhl

+hTknhl

(4.14)

The correlation measure is built in the following fashion. First the pair ofelements with maximal correlation is found then the third element is selected

such that maximizes the sum of correlations with the already selected pair.

The fourth element is found using the same procedure thus selecting the ele-


53/91


ment that maximizes the sum of correlations with respect to the pre selected

triplet. Using the same notations as in the previous chapter the partitioning

algorithm is given by:

Simplified partitioning algorithm for a group size of 4

1) n = 1, n =

{(i, j) :

i < j

}2) hi,j =

hTi hj , (i, j) n3) [in, jn] = arg max(i,j)n (hi,j)

4) kn = argmaxk:(in,k)&(jn,k)n&k=in,jn (hin,k + hjn,k)

5) ln = argmaxl:(in,k)&(jn,k)&(kn,l)n&l=in,jn,kn (hin,l + hjn,l + hkn,l)

6) Gn = {in, jn, kn, ln}

7) n = {(t, jn), (jn, t), (t, in), (in, t), (t, kn), (kn, t), (t, ln), (ln, t) : (t, jn), (jn, t), (t, in), (in, t), (t, kn), (kn, t), (t, ln), (ln, t) n}

8) n+1 = n \ n9) if + + n Ng goto 3 else end

Since matrix HTH is a byproduct from the computation of Wmmse (see

Eq (A-7))it needs not to be recomputed in stage 2. The above algorithm is

very simple and its complexity is that of finding the maximum entry from


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a list. At each one of the Ng stages of the algorithm the list size decreases

drastically. This is a tremendous reduction compared the combinatorial com-

plexity in Eq. (3.15).

In Sec (4.3) we show that under a Gaussian alphabet and Rayleigh chan-

nel assumptions, for NT = 4 the loss of the simplified partitioning algorithm

with respect to optimal partitioning increases with the SNR. When trans-

mitting 16bit/ChannelUse the loss is about 0.25[dB] and when transmitting

30bit/ChannelUse the loss is about 0.35[dB]

4.3 Mutual Information Rayleigh fading chan-

nel

The capacity loss resulting from group detection was computed for the Rayleigh

fading channel, thus the entries of the complex matrix H were independent

Gaussian random variables (Rayleigh Amplitude, uniform phase) with a vari-

ance of 1/NT generated independently at each instant. The expectation of

the capacity is given by

C = E{I

y; a

} = E{1

2log2

2a2z

HHT + INN

} (4.15)

The expectation of the sum rate when using group detection is computed

using Eq. (4.5) and given by:


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E{Ngi=1

I

y; aGi} = NgM

2log

12

2a 1

2

Ngi=1

E{logReGieGi } (4.16)

To probe the loss in capacity incurred by using group detection we turn

to simulations. Since the capacity we are interested in is ergodic we can

approximate the expectation in Eq. (4.15) and Eq. (4.16) by the instant

average. We consider the 2

2 and 4

4 systems and group sizes of 2 and 4

with Optimal Search (OS) partitioning, Simplified Search (SS) partitioning

and simple Per Antenna (PA) partitioning.

Fig. 4.1 summarizes results for a 22 system and shows that for mediumto high SNR group detection has a gain of around 1 .5[dB] over the simple

per antenna partitioning and is only around 0.6[dB] from the capacity.

Fig. 4.3 summarizes results for a 4

4 system for groups of size 4 and 2.

For medium to high SNR, partitioning into groups of size 4 with optimal

group partitioning losses roughly 2[dB] from capacity. Using the simplified

partitioning algorithm losses roughly an extra 0.3[dB]. The simple antenna

partitioning scheme for a group size of 4 (two antennas per group) losses

roughly 3.5[dB] from capacity, thus smart group partitioning shows a gain

of roughly 1 1.5[dB] over simple per antenna partitioning.For medium to high SNR, partitioning into groups of size 2 with optimal

group partitioning losses roughly 3.5[dB] from capacity, using the simplified

partitioning algorithm losses roughly an extra 0.4[dB]. The simple antenna


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0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

18Rayleigh Fading Capacity (Ntx=Nrx=2)

SNR [dB]

Bits/ChannelUse

Full CapacityGD

PA

12 13 14 15 16 17 18 19 20

6.5

7

7.5

8

8.5

9

9.5

SNR [dB]

Bits/ChannelUse

1 6 1 8 2 0 2 2 2 4 2 6 2 8

1 0 . 5

1 1

1 1 . 5

1 2

1 2 . 5

1 3

S N R [ d B ]

Bits/ChannelUse

0.54[dB]

2[dB]

0.64[dB]

2.09[dB]

Figure 4.1: 2 2 Capacity Loss Per Antenna Vs Group Detection.

partitioning scheme (two antennas per group) losses roughly 55.5[dB] from

capacity, thus smart group partitioning shows a gain of roughly 1 2[dB]over simple per antenna partitioning however we pay a price in some increase

in complexity due to the joint detection of the group elements.


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0 5 10 15 20 25 300

5

10

15

20

25

30

35


SNR [dB]

Bits/ChannelUse

Full Capacity

2G_OS_GD

2G_PA_GD

4G_OS_GD

4G_PA_GD

8 10 12 14 16 18 20 22

11

12

13

14

15

16

17

18

19

20

21

Rayleigh Fading Capacity (Ntx=Nrx=4)

SNR [dB]

Bits/Channel

Use

16 18 2 0 22 24 26 28 30

20

21

22

23

24

25

26

27

28

Rayle igh Fading Capaci ty (Ntx=Nrx=4)

SNR [dB]

Bits/ChannelUse

1.7[dB]

2.9[dB]

4.8[dB]

2.2[dB]

3.6[dB]

5.5[dB]

Figure 4.2: 4 4 Capacity Loss Per Antenna Vs Group Detection.


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0 5 10 15 20 25 300

5

10

15

20

25

30

35


SNR [dB]

Bits/ChannelUse

Full Capacity

2G_OS_GD

2G_SS_GD

4G_OS_GD

4G_SS_GD

20 21 22 23 24 25 26

21

22

23

24

25

26

SNR [dB]

Bits/ChannelUse

13 1 4 15 16 17 18 19

13

1 3 . 5

14

1 4 . 5

15

1 5 . 5

16

1 6 . 5

17

1 7 . 5

18

S N R [ d B ]

Bits/ChannelUse

0.26[dB]

0.3[dB]

0.35[dB]

0.45[dB]

Figure 4.3: 4 4 Capacity Loss GD Optimal Search Vs Simple Search.


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Chapter 5

Iterative Group Interference

Cancelation

The detector in Eq. (2.10) does not exploit dependencies between coded bits

which leads to degraded performance. The detector in Eq. (3.23) is an ap-

proximation to Eq. (2.10) and is even more information lossy since informa-

tion is not exchanged between groups. An optimal decoder would regard the

channel code and MIMO channel as serially concatenated codes and would

decode them jointly, such a decoder would have extraordinary complexity.

Many authors [21, 18, 19, 20, 12] propose to use iterative schemes since it

has been shown that such schemes are very effective and computationally

efficient in other joint detection/decoding problems [22, 23]. The iterative

scheme proposed here uses hard decisions from the decoder. Using soft out-puts would result in superior performance however hard output decoders are

commonly implemented in many practical systems and are less complex then

soft output decoders. The iterative scheme proposed here is similar to the

48


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CHAPTER 5. ITERATIVE GROUP INTERFERENCE CANCELATION49

one in [12].

For each group namely group G, hard decoded bits from the decoder are

re-encoded, re-interleaved and used to reconstruct a version of the transmit-

ted MIMO symbol from all symbols but the ones corresponding to group G.

This reconstructed signal is then passed through the effective MIMO channel.

Group Interference Canceling is performed by subtracting the filtered recon-

structed signal from the true received signal. The signal after Interference

Cancelation is given by:

yiG

= HGaG + HG

aG aiG

eG

+z (5.1)

The superscript i in Eq. (5.1) denotes the iteration number. Assuming

correct decisions aiG = aG the above expression is further simplified.

yiG

= HGaG + z (5.2)

The noise after Interference Canceling (assuming correct decisions) is

white and thus a canonical front end matrix is the Matched Filter HTG

aiG = HTGHGaG + H

TGz (5.3)

The group noise covariance matrix after matched filtering is no longer

white and is given by


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+

-

Interleaver

1G

H

i

c

Decoder

LLR1T

GH

2

T

GH

Ng

T

GH

LLR

LLR

De-Interleaver

-

2G

H NgG

H

-

( )ic

( )1

Ng

i

Ga

ya

z

2

i

Ga

1

i

Ga

SymbolMapper

H

Figure 5.1: Iterative Group Detection Scheme.

RGG =12

2z HTGHG (5.4)

5.1 Group Partitioning For Iterative Group

Detection

The partitioning into groups for the iterative stage introduces a new trade

off with respect to the original group partitioning. In the first part of the

decoding process we traded off decoding complexity with performance, where

larger groups resulted in better performance and higher complexity. After

the first decoding pass we have hard estimates for all bits. If one partitions


62/91


the signal into large groups then one is using less new information and at

the extreme not using any new information when no partitioning is done thus

only one group (MAP decoding). On the other hand if one partitions the

signal into very small groups (at the extreme groups of 1 bit each) one may

be more susceptible to error propagation since one only has hard estimates

of the decoded bits with no reliability measure. The partitioning scheme

in Sec. 4 is no longer relevant since it does not take into account the new

information from the initial stage. We thus propose to use the simple antenna

partitioning which is in essence partitioning into groups of size one. The LLR

for group can be efficiently computed by Eq. (3.49) and by setting

P = HTG HGHTG +

2z2a

INN1

HG (5.5)

At the end of each iteration one obtains hard decoded bits that can be

used by the next iteration. Simulation results in chapter 6 suggest that

performing two iterations achieves most of the performance gain.


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Chapter 6

Simulation Results

The performance of the GD scheme for MIMO-BICM was evaluated via

Monte-Carlo simulations. The simulator block diagram for the initial decod-

ing (non iterative) is depicted in Fig 6.1 while the iterative stage is depicted

in Fig 5.1. At the transmitter blocks (packets) of 2000 information bits were

encoded and interleaved using a rate 1/2 64 state convolutional encoder with

octal generators (133, 171) followed by a random per packet interleaver. in-

terleaving between packets would lead to improvement in performance at a

price of large latency. Two antenna configurations were considered, a 2x2 con-

figuration and a 4x4 configuration. For the 2x2 configuration the detection

schemes considered were full MAP detection, Per Antenna group detection

(conventional MMSE) and optimal search Group Detection all with zero,

one and two hard iterations. Most of the performance gain due to iterations

was achieved after two iterations. For the 4x4 configuration two partitioning

schemes were considered namely the partitioning into Ng = 4 groups of size

52


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CHAPTER 6. SIMULATION RESULTS 53

Tx Module

Random RayleighChannel Generator

Find Group PartitionigCompute MMSE Separtion FilterCompute Noise Whitening Filter

AWGN Generator

+

Channel Module

MMSEW

Group Partitioning

Antenna Partitioning

LLR Computer

LLR Computer

GD

LLR Computer

LLR Computer

PA

LLR Computer

MAP

Rx Module

deInterleaver

deInterleaverP

/S

P/S de

Interleaver

ViterbiDecoder

ViterbiDecoder

ViterbiDecoder

MAP

b

G D

b

PA

b

BERCalculator

BERCalculator

BERCalculator

b

b

b

BERMAP

BERGD

BERPA

Whitening FIlter

Whitening FIlter

Whitening FIlter

Whitening FIlter

( )H n

( )H n

( )y n( )z n

( )

mmsea n

( )mmseGa n

( )

mmse

G

a n

( )mmseGa n

Matlab Code C++ Code (Mex)

( )y n

MIMO ChannelGenerator

Rate 1/2 ConvolutionEncoder

MIMO SymbolMapper

RandomInterleaver

b ( )a nP/Sc

Random BitGenerator

Figure 6.1: Simulator Block Diagram (Initial Stage).


65/91


2 each and the partitioning into Ng = 2 groups of size 4 each. For both par-

titioning schemes the detection schemes considered were full MAP detection

(only for fast fading),Per Antenna group detection (PA GD - conventional

MMSE), Optimal Search Group Detection (OS GD),Simplified Search Group

Detection (SS GD) all with zero,one and two iterations.The complex MIMO

channel matrix entries were drawn from a zero mean complex Gaussian dis-

tribution with variance 1/NT in an iid fashion. Simulation results were sum-

marized via average Bit Error Rate (BER) and average Packet Error Rate

(PER) versus SNR1 plots.

6.1 Fast Fading

For fast fading the MIMO channel was independently generated at each in-

stant.

6.1.1 Fast Fading 2x2

Fig 6.2 presents simulation results for the 2x2 configuration for both 16 and

64QAM. Table 6.1 summarizes the gain of the MAP scheme over GD, the gain

of GD over PA and the gain due to iterations for each one of the schemes. The

gain was measured at a BER of 104 - 105. The gains in Table 6.1 correspond

to both 16 and 64QAM since they were found to be similar. Performing

more than two iterations did not show much gain. Fig 6.2 suggests that the

1The SNR is defined as EHa2

E

z2

= 1

2z


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GD gain over PA increases with the SNR. Without iterations GD shows a

substantial gain over PA. Performing iterations closes the gap between GD

and full MAP as well as the gain of GD over PA.

Table 6.1: Fast Rayleigh fading MAP,GD,PA BER Comparison, 2 2Gain [dB] @BER MAP/GD GD/PA

104 105No Iter 0.2-0.3 0.8-1.71 Iter 0.1 0.4-0.8

2 Iter 0.1-0.3 0.35MAP Iter Gain GD Iter Gain

1 Iter 1-1.3 1-1.52 Iter 1 0.8-1

PA Iter Gain1 Iter 2-2.52 Iter 1-1.5

6.1.2 Fast Fading 4x4

Fig 6.3 summarizes simulation results for the 4x4 configuration for 16QAM.

Table 6.2 presents a comparison between the various GD schemes and the

MAP scheme as well as a comparison between GD with group size of 2 to

that of GD with a group size of 4, the gain due to iteration is also included.

The results show that performing iterations reduces the gap between

MAP, the various GD schemes and PA.

Fig 6.4 presents simulation results for the 16QAM 4x4 configuration for

the various GD schemes with the simplified group partitioning (SS GD) al-

gorithms. Results were compared to those of the Optimal Search partition-


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6 8 10 12 14 16 18 20 22 24 26

10-4

10-3

10-2

10-1

Snr

Ber

Ber(Snr) Mimo (2,2) 64QAM Conv K=7 ,Rate = 1/2

16QAM Coded PA Detection16QAM Coded GD Detection

16QAM Coded Map Detection

16QAM Coded IPA Detection

16QAM Coded IGD Detection

16QAM Coded IMAP Detection

16QAM Coded I2PA Detection

16QAM Coded I2GD Detection

16QAM Coded I2MAP Detection

64QAM Coded PA Detection

64QAM Coded GD Detection

64QAM Coded Map Detection

64QAM Coded IPA Detection

64QAM Coded IGD Detection


64QAM Coded I2PA Detection

64QAM Coded I2GD Detection


16QAM

64QAM

2 Iteration

0 Iteration

1 Iteration

2 Iteration

0 Iteration

1 Iteration

Figure 6.2: 2 2 16QAM,64QAM Fast Fading Rayleigh.

ing (OS GD). The simplified partitioning into groups of size 2 (See 4.2.1)

showed a loss of no more then 0.2[dB] with respect to optimal partitioning,

the loss after one iteration dropped to 0.1[dB]. The simplified partitioning

into groups of size 4 (See 4.2.2) showed a loss of no more then 0.35[dB] with

respect to the optimal partitioning, the loss after one iteration remained

around 0.35[dB].


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Table 6.2: Fast Rayleigh fading MAP,GD,PA BER Comparison, 4

4Gain [dB] @BER |G|=2 MAP/GD |G|=2 GD/PA

104 105No Iter 1.5-2 1-21 Iter 1 0.52 Iter 0.6 0.3

|G|=4 MAP/GD |G|=4 GD/PANo Iter 1 1.5-21 Iter 0.4-0.8 0.72 Iter 0.3-0.5 0.5

GD,PA

|G

|=4/2 Iter Gain

No Iter 1-21 Iter 0.4-0.7 1.5-4.52 Iter 0.1-0.3 1-2

6.2 Quasi Static Fading

For quasi static fading the MIMO channel remained constat over a duration

of a block and changed independently from block to block.

6.2.1 Quasi Static Fading 2x2

Fig 6.2 presents simulation results for 16QAM 2x2 configuration. Table 6.3

summarizes the gain of MAP scheme over GD,the gain of GD scheme with

respect to PA scheme as well as the gain due to iterations for each one of the

schemes all at a PER of 102-103.

6.2.2 Quasi Static Fading 4x4

Fig 6.6 and Fig 6.7 presents simulation results for 16QAM 4x4 configuration.

Fig 6.6 summarizes results for the partitioning into 4 groups of size 2 each


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7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 2210

-5

10-4

10-3

10-2

10-1

Snr

B

er


16QAM Coded Map Detection16QAM Coded 4G_PA_GD Detection

16QAM Coded 2G_PA_GD Detection

16QAM Coded 4G_OS_GD Detection



16QAM Coded 4G_PA_IGD Detection


16QAM Coded 4G_OS_IGD Detection


16QAM Coded 4G_PA_I2GD Detection


16QAM Coded 4G_OS_I2GD Detection

16QAM Coded 2G_OS_I2GD Detection


0 Iteration

1 Iteration

2 Iteration

Figure 6.3: 4 4 16QAM Fast Fading Rayleigh.

using the simplified search algorithm, while Fig 6.7 present simulation re-

sults for partitioning into 2 groups of size 4 each using the simplified search

algorithm. Table 6.4 presents a comparison between the various GD schemes

at a PER of 102-103, as well as gain due to iterations.


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5 10 15 2010-5

10-4

10-3

10-2

10-1

Snr

Ber


16QAM Coded 4G_SS_GD Detection16QAM Coded 4G_OS_GD Detection

16QAM Coded 2G_SS_GD Detection


16QAM Coded 4G_SS_IGD Detection




0 Iteration

1 Iteration

Figure 6.4: 44 16QAM Fast Fading Rayleigh Optimal Search Vs Simple Search.

6.3 Simulation Summary

Simulation results suggest that under a fast Rayleigh fading at low BER

GD achieves gains of 1-2[dB] with respect to PA. An extra gain of 1-2[dB]

can be achieved by choosing larger group sizes with a further complexity

price. Under for Quasi static Rayleigh fading at low BER GD achieves gains


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72/91


Table 6.3: Quasi static Rayleigh fading GD,PA PER Comparison, 2 2Gain [dB] @PER MAP/GD GD/PA

102 103No Iter 4-8 3-41 Iter 4-5 2-42 Iter 2-3 2

MAP Iter Gain GD,PA Iter Gain

1 Iter 1-2 3-42 Iter 0.5-1 1

Table 6.4: Quasi static Rayleigh fading GD,PA PER Comparison, 4

4Gain [dB] @BER |G|=2 GD/PA |G|=4 GD/PA

104 105No Iter 3-4 1.5-21 Iter 3.5-4 1.5-22 Iter 2.5-3 1-1.5

PA |G|=4/2 SS |G|=4/2No Iter 7-10 5-81 Iter 4-7 3-52 Iter 3-4 2-3

|G

|=2 Iter Gain

|G

|=4 Iter Gain

1 Iter 5-6 2.5-3.52 Iter 2.5-3.5 1.5-2


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5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 3510

-3

10-2

10-1

100

Snr

PER

PER(Snr) Mimo(4,4) Packet size 2000 infromation bits 16QAM Conv K=7 R=1/2






16QAM Coded 4G_SS_I2GD Detection

Figure 6.6:4

4 16QAM 4 Groups Simple Search Quasi Static Fading Rayleigh.


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5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 3510

-3

10-2

10-1

100

Snr

PER

PER(Snr) Mimo(4,4) Packet size 2000 infromation bits 16QAM Conv K=7 R=1/2






16QAM Coded 2G_SS_I2GD Detection

Figure 6.7: 4 4 16QAM 2 Groups Simple Search Quasi Static Fading Rayleigh.


75/91

Chapter 7

Conclusions

In this thesis we proposed a scalable reduced complexity detection algorithm

for MIMO-BICM. Complexity reduction was achieved by performing detec-

tion in groups instead of joint detection of the entire MIMO signal. A simple

group partitioning algorithm was derived as well as a approximate expression

for the LLR for group size of 2. Performance and complexity were shown to

be traded off by the selection of the group size. By increasing the group size

from 1 to 2 achieves gains of 1-4[dB]. Gains of up to 10[dB] were achieved

by using larger group size. A simple hard iterative interference canceling

scheme was further proposed to enhance performance. Performing hard iter-

ations improved performance of all the schemes as well as reduced the gaps

between them.

64


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Chapter 8

APPENDIX

8.1 Simplification of Q(aG)

To simplify Q (aG) we first derive a closed form expression for 1G . Define

P =

INN +2z2a

HTH

11(A-1)

And note that

G =12

2a

INN + 2z2a HTH1 P1

+VG (I2x2) VTG (A-2)

Then make use of the matrix inversion lemma 1

1G =12

2aP

INN + VG I2x2 VTG P VG1

T

VTG P

(A-3)

Noting that

T =

I2x2 VTG P VG1

=

1 pjj pij

pij 1 pii

(1 pii) (1 pjj ) p2ij

(A-4)

1A1 A1B D1 + CTA1B1 CTA1 = A + BDCT165


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CHAPTER 8. APPENDIX 66

Where pij is the i,jth element of P in Eq. (A-1).

To compute the first term in Eq. (3.33) we evaluate

aTzf1G ei =

12

2aaTzfP

INN + VGT V

TG P

ei (A-5)

Noting that P converts ZF estimation into MMSE estimation since

aTzfP =

PTazfT

=

INN +

2z2a

HTH

11 HTH

1HTy

T= H

TH+ 2z

2aIN

1

HTyT

= Wmmsey

T= aTmmse

(A-6)

The forth equality in Eq. (A-6) follows from the following Eq. (A-7)HTH+

2z

2aI1

HT = 2a

2z

2a2z

HTH+ I1

HT

= 2a

2z

I HT

HHT +

2z

2aI1

H

HT

= 2a

2zHT

I

HHT +

2z

2aI1

HHT

= 2a

2zHT

2a2z

HHT + I1

= HTHHT + 2z2a I1 = Wmmse

(A-7)

The second and forth equalities in Eq. (A-7) follow from the matrix inver-sion lemma while the rest are trivial. Substituting Eq. (A-6) into Eq. (A-5)yields

aTzf1G eiai =

12

2aaTmmse

ei + VGT V

TG P ei

ai

= 12

2a(1pjj)ammsei +pij a

mmsej

(1pii)(1pjj )p2ijai

aTzf1G ej aj =

12

2aaTmmse

ej + VGT V

TG P ej

aj

= 12

2apij ammsei +(1pii)a

mmsej

(1pii)(1pjj)p2ijaj

(A-8)

The last two terms in Eq. (3.33) evaluate to


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79/91

CHAPTER 8. APPENDIX 68

HTH =

hT1 h1 h

T1 h2 h

T1 h3 h

T1 h4

hT2 h1 hT2 h2 h

T2 h3 h

T2 h4

hT3 h1 hT3 h2 h

T3 h3 h

T3 h4

hT4 h1 hT4 h2 h

T4 h3 h

T4 h4

(B-2)Denoting the (i,j)th element ofH by hi,j and by h

Ri,j and h

Ii,j the real part

and imaginary part of the complex H respectively, follows that

hT1 h1 =

hR11

2

+

hR21

2

+

hI11

2

+

hI21

2

= hT3 h3hT

2

h2

= hR122 + hR222 + hI122 + hI222 = hT4 h4hT1 h2 = hR11hR12 + hR21hR22 + hI11hI12 + hI21hI22hT1 h3 = hR11hI11 hR21hI21 + hR11hI11 + hR21hI21 = 0hT1 h4 = hR11hI12 hR21hI22 + hI11hR12 + hI21hR22hT2 h3 = hR12hI11 hR22hI21 + hI12hR11 + hI22hR21 = hT1 h4hT2 h4 = hR12hI12 hR22hI22 + hI12hR12 + hI22hR22 = 0hT3 h4 = h

I11h

I12 + h

I21h

I22 + h

R11h

R12 + h

R21h

R22 = h

T1 h2

(B-3)

Substituting Eq. (B-3) into Eq. (B-2)

HTH =

hT1 h1 hT1 h2 0 h

T1 h4

hT1 h2 hT2 h2 hT1 h4 00 hT1 h4 hT1 h1 hT1 h2

hT1 h4 0 hT1 h2 h

T2 h2

(B-4)

Denoting = 2z

2aand substituting Eq. (B-4) into Eq. (B-1)

Ree = 2A

1 0 0 00 1 0 00 0 1 00 0 0 1

+

hT1 h1 hT1 h2 0 hT1 h4hT1 h2 h

T2 h2 hT1 h4 0

0 hT1 h4 hT1 h1 hT1 h2hT1 h4 0 h

T1 h2 h

T2 h2

1 (B-5)

Denote the scalars a,b,c,d

8/9/2019 Reduced Complexity Demodulation of MIMO

reduced complexity demodulation of mimo bicm using group detection thesis

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