The redundancy afforded by overcomplete expansions have been recently been considered for channel coding. In this paper, we utilise this approach in order to propose a channel coder design to for a correlated additive Gaussian noise channel, of which the noise covariance matrix is assumed to be known. We demonstrate that this approach can lead to a significant reduction of the noise interference by exploiting both the correlation of the channel and the redundancy of the filter banks. Simulation results providing some insight into these mechanisms are provided

Weiss, S.

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Design of overcomplete expansions for channel coding

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Strathprints Institutional RepositoryWeiss, S. (2004) Design of overcomplete expansions for channel coding. In: 6th InternationalConference on Mathematics in Signal Processing, 2004-12-14 - 2004-12-16, Cirencester.Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/DESIGN OF OVERCOMPLETE EXPANSIONS FOR CHANNEL CODINGStephan WeissCommunications Research Group, School of Electronics & Computers ScienceUniversity of Southampton, UKs.weiss@ecs.soton.ac.ukABSTRACTThe redundancy afforded by overcomplete expansions have beenrecently been considered for channel coding. In this paper, weutilise this approach in order to propose a channel coder designto for a correlated additive Gaussian noise channel, of which thenoise covariance matrix is assumed to be known. We demonstratethat this approach can lead to a significant reduction of the noiseinterference by exploiting both the correlation of the channel andthe redundancy of the filter banks. Simulation results providingsome insight into these mechanisms are provided.1. INTRODUCTIONOvercomplete expansion offer redundancy and design freedom,which has recently been used for the reconstruction of erased or er-roneously received expansion coefficients [1], or for the design oferror correction codecs [2]. More recently in [3] a systematic par-allelism between block codes and oversampled filter bank (OSFB)systems implementing overcomplete expansions for channel cod-ing has been drawn, whereby the system design is based on un-quantised “soft-input” signals [4]. The channel coding schemesin [2, 3, 4] are based on an encoding stage using a preset analysisfilter bank. In the decoding stage an OSFB synthesis exploits theredundancy in order to reconstruct the signal — either perfectly orin the mean square error sense — while ideally projecting awayfrom the noise. Most OSFBs consist of simple discrete Fouriertransforms [2, 3, 4], which leads to low cost implementations, ro-bustness towards burst-type errors and compatibility with OFDM-based modulation system.If the additive channel noise is correlated, the projection in [3]is performed in the direction of the principle components of thenoise subspace, which ideally is restricted such that a noise-freesignal subspace exists. Also, in [2, 3, 4] the synthesis design is,despite some degrees of freedoms (DOFs) due to oversampling,limited by the a-priori choice of the analysis filter bank. In [5], thesynthesis filter bank is given more flexibility by aiming the designat the suppression of the channel noise under the constraint of in-vertibility, such that an analysis filter bank encoder can be derivedfrom the synthesis bank. However, the filter bank design in [5] isbased on a crude iterative method that can prove the potential ofthe approach but is otherwise far from optimal.Therefore, in this paper we follow the channel codec schemein [5] for a correlated additive Gaussian noise channel, but applya considerably improved constrained synthesis filter bank designmethod based on the second order sequential best rotation (SBR2)algorithm [6]. By linking the remaining noise variance after de-coding to the covariance matrix of the channel noise in depen-dency of the synthesis filter bank, a suitable broadband eigenvaluedecomposition using SBR2 leads to a paraunitary filter bank de-sign that exploits both the correlation of the channel noise as wellas the DOFs provided by the OSFBs.The paper is organised as follows. Based on an analysis ofthe overall setup in Sec. 2, the codec design is presented in Sec. 3.Simulations discussed in Sec. 4 will provide some insight into theworking and benefits of the proposed method.2. SYSTEMMODEL2.1. Overcomplete ExpansionsFor implementation and analysis purposes, OSFBs implementingovercomplete expansions can be conveniently represented by poly-phase analysis and synthesis matrices [7]. A K channel OSFBdecimated by N < K is thus characterised by a polyphase analy-sis matrixH(z) 2 CKN (z), while a polyphase synthesis matrixG(z) 2 CNK(z) can denote the synthesis operation. Using a de-multiplexing operation for the input signalX(z) intoN polyphasecomponentsX(z) =N 1Xn=0z N+n 1Xn(zN) (1)as shown in 1, a vector Y (z) containing the filter bank outputscan be written as Y (z) = H(z)X(z), whereby X(z) hold theN polyphase components Xn(z), n = 0(1)N   1. A filter banksystem is perfectly reconstructing ifG(z)H(z) = z IN: (2)The redundancy afforded by OSFBs has recently attracted at-tention for channel coding [2]. There, a coding rate N=K < 1can ensure robustness against noise interference, with the aim ofrestoring noise corrupted samples due to the redundant format inwhich the data is transmitted. The analysis and synthesis filterbanks function as encoder and decoder, while the filters in H(z)and G(z) are no longer limited to a bandpass design, but willrather be selected according to the characteristics of the interferingnoise.2.2. Channel CodecThe general channel coding system using OSFBs is given in Fig. 1.The polyphase components ^Xn(z) of the received signal in Fig. 1can be collected similar to X(z) in a vector ^X(z), which is givenby^X(z) = G(z) (Y (z) +W (z)) (3)Y0Y1Y01010Y1YYW0 WW1H G)(z1K−)(z)(z)(zX)(zX)(zX 1N−)(z )(z)(zX)(zX)(zX 1N−)(z)(z)(z1K−)(z )(z1K−)(z)(zXNNNNNN)(zXwhereby Y (z) = H(z)X(z) 2 CK (z) and W (z) 2 CK (z) con-tain the subband signal components of the transmitted data and thenoise, respectively. Selecting perfect reconstruction filter banksG(z)H(z) = IN,E(z) = X(z) ^X(z) =  G(z)W (z) (4)is obtained.To assess the total received noise variance 2ein ^X(z), let theN -element vector e[m℄ contain the N time series associated withthe z-domain quantities in E(z) —Æ e[m℄, which depend on thetime index m in the decimated domain. Thus we have2e=1NtrfEne[m℄ eH[m℄og ; (5)where trfg denotes trace and Efg is the expectation operator.Defining the auto-correlation matrixRee[ ℄ = Ene[m℄ eH[m   ℄o(6)and its z-transformRee(z) —Æ Ree[ ℄ denoting the power spec-trum of the process e[m℄ [7], the noise variance is given by2e=1NtrfRee[0℄g =1NtrfRee(z)gz=0(7)=1NtrfG(z)Rww(z)~G(z)gz=0; (8)Note that (4) has been exploited to trace the noise variance backto the power spectrum Rww(z), which is the z-transform of thecovariance matrix of the channel noise,Rww[ ℄ = Enw[m℄ wH[m   ℄o(9)withw[m℄ Æ— W (z) as defined in Fig. 1.3. CODEC AND FILTER BANK DESIGNBased on the idea of the channel codec outlined in Sec. 3.1, thissection considers a suitable factorisation of the power spectrum atthe decoder output in Sec. 3.2, admitting a useful codec designin Sec. 3.3. An algorithm to construct filter banks achieving thisdesign is reviewed in Sec. 3.4.3.1. Proposed Coding ApproachNote that the current standard approach [3] exploits the degrees offreedom in the design of G(z) for a given H(z) in order to min-imise the noise power 2ein (5). The proposed method aims tominimise 2eby optimising G(z) without restrictions. The onlycondition placed on G(z) is that it admits a right inverse Gy(z)Fig. 1. General setup of a channel coderbased on K channel analysis and syn-thesis filter banks, arranged around thetransmission over K additive Gaussiannoise channels.such that G(z)Gy(z) = z . A stronger restriction placed onG(z) is paraunitarity, which however has two important advan-tages: (i) the analysis filter banks is immediately given byH(z) =~G(z), and (ii) paraunitarity provides a minimum norm solutionsuch that the transmit power is limited. As a counter example, aninvertible G(z) might elicit an ill-conditioned H(z) which mayencourage to transmit highly powered signals over subspaces as-sociated with near rank deficiency.3.2. Factorisation of the Noise Covariance MatrixWe approach the minimisation of (8) via a factorisation of thepower spectrumRww(z) = U(z) (z)~U(z) (10)such that U(z) 2 CKK (z) is paraunitary and strongly decorre-lates Rww(z), i.e. (z) = diagf 0(z);  1z; : : :  K 1(z)g (11)is a diagonal matrix with polynomial entries  k(z). This factorisa-tion presents a broadband eigenvalue decomposition, which can befurther specified by demanding  (z) to be spectrally majorised [6]such that the power spectral density of the kth noise component k(ej) =  k(z)z=ejevaluated on the unit circle obeys k(ej)   k+1(ej) 8 and k = 0(1)K   2; (12)similar to the ordering of the singular values in a singular valuedecomposition. Note that paraunitarity or losslessness of U(z)conserves power, i.e. trf (z)gjz=0= trfRww(z)gjz=0.3.3. Codec DesignUsing the redundancy N < K due to oversampling, we can con-structG(z) fromU(z) to select the lower (and therefore smallest)N elements on the main diagonal of  (z). LetU(z) = [U0(z) U1(z)    UK 1(z)℄ ; (13)thenG(z) =26664~UK N(z)~UK N+1(z)...~UK(z)377752 CNK(z) ; (14)such that G(z)U(z) = [0NK NIN℄. If (z) = 00(z)  01(z) 10(z)  11(z)(15)0 0.1 0.2 0.3 0.4 0.5−15−10−50510152025noise PSD / [dB]normalised angular frequency Ω / 2piR0(ejΩ)R1(ejΩ)R2(ejΩ)R3(ejΩ)R4(ejΩ)R5(ejΩ)0 0.1 0.2 0.3 0.4 0.5−15−10−50510152025normalised angular frequency Ω / 2piΓ0(ejΩ)Γ1(ejΩ)Γ2(ejΩ)Γ3(ejΩ)Γ4(ejΩ)Γ5(ejΩ)Fig. 2. PSDs on the main diagonals of (top) the power spectrumRww(z) of the channel noise consisting of the Rk(ej) of (18)and (bottom)  (z) after application of the SBR2 algorithm.with  11(z) 2 CNN and the remaining sub-matrices of appro-priate dimension, then the noise power at the decoder output be-comes2e=1Ntrf 11(z)gz=0(16)=1NK 1Xi=K N2Z0 i(ej) d : (17)Therefore, the spectral majorisation in the broadband eigenvaluedecomposition (10) is essential to the success of the proposed chan-nel coder design.3.4. Sequential Best Rotation AlgorithmIn order to achieve the factorisation in (10) fulfilling spectral ma-jorisation according to (12), we use the second order sequentialbest rotation (SBR2) algorithm [6]. SBR2 is an iterative broadbandsingular value decomposition technique, which finds a sequenceof paraunitary matrix operations, each eliminating the largest off-diagonal element of the remaining power spectrum matrix. In ex-tensive simulations, SBR2 has proven very robust and stable inachieving both a diagonalisation and spectral majorisation of anygiven covariance matrix, whereby the algorithm is stopped eitherafter reaching a certain measure for suppressing off-diagonal termsor after exceeding a defined number of iteration [6].4. SIMULATIONS AND RESULTS4.1. Multichannel TransmissionIn a first transmission scenario as shown in Fig. 1, K sub-channelsare corrupted by Gaussian noise processeswk[m℄, k = 0(1)K 1,such thatEfwk[m℄wj[m  ℄g =0 for k 6= j;rk[ ℄ Æ— Rk(ej) for k = j:(18)Specifically, power spectrumRww(z) is diagonal with power spec-tral densities (PSD)Rk(ej) as defined in (18) and shown in Fig. 2(top).In order to run the SBR2 algorithm on Rww(z) requires theapplication of an arbitrary paraunitary matrix to perturb its purelydiagonal structure. Thereafter, running SBR2 results in a diag-onalisation after approximately 250 iteration, whereby a ratio of10 3 between the energy of off-diagonal and on-diagonal terms is0 1 2 3 4 501020index k10log 10 r k[0]0 1 2 3 4 501020index k10log 10 γ k[0]Fig. 3. Variances of (left) uncoded noise, rk[0℄, and (right) codedsub-channels, k[0℄ =12R20 k(ej)d.reached. The PSDs of the main diagonal elements,  k(ej), aredepicted in Fig. 2(bottom). Except for a low-power region of thebands  4(ej) and  5(ej) near  = , spectral majorisation hasbeen achieved in the sense of (12).Integrating over the PSDs in Fig. 2 provides the noise vari-ance of the various sub-channels, which are illustrated in Fig. 3for Rww(z) and  (z) without and with coding, respectively. Thecoder would then be utilise the N coded sub-channels representedin (z) that carry the lowest noise power. These N coded sub-channels then convey the N polyphase components of the trans-mitted signal X(z), which according to Fig. 3 are subject to differ-ent levels of noise. Note that the polyphase component transmit-ted over the lowest sub-channel provides the best protection againnoise, while noise introduced on higher sub-channels increases inpower. This fact can be exploited for unequal error protection for,for example, high quality high-speed video transmission.If a decimation factor of N = 2 is chosen for the filter banks,only the two coded sub-channels with the lowest noise variancein Fig. 3(right) will be utilised. The reduction in noise power re-sults in an SNR enhancement of the coded scheme with respect toa transmission scenario of identical symbol throughput based onmaximum ratio combining of the K = 6 channels in Fig. 3(left)of 7.5 dB. Note that a maximum ratio combiner uses a zero orderdiagonal G(z) and accordingly H(z), with the elements inverselyproportional to the noise variance in the sub-channels.Some insight into how the reduction of noise power is gainedby the proposed coding method for the case N = 2 is demon-strated in Fig. 4, where the resulting characteristics of a K = 6channel filter bank decimated by N = 2 are shown. The displayedmagnitude response of the filter bank are plotted against the PSDsof the channel noise after N = 2 fold expansion. Fig. 4 veryclearly underlines the functioning of the coder, which effectivelyexcludes the two sub-channels with high noise power from trans-mission, while in all other sub-channels the transmitted power isconcentrated in frequency bands where the noise PSD takes on itslowest values.4.2. Time-Multiplexed TransmissionThrough a time multiplexed transmission of the K encoded sym-bols over the same channel, the additive Gaussian noise w[m℄ onthe system results in polyphase components which are likely to becorrelated. Hence the covariance Rww(z) will not be of diagonalform, but can be shown to be pseudo-circulant.We assume that the innovation filter is a 24th order finite im-pulse response (FIR) system with coefficients drawn from a zeromean unit variance white Gaussian process, resulting in a channelnoise PSD of w[m℄ as shown in Fig. 5. The noise and PSD withineach of the sub-channels described byRww(z) for any given K isidentical. Therefore, different from Sec. 4.1 it is additionally theinter-channel correlation that the coder has to exploit. After ap-0 0.1 0.2 0.3 0.4 0.5−40−20020 log10 |G0(ejΩ)|noise PSD / [dB]0 0.1 0.2 0.3 0.4 0.5−40−20020 log10 |G1(ejΩ)|noise PSD / [dB]0 0.1 0.2 0.3 0.4 0.5−40−20020 log10 |G2(ejΩ)|noise PSD / [dB]0 0.1 0.2 0.3 0.4 0.5−40−20020 log10 |G3(ejΩ)|noise PSD / [dB]0 0.1 0.2 0.3 0.4 0.5−100−500normalised angular frequency Ω / 2pi20 log10 |G4(ejΩ)|noise PSD / [dB]0 0.1 0.2 0.3 0.4 0.5−100−500normalised angular frequency Ω / 2pi20 log10 |G5(ejΩ)|noise PSD / [dB]Fig. 4. PSDs of channel noise processes wk[m℄, k = 0(1)K  1, decimated by N = 2 and magnitude responses of the filtersjGk(ej)j = jHk(ej)j.plication of the SBR2 algorithm, the reduction in noise power —the ratio between the output power of the coder to the power of thechannel noise process w[m℄ — for various choices of K and Nis depicted in Fig. 6. In comparison to maximum ratio combiningwith identical symbol throughput, the proposed coder in generalperform consistently and considerably better, whereby an increasein K permits both a finer resolution to exploit spatial correlationas well as the use of more flexible code rates N=K.5. CONCLUSIONSIn this paper we have proposed a channel coding approach basedon OSFBs by first designing a decoder that minimises the influenceof correlated channel noise in the receiver, and thereafter derivesthe encoder. By demanding paraunitarity for the decoding OSFB,the latter step is trivial and ensures a strict bound on the transmit-ted power. An OSFB design method has been proposed, whichis based on a broadband eigenvalue decomposition and demon-strates good performance in suppressing the correlated channelnoise. Some insight into the effects of the design have been givenby considering transmission scenarios over K independent chan-nels or by time multiplex transmission, where the exploitation ofspatial or spectral correlations can bring substantial benefits over atransmission of identical symbol throughput using maximum ratiocombining of the sub-channels.The SNR enhancement gained from the proposed coding ar-chitecture can be utilised in conjunction with the transmission ofquantised data such as found in binary phase shift keying or multi-level quadrature amplitude modulation symbols, such that the oc-currence of symbol or bit errors is reduced.6. ACKNOWLEDGEMENTSThe author would like to express his deepest gratitude to JohnMcWhirter, Soydan Redif, and Tom Cooper of the Advanced Sig-nal Processing Group, QinetiQ Ltd, Malvern, for their help and0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−30−20−10010normalised angular frequency Ω/2piS ww(ejΩ) /[dB]Fig. 5. Channel noise PSD in time multiplex channel I.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110−1100code rate N/Kσ2 e/σ2 w K=2K=3K=4K=5K=6K=8K=10K=12K=15K=20max. ratioFig. 6. Noise reduction achieved by the proposed coding schemeover channel I characterised in Fig. 5.considerable technical input to this work.7. REFERENCES[1] F. Labeau, L. Vandendorpe, and B. Macq, “Structures, Fac-torizations, and Design Criteria for Oversampled ParaunitaryFilterbanks Yielding Linear Phase Filters,” IEEE Trans. SP,48(11):3062–3071, 2000.[2] F. Labeau, “Design and Implementation Issues in Over-sampled Filter Banks,” in Asilomar Conf. Signals, Systems,Comp., Pacific Grove, CA, 2002, 1:328–332.[3] F. 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Weiss, S

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