A New Quasi-Optimal Detection Algorithm for aNon Orthogonal Spectrally Efficient FDM

Ioannis Kanaras1, Arsenia Chorti2, Miguel Rodrigues3 and Izzat Darwazeh1

1EEE Dep., University College London, London WC1E 7JE, UK, {i.kanaras, i.darwazeh}@ee.ucl.ac.uk2CCM Dep., EIS, Middlesex University, Hendon, London, NW4 4BT, UK, a.chorti@mdx.ac.uk, a.chorti@ee.ucl.ac.uk

3Dep. of Computer Science, University of Porto, 4169 - 007 Porto, Portugal, mrodrigues@dcc.fc.up.pt

Abstract—Non-orthogonal Spectrally Efficient Frequency Divi-sion Multiplexing (SEFDM) signals of a small dimensionality canbe optimally detected using the Sphere Decoder (SD) algorithm.However, the employment of such detectors is restricted by twofactors; the ill-conditioning of the SEFDM projections matrix inthe system linear statistical model and the sensitivity of the SDcomplexity to noise. A solution to the latter could be given by afixed complexity detection based on the Semidefinite Program-ming (SDP). Notwithstanding, SDP error performance appearsto be suboptimal. In order to diminish the error performance gapbetween the SDP and the optimal detector we propose a modifiedSD that investigates only the points of the SEFDM lattice withina hypersphere whose size is determined by a first SDP estimate.In addition, the new SD tree is pruned to include only thebranches that have a heuristically predefined Hamming distancefrom the SDP estimate. We show that the introduced schemeachieves a quasi optimal Bit Error Rate (BER) for an SEFDMscheme with 20% spectral gain compared to Orthogonal FDM(OFDM). Moreover, we demonstrate by simulation that the newscheme is superior in terms of computational effort comparedto an equivalent SDP - brute force Maximum Likelihood (ML)scheme. Finally, it is shown that the new pruned SD reduces bymore than 30% the number of the visits to the nodes of the SDtree made by the conventional SD using the Schnorr Euchner(SE) reordering strategy.

I. INTRODUCTION

Sphere Decoder (SD) was first proposed in [1] as animproved method to find the shortest vector in a given lattice.The aforementioned problem is equivalent to an Integer LeastSquares (ILS) optimisation problem that is well known to be ofnon polynomial complexity when solved using an exhaustivesearch over the entire feasibility set [2]. Interestingly, SDaccomplishes, under specific conditions, the optimal solutionwith a reduced complexity of polynomial order [3].

Hence, the SD algorithm was proposed in the communi-cations as an efficient solution for detection problems thatcould be reduced in ILS. Examples of SD investigations areits implementation for the detection of lattice codes [4], Multi-User Code Division Multiplexing Access (MU-CDMA) [5] aswell as Multiple Input Multiple Output (MIMO) [6] systems.

Recently, efforts have been made in order to examinethe possibility of a tangible detection for a non orthogonalSpectrally Efficient FDM (SEFDM) system [7]. It has beenshown that a Regularised version of SD (RSD) could recoverthe SEFDM signal in moderate and high Signal to Noise Ratio(SNR) regimes [8]. Nevertheless, the complexity of the RSD is

random, depending on the noise random variable realization.Therefore, reducing the complexity of the SEFDM detector isstill an open area of research.

Very recently, it was demonstrated that the application ofa boxed ML detection around an SDP estimate could offera near optimal solution with an almost cubic complexity [9].Motivated by this work, we introduce in this paper a modifiedSphere Decoder that could replace the brute force ML partof the combined SDP-ML method. Our goals are twofold; toimprove the error performance by decreasing the gap betweenthe optimal and the SDP-ML approach, and to substantiallydecrease the algorithmic complexity by reducing the numberof the points that are investigated by the exhaustive search partof the SDP-ML algorithm.

This paper is organised as follows: In section II we describethe linear model of the SEFDM system. In section III wediscuss the detection of the SEFDM signal and provide thebasic formulation of the simple SDP and SD SEFDM detec-tion problems. In section IV we explain the concept of theintroduced Pruned Sphere Decoder (PSD). Finally, in sectionsV and VI we cite our simulation results and conclusions,respectively.

II. SEFDM SYSTEM MODEL

The FDM transceiver is originally described in [7]. Incontrast to the typical OFDM, the frequency separation ∆fbetween the FDM subcarriers is deliberately set to be equalto a fraction α of the FDM symbol period T , i.e.,

∆f =α

T, with α < 1. (1)

Hence, we term α = ∆fT the FDM normalised subcarrierfrequency separation.

The baseband FDM signal, in T , is then given by

s (t) =1

√

T

N−1�

n=0

Snfα,n(t) =1

√

T

N−1�

n=0

Snej2πn∆ft, 0 ≤ t ≤ T

(2)where N is the number of the FDM subcarriers fα,n(t) and Sn

are the data symbols that take values over an M -ary alphabet.Thanks to the overlapping of the subcarriers individual

bands, the overall bandwidth occupied by the FDM signal isreduced by a factor 1−α as opposed to an equivalent OFDM.The FDM spectral gain is demonstrated in Fig. 1 where the

978-1-4244-4522-6/09/$25.00 ©2009 IEEE ISCIT 2009460

−1 −0.5 0 0.5 1

x 107

−50

−40

−30

−20

−10

0

10

Frequency in Hz

Nor

mal

ised

PS

D in

dB

OFDMα=0.75 FDMα=0.5 FDM

Fig. 1. Spectrum of N = 32 and T = 4 µsec FDM schemes.

spectrum of the transmitted FDM signal is depicted for α = 1(OFDM), α = 0.75 (33% spectral gain) and α = 0.5 (100%spectral gain).

We assume that the communications channel introducesonly AWGN n(t) with Power Spectral Density (PSD) N0

2.

As a result, the received signal r(t) is given by

r(t) = s(t) + n(t). (3)

The receiver consists conceptually of two stages. The firststage uses a bank of N correlators to extract N sufficientstatistics from the received signal. The second stage comprisesa detector that recovers estimates of original data symbols.

The particular choice of the correlation functions at thereceiver first stage is driven by two important requirements: (i)The correlation functions bi(t), with i = 0, . . . , N−1, shouldbe orthonormal in order to prevent noise coloring, and (ii) easeof computation. Both requirements are met by generating anorthonormal base that spans the FDM signal space using theIterative Modified Gram Schmidt (IMGS) orthonormalisationmethod [10].We also note it is possible to relate the vector of sufficient

statistics to the vector of information symbols as follows

R =MS+N, (4)

where R = [Ri] is the vector of the N observation statistics,S = [Si] is the vector of the N transmitted symbols and M =[Mij ] is the N ×N covariance matrix of the FDM subcarriersand the orthonormal base. Furthermore, N = [Ni] is a vectorcontaining N independent Gaussian noise variables of zeromean and covariance matrix σ2

IN (IN being an N×N identitymatrix and σ2 = N0

2).

The elements of R and M are given by

Ri =

� T

0

r(t)b∗i (t)dt, i = 0, . . . , N − 1, (5)

Mij =

� T

0

fα,i(t)b∗

j (t)dt, i, j = 0, . . . , N − 1. (6)

Fig. 2. FDM modem.

Finally, it is interesting to note that the typical imple-mentation of an FDM system mimics the IFFT-FFT basedimplementation of an OFDM system [11]. For example, anInverse Fractional Fourier Transform (IFrFT), with quadruplethe complexity of a conventional IFFT, could be used for theFDM signal generation [8]. A block diagram of a possibleFDM modem is illustrated in Fig. 2.

III. SEFDM DETECTION PRELIMINARIES

Before proceeding with the detection issue, we need tomention a few important properties of the covariance matrixM. First, M is upper triangular and therefore its eigenvaluesare equal to its diagonal elements. Second, its eigenvaluesare reals that overly degrade as the number of the SEFDMcarriers N and/or their frequency separation ∆f decreases.Consequently, M becomes a severely ill conditioned matrixand turns the SEFDM detection to an ill posed problem.

A. Optimum Detection

In the presence of AWGN, the optimum detection of theSEFDM signal, based on equation (4), reduces to the followingcombinatorial Least Squares problem [7], [12]:

minimise �R − MS�2,

subject to S ∈ QN , (7)

where �·� denotes the Euclidean norm and QN is the set ofall the possible data symbols N -tuples.

For implementation reasons, we apply a real decomposition[13] of all the matrices of equation (7) so that the problemcan be solved using algorithms applied for real LS problems.Hence, the problem dimension is doubled and equal to 2N .

In addition, we assume that the feasible set is furtherconstrained so that it contains only binary (±1) 2N -tuples (i.e.4-QAM symbols). Consequently, the SEFDM ML detection

461

reduces to an ILS problem that can be solved using SD as:

minimise �R − MS�2,

subject to �R − MS�2≤ C,

S ∈ {±1}2N

, (8)

where C is the radius of the SD hypersphere.It has been shown [8] that the typical SD implementation is

affected by a substantial increase of the complexity due to theprojections matrix M ill conditioning. Furthermore, when thematrix becomes numerically singular the SD cannot be applieddue to the zero diagonal elements of matrix M. To overcomethese difficulties a Tikhonov regularisation is applied on theILS cost function so that (8) reduces to:

minimise �D (P − S)�2,

subject to �D (P − S)�2≤ C ′,

S ∈ {±1}2N

, (9)

with

D = chol�

MTM+ ǫI

�

,

P = (MTM+ ǫI)−1

MTR, (10)

where the operators chol {·} and {·}

T denote the Choleskydecomposition and the transpose of a matrix, respectively, theregulator ǫ is a real positive term and I is the identity matrixof 2N × 2N dimension. Finally, the radius C ′ of the newproblem is derived according to the following:

C ′ = C + ǫSTS − R

TR+P

TD

TDP. (11)

Although, it has been shown that RSD improves the com-plexity of the SEFDM detection, as opposed to a typicalSD [8], it is still sensitive to the noise level in the system.For this reason, in [9] we explored the possibility of usingconvex optimisation techniques, trading error performancefor algorithmic complexity. For clarity, we outline the SDPdetector briefly.

B. Semidefinite RelaxationThe problem described in (8) is equivalent to:

minimise STM

TMS − R

TMS − S

TM

TR,

subject to S ∈ {±1}2N

. (12)

Adding an extra slack variable S2N+1 = 1, the cost functionof equation (12) becomes

STM

TMS − R

TMS − S

TM

TR = x

TLx, (13)

where

L =

�

MTM −M

TR

−RTM 0

�

, x =

�

S

S2N+1

�

. (14)

Thanks to the symmetric nature of L the Right Hand Side(RHS) of (13) is equal to Tr{Lxx

T}, where the operator Tr{·}

denotes the ‘trace’ function. Consequently, (12) reduces to

minimise Tr{LX},

subject to diag{X} = e,

X � 0, rank{X} = 1, (15)

where the curly inequality � indicates that X = xxT is a

positive semidefinite matrix. In addition, e is a (2N + 1)× 1vector of ones and the diag{·} operator generates a (2N +1) × 1 vector that includes all the diagonal elements of theargument matrix. Finally, the rank{·} operator provides therank of the matrix X.

Similar to the ML problem in (7), the equivalent problem of(15) is not convex. However, convexity can be accomplishedby relaxing the feasible set of the problem after discarding thenon affine rank constraint

minimise Tr{LX},

subject to diag{X} = e,

−X � 0. (16)

This problem constitutes a Semidefinite Program and canbe very efficiently solved using well known Interior PointMethods (IPM) [14]. Yet, a heuristic is needed so that anestimate S of the FDM symbol is recovered from the solutionX of the SDP problem. In our implementation, we used arandomisation technique [15].In [9] we demonstrated that although SDP has a fixed

complexity, its error performance is suboptimal and degradeswith the ill conditioning of the M matrix. In order to mitigatethis effect, a combination of SDP and ML was introduced.

In particular, a brute force ML was applied in a subset Dof the entire feasible set Q2N . The elements of D were allthe possible SEFDM symbols that had a predefined Hammingdistance dH from a first SDP estimate S. It was shown that(even for dH = 1) the combined SDP-ML method performedbetter than a simple SDP for the detection of SEFDM signalswith N ≤ 32 carriers. Nevertheless, the main drawback ofthe algorithm is that the performance improvement requires alarger dH as N increases. Unfortunately, that also results intoan overly increase of the SDP-ML complexity.

IV. PRUNED SPHERE DECODER (PSD)

Motivated by the last finding, we propose a modified SDversion that could implement faster the SDP-ML methodthanks to the reduction in the computation of the bruteforce ML part. In particular, the proposed algorithm involvestwo consecutive steps: Initially, the SDP estimate S of thetransmitted SEFDM symbol is calculated. Furthermore, thenew radius C ′ of the regularised SD hypersphere is derivedaccording to eq. (11) as follows:

C ′ =�

�

�R − MS

�

�

�

2

+ ǫSTS − R

TR+P

TD

TDP. (17)

Consequently, S lies on the surface of the RSD sphere.Then, an extra condition is added in the RSD implementa-

tion so that the algorithm never traces the RSD tree pathsthat correspond to the SEFDM symbols that have largerHamming distance from the calculated S than a selected ρvalue. Thus, the proposed pruned tree RSD reduces to the

462

Fig. 3. Flow chart of the modified pruned Sphere Decoder algorithm. S represents the outcome of the proposed pruned SD.

following problem:

minimise �D (P − S)�2,

subject to �D (P − S)�2≤ C ′,

S ∈ {±1}2N

,

dH = HD�

S, S�

≤ ρ, (18)

where the HD {·, ·} operator calculates the Hamming distancebetween the argument vectors and ρ is the heuristically prede-fined value of dH that could range from 0 to N log2M (i.e.,the length of the binary representation of S).Fig. 3 provides a flow chart of the pruned SD algorithm. The

red dotted lined boxes correspond to the add-in modificationsof the algorithm. It is obvious that the main alteration isthe addition of an if loop that checks the Hamming distancebetween the path and the respective part of the SDP estimateat each visited tree node. Should dH ≥ ρ at a node, thealgorithm cuts the attached subtree and continues the searchgoing backwards.

In order to simplify the understanding of the pruning of the

PSD tree we provide the following example: Assuming thatwe have a BPSK SEFDM signal of N = 3 carriers. The fullRSD tree has 23 = 8 different paths that correspond to the fullfeasible set of the ML optimisation problem. If the derived S

is [−1,+1,+1]T and the Hamming distance is ρ = 1 thenthe pruned SD tree includes only 4 over 8 paths as shownin Fig. 4. As a consequence, it is heuristically expected thatthe number of the visits to the SD nodes will be significantlyreduced as opposed to the full SD at the expense of a penaltyin the optimality of the achieved Bit Error Rate.

Moreover, as opposed to the SDP-ML, the pruned SD isexpected to achieve the same error performance for the samegiven dH without necessarily tracing all the 4 paths of thepruned SD tree as in the exhaustive search ML case.

V. RESULTS

In order to evaluate the performance of PSD, we ran a setof simulations for varying number of SEFDM carriers N andfrequency separation ranging from α = 1 (OFDM) to α = 0.5

463

Fig. 4. Full (solid lines) and Pruned (dotted lines) SD tree. The SDP estimateis represented by the bold solid line.

(half OFDM). We also tested the proposed scheme for variousvalues of the parameter ρ of the added constraint of eq. (18).

In all cases, we compared through simulation the errorperformance and complexity of the proposed PSD algorithm tothose of the SDP-ML approach, as well as to the optimal RSD.As far as the error performance is concerned, the Bit ErrorRate (BER), versus the normalised frequency separation α orthe Energy of the bit over the Noise power density Eb/N0, wasused as a measure of comparison. Moreover, the complexitywas evaluated in terms of simulation time and visits to theRSD tree nodes for the comparison with SDP-ML and simpleRSD, respectively.

Finally, we need to cite that for the modeling and simulationof the SDP part of the algorithm we used the CVX optimisa-tion tool [16], [17]. However, in all cases the simulation timesof our results involved only the time consumed by the SD orthe ML parts and not the entire algorithms.

A. PSD Error PerformanceFig. 5 depicts the error performance of the PSD scheme ver-

sus the normalised frequency separation α of the SEFDM car-riers. The PSD results are compared to the SDP-ML detectionerror rates for different number of carriers N and values of theHamming distance parameter ρ. In addition, the BER curvesof the simple RSD detection for N = 8 and N = 16 representthe optimal detection for these SEFDM signal dimensions. It isapparent that in all cases the performance of the PSD method isequivalent to the performance of the SDP-ML scheme with thesame ρ. Furthermore, the proposed scheme offers a suboptimalsolution since it diverges from the RSD curves especially afterthe α = 0.8 point. However, as the condition for the Hammingdistance relaxes (i.e. ρ becomes larger) the difference betweenRSD and PSD results decreases. In particular, we note that forN ≤ 32 and ρ ≤ 2 PSD approximates the optimal detection.

Motivated by the previous results, we repeated the simula-tions for α = 0.8 and Eb/N0 that ranged from 0 to 7 dB. Theresults that are illustrated in Fig. 6 show that PSD achieves aquasi optimal BER for N ≤ 32 and ρ ≤ 2. In addition, fora N = 32 carriers SEFDM signal, PSD with ρ = 2 offeredan almost 2 dB gain as opposed to the simple SemidefiniteProgramming (SDP) detection.

0.5 0.6 0.7 0.8 0.9 110−3

10−2

10−1

100

Normalised Frequency Separation − α

BE

R fo

r Eb/N

0 of 5

dB

Error Performance of PSD 4−QAM SEFDM Detection

08RSD08PSD, ρ=116RSD16SDPML, ρ=116PSD, ρ=132SDPML, ρ=132PSD, ρ=132PSD, ρ=232PSD, ρ=3

(OFDM)

Fig. 5. Error Performance of the proposed PSD scheme versus the SEFDMcarriers normalised frequency separation α.

0 1 2 3 4 5 6 710−4

10−3

10−2

10−1

Eb/N

0 (dB)

BE

R w

ith α

=0.8

Error Performance of PSD 4−QAM SEFDM Detection

SC−theoretical08RSD08PSD, ρ=116RSD16SDPML, ρ=116PSD, ρ=132SDPML, ρ=132PSD, ρ=132PSD, ρ=232SDP

Fig. 6. Error Performance of the proposed PSD scheme versus the Eb/N0.

B. PSD Complexity

We evaluated the computational complexity of the proposedmethod as opposed to the SDP-ML and the full tree RSDmethods. Fig. 7 shows the simulation time required by theSD or the ML parts of the pruned SD and SDP-ML schemes,respectively. All the results were normalised over the valuesof the 32SDP-ML with ρ = 1 scheme. It is clear that PSDperforms faster than an equivalent, i.e. using the same ρ, SDP-ML scheme. For example, we see that for the detection of anSEFDM signal with N = 32 and α = 0.8, PSD performsalmost 15 times faster than the SDP-ML with ρ = 2. Thisimprovement is due to the SD that investigates only the frac-tion of the SEFDM lattice points [8] with a specific Hammingdistance dH from S that are within the SD hypersphere.

In addition, in Fig. 8 we compare the complexity of thepruned SD as opposed to a simple RSD that traces a full tree.

464

0.5 0.6 0.7 0.8 0.9 110−2

10−1

100

101

102

103

Normalised Frequency Separation − α

Nor

mal

ised

sim

ulat

ion

time

in a

rbitr

ary

units

Complexity of PSD 4−QAM SEFDM Detection for Eb/N

0 = 5dB

08SDPM, ρ=108PSD, ρ=116SDPML, ρ=116PSD, ρ=132SDPML, ρ=132PSD, ρ=132PSD, ρ=232PSD, ρ=332SDPML, ρ=2N=32

N=08

N=16

Fig. 7. Complexity Comparison of PSD with the SDP-brute force ML scheme.

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1101

102

103

104

105

106

Normalised Frequency Separation − α

Ave

rage

Vis

its n

umbe

r to

the

prun

ed S

D tr

ee

Complexity of PSD 4−QAM SEFDM Detection for Eb/N

0 = 5dB

08RSD08PSD, ρ=116RSD16PSD, ρ=132PSD, ρ=132PSD, ρ=232PSD, ρ=3

N=16

N=08

Fig. 8. Complexity Comparison between the pruned tree SD and a full treeSD implementation.

We see that in the PSD case the number of the visits to thetree nodes is significantly lower than in the RSD case, eventhough a Schnorr Euchner [18] reordering strategy is used.In particular, for all combinations of N and α, our resultsshow that the number of the node visits is reduced by at least30% with respect to the single RSD algorithm. This is clearlydue to the addition of the Hamming distance dH constraintas described in equation (18). Therefore, the relaxation of thisconstraint (i.e. the increase of the parameter ρ) results into adegradation of the PSD computational complexity.

VI. CONCLUSIONS

We proposed a new modified Sphere Decoder algorithm(PSD) that implements faster a combined Semidefinite Pro-gramming and brute force Maximum Likelihood detection fora non orthogonal spectrally efficient FDM system. The numberof branches of the PSD tree is restricted due to the addition

of an extra constraint so that the feasibility set includes onlythe SEFDM vectors that have a predefined Hamming distancedH from an initial SDP estimate. It was shown by simulationthat for dH ≤ 2 the new scheme achieves a quasi-optimalerror performance for SEFDM systems with N ≤ 32 and∆fT = 0.8 in low SNR regimes. In addition, PSD offersthe same solution with an equivalent combined SDP-MLscheme but in a fraction of the computational effort of thelatter. Finally, simulation results showed that PSD significantlyreduces the required effort for the tree search of an optimalSD at the expense of a small penalty in the BER.

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