Memory ecient adaptation of vector quantizers to time-varying channels

Transcription

1 Sgnal Processng 83 (3) Memory ecent adaptaton of vector quantzers to tme-varyng channels orbert Gortz a;,jorg Klewer b a Insttute for Communcatons Engneerng (LT), Munch Unversty of Technology (TUM), 89 Munch, Germany b Insttute for Crcuts and Systems Theory, Unversty of Kel, Kaserstr., 443 Kel, Germany Receved 9 September ; receved n revsed form 6 February 3 Abstract Channel-optmzed vector quantzaton (COVQ) s approxmated by the novel channel-adaptve scaled vector quantzaton (CASVQ). Ths new method uses a reference codebook that s optmal for one specc channel condton. However, for a bt-error rate beng derent from the desgn assumpton for the reference codebook, all codevectors are scaled by a common factor, whch depends on the channel condton. It s shown by smulatons that a performance close to that of COVQ can be acheved n many practcally mportant stuatons. Wthout a sgncant ncrease n complexty, the new CASVQ-scheme can be adapted to tme-varyng channels by adjustng the scalng factor to the current bt-error probablty. Another advantage s that only one codebook needs to be stored for all error probabltes, whle for COVQ ether the performance degrades sgncantly due to channel msmatch, or a large set of codebooks must be avalable at the encoder and the decoder.? 3 Elsever Scence B.V. All rghts reserved. Keywords: Jont source-channel codng; Channel-optmzed vector quantzaton; Tme-varyng channels; Complexty and memory requrements. Introducton Channel-optmzed vector quantzaton (COVQ) [4,5] acheves strong qualty-mprovements over conventonal vector quantzaton (VQ) f the transmsson channel s nosy. Varatons of COVQ n whch smulated and determnstc annealng are used have been proposed, e.g., n [6,]; the algorthms work Ths paper was presented n part at the IEEE Internatonal Symposum on Informaton Theory, Washngton, DC, USA, June. Correspondng author. Tel.: ; fax: E-mal address: norbert.goertz@e.tum.de (. Gortz). superor to normal COVQ due to mprovements n the codebook desgn that avod gettng stuck n poor local optma. Although a great deal of the work on COVQ has been done for the bnary symmetrc channel, the codebook tranng and the desgn of optmal encoder-decoder pars for soft source decodng after the transmsson wth soft-decson demodulaton have also been consdered, e.g., n [,]. Recently, the adaptaton of COVQ to tme-varyng channels has become a subject of nterest [8], where COVQ desgn approaches are proposed for derent types of mssng channel nformaton. In ths paper a new memory-ecent COVQ approxmaton for tme-varyng channels s presented. More precsely, we address the problem how to /3/$ - see front matter? 3 Elsever Scence B.V. All rghts reserved. do:.6/s65-684(3)7-9

2 5. Gortz, J. Klewer / Sgnal Processng 83 (3) lmt the memory and complexty requrements n a COVQ-framework such that these are almost the same as n the conventonal VQ case, whle keepng good performance for all (tme-varyng) channel condtons. It s shown that ths goal can be acheved by smply scalng a COVQ reference codebook dependng on the actual channel bt error rate, whch leads to a strong reducton of the overall memory and complexty requrements compared to the COVQ approach. A closely related method has been stated n [], where scalng s ntroduced to decrease the dstances between the codewords and thus to allevate the qualty decrease at the decoder output due to ndex permutatons caused by bt errors. In our work, however, we employ codevector-scalng to obtan a memory ecent COVQ codebook approxmaton whch s desgned for a bt-error probablty beng derent from that used for the reference codebook. Ths paper s organzed as follows. After a dscusson of complexty and memory ssues for VQ and COVQ n Secton, the new approxmaton of COVQ called channel-adaptve scaled vector quantzaton (CASVQ) s ntroduced n Secton 3. In Secton 4 the performance of the new scheme s compared wth COVQ.. Memory and complexty ssues for VQ and COVQ Fg. shows the model of the used transmsson system. It conssts of a channel-optmzed vector quantzer wth the codebook C, a bnary symmetrc transmsson channel that causes ndex-permutatons by bt errors, and a decoder that essentally performs a table-lookup. The bt-error probablty s assumed to be known at both the encoder and the decoder. The dea of channel-optmzed vector quantzaton (COVQ) s to explot the knowledge about the channel n the desgn of the codebook and n the encodng algorthm. The COVQ codebook desgn algorthm [4] can be regarded as an extenson of the classcal LBG-algorthm [9] for noseless channels. In the followng, we use the performance of COVQ [4] asa reference for the approxmatons that are nvestgated n ths paper. ote that the COVQ dstance measure, whch s used n both the codebook desgn procedure and the encodng operaton, s derent from that n x COVQ Encoder Codebook C b Bts Channel (BSC) j b Bts C ={y, =,,..., b } y : codevector, dmenson Decoder Codebook C Fg.. Transmsson system wth a channel-optmzed vector quantzer. conventonal VQ. Ths s due to the fact that transtons from the transmtted ndces to some ndces j at the recever occur wth probabltes P j that depend on the channel. For nstance, f we transmt over a bnary-symmetrc channel (BSC) wth bt-error probablty, the transton probabltes are gven by P j = p h(;j) e ( ) b h(;j) ; () where b s the number of bts requred to encode the ndex and h(; j) s the Hammng dstance between the transmtted and the receved ndex,.e., h(; j) s the number of bt errors nserted by the channel. The probabltes P j are used n the COVQ-encoder to mnmze the expected dstorton at the recever output due to the quantzaton decson. The expected dstorton, the COVQ dstance measure, s gven by [4] C d covq (x; y )= P j d vq (x; y j ); () j= where x s the data-vector to be quantzed and y the codevector wth the ndex =; ;:::; C. : The number of codevectors s denoted as C = b (sze of the codebook), and d vq (x; y j ) s the dstance measure that would be used n conventonal VQ. In COVQ-encodng, () s computed for each codevector y, and the ndex opt correspondng to the codevector that produces the mnmum value d covq (x; y opt ) s selected for the transmsson over the channel. Besdes, () s also used n the codebook tranng process. Often (and therefore also n ths paper) the meansquared error d vq (x; y )= (x l y ;l ) (3) y j

3 . Gortz, J. Klewer / Sgnal Processng 83 (3) s used as the VQ dstance measure. The vector dmenson s denoted by and the vector components of y and x are ndexed by l. The computaton of (3) requres 3 oatng-pont operatons per codevector of dmenson, where oatng-pont operaton wll be used as a generc term for addton, subtracton, multplcaton, or dvson n the followng. It s a standard technque to smplfy the mplementaton by expandng the sum n (3) as follows: d vq (x; y )= xl x l y ;l + y;l = xl + d vq(x; y ): (4) The rst term s non-negatve and does not depend on the codevector. Snce, furthermore, = s a constant factor t s sucent to mnmze d vq(x; y )=w(y ) x l y ;l (5) wth w(y ) = : y;l (6) by a proper selecton of the codevector ndex. In what follows, d vq(x; y ) wll be referred to as the smpled VQ dstance measure. The (halves of the) energes w(y ) of the codevectors can be precomputed and stored n advance (snce they do not depend on the data x), so by use of C addtonal scalar memory locatons we may reduce the complexty of the codebook search to oatng-pont operatons per codevector. For COVQ, the calculaton of the expected dstorton () for each of the codevectors requres the values of the VQ dstance measure (3) for all codevectors and the correspondng ndex transton probabltes P j. Smpled mplementatons, whch are algorthmcally equvalent to the mnmzaton of (), have been stated n [5]: whle the decoder uses the COVQ-codebook drectly, the encoder uses a transformed codebook that ncludes the ndex transton probabltes. We wll brey descrbe the method here: f we nsert (3) nto () and expand the sums we obtan C d covq (x; y )= P j d vq (x; y j ) j= C = j= P j (x l y j;l ) = xl + C P j yj;l j= }{{} j= w(y ) C x l P j y j;l : (7) }{{} Thus, smlar as n the VQ-case, t s sucent to mnmze the smpled COVQ dstance measure d covq(x; y )=w(y ) x l y ;l ; (8) wth w(y ) : = and C j= P j y ; l C yj;l = j= P j w(y j ) (9) : C y ;l = P j y j;l ; () j= nstead of the mnmzaton of (). The transformed codevectors y = {y ;l ; l =;:::; } from (), whch can be nterpreted as the expectaton of the codevector at the recever condtoned on the possbly transmtted ndex, and the energes w(y ) from (9) may be stored at the encoder n place of the actual codebook, whch s only requred at the decoder. Thus, the memory requrements and the complexty of COVQ encodng by mnmzaton of (8) over

4 5. Gortz, J. Klewer / Sgnal Processng 83 (3) are not larger than n the conventonal VQ-case (where (5) s mnmzed) as long as the channel s tme-nvarant. If the channel s tme-varyng the channel-optmzed codebook mght not be matched to the current channel statstcs (channel msmatch),.e., the performance degrades, compared to the optmal case. If the assumpton of the bt-error probablty for COVQ-codebook tranng ders only slghtly from ts true value on the channel, the performance of unmatched COVQ s not sgncantly worse compared to optmally matched COVQ [4]. But f, for nstance, a COVQ-codebook desgned for a bt-error probablty of =:5 s used on an uncorrupted channel ( = ), a sgncant loss n the clean-channel performance can be observed. One way to mprove the performance n such a stuaton s to swtch between a nte number of codebooks at the encoder and the decoder dependng on the current channel state [7]; the codebooks can cover the range of possble channel varatons (although no perfect match s possble snce, n general, s a real number). The robustness of COVQ aganst small varatons of preserves close-to-optmum performance f the number of codebooks s hgh enough. However, ths strategy requres the storage of several codebooks at both the encoder and the decoder, whch may be qute memory consumng. In the next secton a new algorthm s stated that does not show ths dsadvantage by reducng the number of requred codebooks to just one for all channel condtons. 3. Channel-adaptve scaled vector quantzaton (CASVQ) 3.. Basc prncple In Fg. codebooks wth C = 3 two-dmensonal codevectors are depcted that were traned for a Gauss Markov source wth a correlaton coecent of For COVQ on very nosy channels (e.g., bt-error probabltes :5) empty encodng regons (correspondng to mplct error control codng) can be observed: certan ndexes are never selected by the encoder for transmsson,.e., the correspondng transformed codevectors do not need to be consdered n the dstance computaton (8). If many empty regons occur, ths can lead to a sgncantly lower complexty [5] compared to the standard VQ-case. =:9. Such a source model s representatve for the long-term statstcs of many source sgnals such as speech, audo, and mages but also for the parameters that are extracted from a block of nput samples by advanced source codng schemes. The expected mean-squared error () was used as a dstance measure for COVQ-codebook tranng. The plots show the codevectors (marked by ) that result from the tranng procedure for the COVQ-codebooks [5] (wth splttng for the ntalzaton) for several assumptons of the bt-error probablty on a bnary symmetrc channel (BSC). For = the codebook s equal to a conventonal VQ-codebook (Fg. (a)). Fg. shows that the codevectors are placed closer to the all-zero vector (.e. the average of all tranng-data vectors) when the channel qualty gets worse. Ths s because a permutaton of a transmtted ndex (whch, for nstance, may quantze the data pont Z ) to some other ndex j (caused by channel errors) leads to a large dstorton, f the assocated codevectors have a large dstance n the sgnal space; ths s the case for the codevectors marked by and j n Fg. (a). The dstance between the vectors ndexed by and j n Fg. (b) s smaller,.e., the dstorton caused by channel errors that permute both vectors s also smaller. On the other hand, the quantzaton of data-ponts wth large dstance from the orgn (e.g., Z ) s less accurate n Fg. (b); thus, we have a loss n clean-channel performance ( =) usng COVQ-codebooks optmzed for. In other words: a qualty decrease due to hgher quantzer overload-dstorton s traded for the qualty mprovement due to a hgher robustness aganst ndex permutatons. A precse analyss of the postons of the codevectors reveals (Fg. (c) and (d)) that the codevectors form clusters,.e., the codevectors are not only shrunken f ncreases but they also have new relatve locatons. Ths clusterng corresponds to empty codng regons that cause an mplct error control codng (redundant codng levels) as reported n [5]. However, a COVQ-codebook for = p may be vewed as a shrunken verson of COVQ-codebook It s known from lterature [3] that the splttng method n VQ-codebook tranng leads to a relatvely good ndex assgnment. In case of COVQ-codebook tranng wth the ndex assgnment s automatcally optmzed.

5 . Gortz, J. Klewer / Sgnal Processng 83 (3) y,, =,,...,3.5 pe=. α=. * Z (data pont) j j * * Z (data pont) Z (data pont) (a) y,, =,,...,3 (b) y,, =,,...,3 y,, =,,...,3.5 pe=. α=.75 y,, =,,..., pe=.5 α=.57 y,, =,,..., pe=. α=.5 (c) y, =,,...,3, (d) y, =,,...,3, Fg.. Comparson of COVQ-codebooks ( ) and CASVQ-codebooks ( ), both wth C = 3 codevectors. The codebooks n plot (a) are dentcal to a conventonal VQ-codebook desgned by the LBG algorthm [9] (wth splttng ). optmzed for p, f only the rough shape s consdered. Ths observaton leads to the basc dea of channel-adaptve scaled vector quantzaton (CASVQ): the codevectors y (r) from a reference COVQ-codebook are scaled by a channel dependent factor ( ), where ( ) f the channel does not match the tranng assumpton of the reference codebook. ote that ( ) fthe reference codebook s obtaned for a smaller BER compared to the actual one (as n the examples of Fg. ) and ( ) f a larger BER s used for the desgn of the COVQ reference codebook. Thus, the channel-matched COVQ-codevectors y ( ) are approxmated by the CASVQ-codevectors y (p e )=( ) y (r) : () For the sake of brevty we omt the dependence on for y (p e ) and ( ) n the followng. As an example, CASVQ-codevectors (marked by ) are ncluded n Fg.. They have been derved from the VQ reference codebook (COVQ-codebook wth = as the desgn assumpton) n Fg. (a) by the scalng factors stated n the legends. It s obvous that the rough shape of the COVQ-codebooks (marked by ) can be approxmated by the CASVQ-codebooks qute well. 3.. Optmzaton of CASVQ The queston arses, how to nd approprate scalng factors for each channel condton,.e., the functon = f( ) s requred. Unfortunately, there s no way to nd the optmal factor analytcally by varatonal technques, because the average dstorton for a tranng-set of source vectors depends on the ndvdual quantzaton decsons that agan depend on the scalng factor. Hence, the quantzaton decsons change, as the scalng factor s vared to mnmze the average dstorton. However, snce the functon = f( ) s ndvdual for a codebook and ts source sgnal, t may be determned n advance before the system s used. Thus, we can accept some computatonal complexty n the o-lne optmzaton of the CASVQ-codebook for a reasonable number of samples from and. As an example, let us thnk of a correlated Gaussan source sgnal, whch s vector-quantzed by a codebook wth c = 3 two-dmensonal codevectors; () s used as a dstance measure for the quantzaton

6 54. Gortz, J. Klewer / Sgnal Processng 83 (3) SR n db p. e α Fg. 3. Performance of CASVQ for codebooks wth C =3 two-dmensonal codevectors whch are used to encode a strongly correlated Gauss Markov source (correlaton coecent = :9). The SR-value s plotted versus the bt-error probablty and the scalng factor. A COVQ-codebook for = (.e., a conventonal VQ-codebook) s used as a reference codebook. and a conventonal VQ codebook s used as a reference codebook for CASVQ. Thus, the range of the scalng factors s lmted to 6 n ths case. The SR-values, measured between the nput of the quantzer and the output of a table-lookup decoder, are depcted n Fg. 3 n a three-dmensonal plot, for a smulaton of several (; )-pars. Fg. 3 shows that the SR-surface s qute smooth,.e., a lmted number of (; )-pars s sucent to acheve a good approxmaton of the best possble performance. The dashed curve on the SR-surface ndcates the best SR (and the assocated -value) for each smulated value of ; the requred functon = f( )s gven by the projecton of the dashed curve nto the -plane. The projecton of the dashed curve nto the SR- -plane ndcates the best SR-values achevable by CASVQ for each value of. Ths curve appears agan n the smulaton results n Fg. 5 (labeled CASVQ, covq-dst. ). ote that the smooth behavor of the SR surface n Fg. 3 and the functon = f( ), resp., holds for arbtrary nput source processes. Ths s due to the fact that the COVQ codevectors may be nterpreted as pre-computed mean-square estmates of the vector-quantzed transmtted source symbols. If the channel s strongly dstorted the mean-square estmate at the output of the decoder tends to zero for any zero-mean nput source, snce the COVQ code vector locatons approach the zero vector. The CASVQ approach approxmates ths behavor for large by ( ). Snce the mean-square error between the source sgnal ponts and the CASVQ codevector locatons s a contnuous functon n the SR surface and thus also = f( ) are smooth functons. By means of a smulaton as descrbed above, a close approxmaton of the functon = f( ) can be found for any practcally relevant codebook n reasonable tme Complexty and memory requrements of CASVQ on tme-varyng channels If the CASVQ-codebook s used, the smpled COVQ dstance measure (8) corresponds to d covq(x;y (r) wth C w(y (r) )= and C y (r) ;l = )=w(y (r) j= P j ) x l y (r) ;l ; () (y (r) j;l ) C = P j w(y (r) j )= w(y (r) ) (3) j= j= P j y (r) j;l C = P j y (r) j;l = y(r) ;l ; (4) j= where (9) and () have been used. ote that both and P j are functons of the current bt-error probablty, whch s dstnct from the reference bt-error probablty. Thus, () may be wrtten as d covq(x;y (r) )= w(y (r) ) x l y (r) ;l : (5) As we can observe from (5) t s possble to adapt CASVQ to any value of by use of an approprate

7 . Gortz, J. Klewer / Sgnal Processng 83 (3) Reference Codebook C (r) C (r) ={y (r), =,,..., b } y (r) : reference codevector Reference Codebook C (r) x x α VQ Encoder b Bts Channel (BSC) j b Bts Decoder y (r) j α y (r) j α( ) α ( ) Fg. 4. CASVQ system wth VQ dstance measure for encodng. factor and only one reference codebook for calculatng the transformed reference codevectors y (r) and the energes w(y (r) )n(5). Whenever changes, y (r) and w(y (r) ) have to be recomputed from the reference codebook by applyng (9) and (), respectvely. Thus, the proposed CASVQ method wth COVQ dstance measure for encodng, whch wll be denoted wth CASVQ, covq-dst. n the followng, s more complex than conventonal VQ. Furthermore, memory for two codebooks, the reference codebook y (r) and the current transformed codebook y (r) used for encodng, s requred at the encoder. ote that the energes w(y (r) ) only need to be stored for the current bt-error probablty, so here, no addtonal memory s requred for tme-varant channels. In contrast, for perfectly channel-matched COVQ a new codebook has to be traned and stored n both the encoder and the decoder for each value of. In order to save the complexty for the recomputatons (due to a change of ) and the memory for the second codebook n the CASVQ approach, one may use the conventonal VQ dstance measure nstead of the COVQ dstance measure for encodng. Certanly, ths s another approxmaton that wll decrease the performance compared to the optmal COVQ; however, the degradaton s only moderate but the smplcaton of the encodng s sgncant as we wll see n the followng. From (5) and (6) we obtan d vq(x;y (r) )= (y (r) ;l ) } {{ } w(y (r) ) x l y (r) ;l ( ) = w(y (r) x l ) y(r) ;l : (6) Snce n (6) s equal for each codevector, t s sucent to mnmze vq(x;y (r) ) = : ( w(y (r) xl ) ) y (r) ;l (7) d over n place of (6). ow, all we have to do addtonally (compared wth conventonal VQ) s to scale the components of the nput data vector x by = and to rescale the output codevector y (r) j by ; a small table to represent the functon = f( ) s also requred. The rest s the same as n conventonal VQ ncludng the memory and complexty requrements. The CASVQ-system wth the VQ-dstance measure for encodng s depcted n Fg. 4. If ths system s used, the correspondng smulaton results are labeled CASVQ, vq-dst.. However, the smulaton results n Secton 4 show that the qualty decrease obtaned by replacng the COVQ dstance measure wth the smple VQ dstance measure s only small. ote that a change of due to varyng channel statstcs aects both the encoder and decoder codebook. Therefore, we assume that a backward channel s avalable (whch holds true for many wreless and wrelne communcaton systems) such that the estmated bt-error rate at the decoder can be communcated to the encoder. 4. Smulaton results In Fg. 5 the performances of COVQ and CASVQ for the transmsson of a strongly autocorrelated

8 56. Gortz, J. Klewer / Sgnal Processng 83 (3) COVQ, covq dst. CASVQ, covq dst. CASVQ, vq dst. VQ, vq dst. 8 6 COVQ, covq dst CASVQ, covq dst CASVQ, vq dst VQ, vq dst SR n db 8 6 SR n db Fg. 5. Performance of COVQ, CASVQ, and VQ for the quantzaton of a strongly correlated Gauss Markov source; codebooks wth C = 3 two-dmensonal codevectors are used. Fg. 6. Performance of COVQ, CASVQ, and VQ for the quantzaton of a strongly correlated Gauss Markov source; codebooks wth C = 8 three-dmensonal codevectors are used. Gauss Markov source sgnal (correlaton coecent =:9) over a bnary symmetrc channel are depcted; the quantzatons are carred out by codebooks wth 3 two-dmensonal codevectors. The COVQ-codebooks are optmally matched to the true value of on the channel; the scalng factor ( )of the CASVQ-codebooks s adapted to as descrbed n Secton 3. As expected, the COVQ codebook wth COVQdstance measure for encodng (curve labeled COVQ, covq-dst. ) works best of all, but there s only a moderate loss for CASVQ, covq-dst.. Moreover, the performance of CASVQ wth the conventonal VQ dstance measure for encodng ( CASVQ, vq-dst. ) s only slghtly nferor to CASVQ, covq-dst.,.e., the better dstance measure does not sgncantly mprove the performance of CASVQ. The results for conventonal VQ wth VQ dstance measure for encodng ( VQ, vq-dst. ) have also been ncluded n Fg. 5 to evaluate the derence between COVQ and CASVQ: when a moderate loss n performance compared to optmally channel-matched COVQ can be accepted, t s possble to apply a CASVQ-codebook wth the smple VQ dstance measure for encodng. Ths allows a very smple and memory-ecent adaptaton of the codng scheme to tme-varyng channels whle keepng most of the performance-gan of optmally matched COVQ. SR n db COVQ, covq dst. CASVQ, covq dst. CASVQ, vq dst. VQ, vq dst Fg. 7. Performance of COVQ, CASVQ, and VQ for the quantzaton of an uncorrelated Gaussan source; codebooks wth C =3 two-dmensonal codevectors are used. Qualtatvely, the same mplcatons hold for the second smulaton, whch agan was carred out for the correlated Gaussan source sgnal, but wth codebooks contanng 8 three-dmensonal codevectors. The results are depcted n Fg. 6. The performances for an uncorrelated Gaussan source sgnal quantzed wth 3 two-dmensonal codevectors are shown n Fg. 7. As above, CASVQ wth the conventonal VQ dstance measure performs close to COVQ.

9 . Gortz, J. Klewer / Sgnal Processng 83 (3) =.5, C =64 codevectors 9 =.5, vector dmenson = SR n db SR n db COVQ, corr. CASVQ, corr. COVQ, uncorr. CASVQ, uncorr. VQ, corr. VQ, uncorr (vector dmenson) 3 COVQ, corr. CASVQ, corr. COVQ, uncorr. CASVQ, uncorr. VQ, corr. VQ, uncorr b (number of quantzer bts) Fg. 8. Performance of COVQ, CASVQ (wth VQ dstance measure), and conventonal VQ for the quantzaton of strongly correlated and uncorrelated Gaussan sources for varous codevector dmensons ; codebooks wth C = 64 codevectors are used; the bt-error probablty s =:5. Fg. 9. Performance of COVQ, CASVQ (wth VQ dstance measure), and conventonal VQ for the quantzaton of strongly correlated and uncorrelated Gaussan sources for varous codebook szes C = b; codevectors wth dmenson = are used; the bt-error probablty s =:5. Fg. 8 shows the performance dependences of COVQ and CASVQ on the codevector dmenson for a xed number C = 64 of codevectors and a bt-error probablty of =:5. Snce s recprocally proportonal to the source codng rate t s clear that the SR s essentally decreasng for ncreasng. However, the largest gan of COVQ and CASVQ over conventonal VQ s obtaned for small vector dmensons. For strongly correlated ( = :9) source sgnals (where the use of vector quantzaton really pays o) the gan of CASVQ over VQ remans sgncant also for moderate vector dmensons. For uncorrelated source sgnals and hgh vector dmensons there s no sgncant gan by CASVQ over VQ, but n ths case even COVQ works only moderately better than conventonal VQ. In Fg. 9, n contrast to the prevous smulaton, the vector dmenson s xed ( =) but now the number b of quantzer ndex bts (and thus the codebook sze C = b ) s varable, agan for a bt-error probablty of =:5. For both strongly correlated (=:9) and uncorrelated sources the gans of COVQ and CASVQ over VQ ncrease wth the codebook sze. In summary, CASVQ works sgncantly better than conventonal VQ, especally f the source sgnal s correlated and the bt-rate s hgh or, equvalently, when the vector dmenson s small and the number of codevectors s large. The loss of CASVQ compared to COVQ gets larger wth an ncreasng number of codevectors, but at the same tme the gan of CASVQ over conventonal VQ ncreases. 5. Conclusons As a new result we have proposed channel-adaptve scaled vector quantzaton (CASVQ) as a substtute for channel-optmzed vector quantzaton. The advantage of CASVQ s that on tme-varyng channels the memory- and complexty requrements are practcally the same as for conventonal VQ, but, especally for correlated source sgnals, the performance s close to that of optmally channel-matched COVQ, whch would requre to store several codebooks for the adaptaton to tme-varyng channels. The CASVQ codebook s generated by scalng all the codevectors n a reference codebook wth a channel-dependent factor, whch approxmates the shape of the COVQ codebook for the current bt-error probablty. If, addtonally, the conventonal VQ dstance measure s used for encodng, the scalng may be moved from the codevectors to the nput source sgnal, yeldng a further complexty reducton as the normal VQ encodng algorthm can be used

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