Analysis/Synthesis Filter Bank Design Based on Time Domain Aliasing Cancellation

IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL.ASSP-34, NO. 5, OCTOBER 1986 1153 Analysis/Synthesis Filter Bank Design Basedon Time Domain Aliasing Cancellation

JOHN P. PRINCEN, STUDENT MEMBER, IEEE, AND ALANBERNARD BRADLEY

Abstract-A single-sideband analysis/synthesis system is proposed which provides perfect reconstruction aof signal froma set of critically sampled analysis signals. The technique is developed in terms of a dTr weighted overlap-add methodof analysis/synthesis and allows overlap q-1 x(n) %nl between adjacent time windows. This implies that time domain aliasing is introduced in the analysis; however, this aliasing is cancelledin the synthesis process, and the system can provide perfect reconstruction. k=O... K-1 Achieving perfect reconstruction places constraints on the time domain window shape which are equivalent to those placed on the frequency Fig. 1. Analysidsynthesis system framework. domain shape of analysis/synthesis channels used in recently proposed ily expressed in the frequency domain. Reconstruction can critically sampled systems based on frequency domain aliasing cancellation [7], [8]. In fact, a duality exists between the new technique and be obtained if the composite analysidsynthesis channel the frequency domain techniques of [7] and 181. The proposed tech- responses overlap and add such that their sum is flat in nique is more-efficient than frequency domain designs fora given num- the frequency domain. Any frequency domain aliasing in- ber of analysis/synthesis channels, and can provide reasonably band- troduced by representing the narrow-band analysis signals limited channel responses. The technique could be particularly useful at areduced sample rate mustbe removed in the synthesis in applicationswhere critically sampled analysislsynthesis is desirable, e.g., coding. bank [2], [4]. A second description of an analysislsynthesis system is in ‘terms of a block transform approach (Fig. 2). Analysis I. INTRODUCTION involves windowing the signal and transforming the win- NALYSIS/SYNTHESIS techniques have found wide dowedsignal using a “frequency domain transform.’’ A application and areparticularly useful in speech cod- Note that the transform is restricted to be one which has ing [l]. The basic framework for an analydsynthesis an interpretation in the frequency domain [l]. This class system is shown in Fig. 1. The analysis bank ,segments includes the discrete Fourier transform (DFT), discrete the signal x(n) into a number of contiguous frequency cosinetransform (DCT),and discrete sine transform bands, or channelsignals X,(m). The synthesis bank forms (DST). The transform domain samples are the channel a replica of the original signal based on the channel sig- signals. To synthesize, the inverse transform is applied nals $,(rn), which are related to, but not always equal to, and the resulting time sequence is multiplied by a synthe- the signals X, (m).For example, implementation of a fre- sis window and overlapped and added to a buffer of pre- quency domain speech coder [l] involves coding and de- viously accumulated signal segments [2], [3], [5]. Again coding the channel signals, and this process generally in- a straightforward statementof the requirements of the troduces some distortion.An important requirement of any analysis/synthesis system,so that reconstruction of x (n)is analysis/synthesis system used in coding is that, in the possible, can be made. Reconstruction of x(n) will occur absence of any coding distortion, reconstruction of the if thecomposite analysis/synthesis window responses signal x(n) should be possible. Historically, there have overlap and add in the time domain so that the result is been two approaches to the design and implementation of flat and any time domain aliasing introduced by the fre- analysis/synthesis systems. quency domain representation is removed in the synthesis The first is based on a descriptionof the systemin terms process. of banks of bandpass filters, or modulators and low-pass It has been recognized that a formal duality exists be- filters [2]. Using this description, the designof the system tween the two descriptions when the filter bank described so that reconstruction of x (n)is achieved canbe most eas- has uniformly spaced channels and the shapeof the channel responses are derived from a single low-pass proto- Manuscript received October 30, 1984; revised March 7, 1986. type [2], 141. This duality has been discussed with partic- J. P. Princen was with the Department of Communication and Elec- ular referenceto filter banks which use a channel structure tronic Engineering, Royal Melbourne Instituteof Technology, Melbourne, Australia 3001. He is now with the Department of Electronic and Electri- based on complexmodulation and block transform ap- cal Engineering, University of Surrey, Guildford, England, GU2 5XH. proaches utilizing the DFT[ 11-[4]. It has been shown that A. B. Bradley is with the Department of Communication and Electronic the approaches arein fact mathematically equivalent. This Engineering, Royal Melbourne Institute of Technology, Melbourne, Aus- tralia 300 1. means thatfilter banks which use complex modulation can IEEE Log Number 8609632. be designed and implemented using either interpretation.

0096-3518/86/1000-1153$01.00 0 1986 IEEE 1154 TRANSACTIONSIEEE ACOUSTICS,ON SPEECH, SIGNALAND PROCESSING. VOL. ASSP-34, NO. 5, OCTOBER 1986

scribed in terms of the complex channel structure. They Analysls window produce real channel signals and can be described using Input signal , I : ;/ channel structures based on single-sideband (SSB) modu- * lation [2]. n=mM n An SSB analysis/synthesis system is shown in Fig. 3. U As will be shown, it is possible to develop a time domain /=I /=I Analysis description of the SSB system which results in a block transform implementation that is basically the same as the transform implementation of complex filter banks; however, the DFT is replaced by appropriately defined DCT and DST. Just as a duality exists between the time and frequency domain descriptions of complex analysis/syn- [ INV. TRANSFORM I thesis systems, a duality also exists between time and frequency domain descriptions of SSB systems. This means that the form of the time domain aliasing introduced in a critically sampled SSB system with overlapped windows

Overlap and Add is the same asthat of the frequency domain aliasing introduced in a critically sampled SSB system with overlap-

Reconstructed signal ped channel responses. --- In this paper a new critically sampledSSB analysidsynthesis system is described which allows overlapped win- n=rnoM n dows to be used. Overlap occurs between adjacent win- Fig. 2. A block transform description of an analysisisynthesis system. dows.Time domain aliasing distortion is introduced in the analysis; however, this distortion is cancelled in the In coding applications it is desirable that the analysis/ synthesis process. synthesis system be designed so that the overall sample Since the technique is a critically sampled SSB system, rate at the output of the analysis bank is equal to the sam- the second section of this paper defines such a system and ple rate of the input signal. In the case ofa uniform filter states the efficient weighted overlap-add (OLA) [2], [3], bank with K unique channel signals all of equal band- [5] analysis and synthesisequations. The analysis and width, each channel signal must be decimated by K. Sys- synthesis involve both sine and cosine transforms, and it tems which satisfy this condition are described as being is shown that representation and subsequent reconstruc- critically sampled. tion of a finite sequence from the appropriately defined Consider the design of critically sampled analysis/syn- sine or cosine transforms results in time domain aliasing. thesis systems using the two approaches outlined above. The form of this aliasing and the effect of the introduction For the frequency domain approach, if a system satisfies of a time offset in the modulation functions is described the overlapcondition for reconstruction, then critical in Section 111. sampling will always introduce frequency domain alias- Based on this discussion, it is shown that perfect recon- ing, except in the case where the analysis and synthesis struction by time domain aliasing cancellation is possible filters have rectangular responses of width equal to the from a critically sampled set of analysis signals. The con- channel spacing. A dual condition exists for the time do- straints on the time domain windows are emphasized, and main approach. If the time domain overlap requirement is the final section discusses the design of suitable windows. satisfied, a critically sampled system will always intro- Two examples are given. The first is similar to the win- duce time domain aliasing, exceptif the analysis and syn- dows commonly used in transform coding [l] and has a thesis windows are rectangular and their widths (in sam- small amount of overlap. The second contains maximum ples) are equal to the decimation factor. allowable overlap, and the resulting channel frequency re- It is possible, however, to develop critically sampled sponse shows that the bandwidth of such designs is close analysis/synthesis systems which have overlapped chan- to the bandwidth of designs used in subband coding of nel frequency responses and also providereconstruc- speech. tion. One such technique is known as quadrature mirror The advantage of the new technique is that critically filtering (QMF) and was originally proposed by Croisier sampled analysis/synthesis can be performed with overlap et al. [6] for the case of a two-channel system. Using the between adjacent analysis and synthesis windows. This is basic QMF principle, a number of authors have extended not possible using conventional block transform tech- the result to allow an arbitrary number of channels [7], niques and implies that narrower channel frequency re- [8]. Overlap is restricted to occur only between adjacent sponse can be obtained, while allowing critical sampling. channel filters. For these techniques, frequency domain aliasing is introduced in the channel signals; however, the 11. SSB ANALYSI~SYNTHESIS systems are designed so that this distortion is cancelled in Consider a uniformly spaced, evenly stacked, critically the synthesis filter bank. The techniques cannot. be de- sampled SSB analysis/synthesis (Fig. 3) [l], [2]. The PRINCEN ANDBRADLEY: ANALYSISiSYNTHESIS FILTERBANK DESIGN 1155

-1 cos(mn/l) K cos(wo(n+nO)i

sin(wo(n+no)) sin(mni2) -1'

0 A set of K rin(wo(ntno)) Unique channel . - signals for an eveniy stacked system . cos(mW2) K cos(wKlq(n+no))

Note: M = K/2 fol critical sampling

Fig. 3. A single-sideband analysiskynthesis system.

Magnitude Only K/2 + 1 of the K analysis channels are unique; how - ever, in the following development the system will be described as a K-channel system, and the full K channels will be retained. This is more convenient when consid- ering a block transform implementation.As shown in Fig. 4, half-width channels occur at k = 0 and k = K/2. If the full-width channels are decimated by M, then the half- width channels can be decimated by 2M. Critical sam- -n -211 /K 0 211 /K n Freq. .piing implies that the overall output sample rate equals Fig. 4. A bank of real bandpass filters. the input sample rate, so

M=-K channelresponses are the low-pass equivalent of the 2' bandpass filter bank shown in Fig. 4. The analysis channel signals are of course real, and the center frequencies From Fig. 3 and (1) the analysis signals can be written as of the K channels are

The modulation functions have been generalized to include a time shift (no).The reason for this will become apparent later. h(P - 1 + mM - n) sin - (n + no) . As shown in Fig.3, each analysis and synthesis channel ?ik ) has two paths. These paths are time multiplexed [l], [2] (3) since Both h (n) and f(n) are assumed to be finite impulse re- sponsefilters and exist only for 0 s n s P - 1. The sin E)= 0, m even delay of P - 1 has been added to make the development straightforward. cos = 0, m odd. To express the analysis as a block transform operation, E) make the substitution - Y = mM - n [2], [3], [5] and use 1156 IEEETRANSACTIONS ON ACOUSTICS.SPEECH. SIGNALAND PROCESSING. VOL. ASSP-34, NO. 5, OCTOBER 1986

(2)to give the OLA analysis equations

P- 1 * c x(mM + r)h(P - 1 - r) Substituting r = n - moM, the synthesized signal at out- r=O put block time mO can be determined as the windowed inverse transform of Xk(mO)plus overlapping terms from sin r?K (r + no + s)).2 other time segments [2],[3], [5],i.e., m At this point it is useful to introduce the notation m = -m xm(r)= x(mM + r) K- 1

-%n(r)= xm([rlK) where [rIKrepresents the residue of r modulo K. Note that

Tm(r) = xm(r), r = 0 . * * K - 1. * (r + moM + no) . (8) Using the above notation and recognizing that 27rk/K X )I To interpret the term between the braces as a block trans- mKf2 = Tmk, (4) can be reduced to

Xk(m) = (- cos (y);.$ x,@) h(P - 1 - r)

(5) It can be seen that for a given value of the block time m only one of the terms in (5) is nonzero. For m even, the channel signals Xk(m) are the appropriately defined DCT of the windowed signal segment near time n = mM, modified by (- l)mk. Similarly, form odd, the channel signals are the modified DST of the windowed sequence near n = mM. A block transform implementation of the synthesis procedure is also possible. Assuming that &(m) = Xk(m) and That is, form even,Ym(r) is the inverse DCT of the mod- using Fig. 3 and (l),the synthesis procedure can be writ- ified channel signals; while for m odd, it is the inverse ten as DST of the modified signals. The index r can be interpreted as reduced modulo K. m Using (9) or (lo), we can write

m = -m Ym ((mo - m)M + r)

m m =c -m Xk(m)sin @)f(n - mM)]. (6)

Interchanging the order of summations gives AND BRADLEY: ANALYSISISYNTHESISBRADLEY: PRINCEN AND DESIGN BANK FILTER 1157

This can be identified as the transform operation in (8), be thought of as time domain aliasing which results from andhence, from (8) and (1 1) inadequatelyan sampled frequency domain. Thetimedo- main aliasing which is introducedcan be controlled to m some extentby introducing a phase factor in the transform a,,(r) = C f((m0 - rn)~+ r) m= -m kernels. Consider first the DCT of a K-point sequence, defined ’ ym((mo- m)M + r), r = 0 M - 1. as (12) K- 1 xk = x(n) cos f* (n f no)), It can be seen that sincef(n) is finite, the summationin n=O (12) includes only a small number of terms. In this in- k=O**.K- 1, (15) stance we limit both the analysis and synthesis windows to be of length P where K/2 5 P 5 K. This implies that and the inverse transform, which gives a distorted replica overlap occurs only between adjacent time segments, and of x(n), -at most two terms in (12) are nonzero and contribute to 1 K-l the final output atgiven ablock time,i.e., R‘(n) = - c x, cos (T (n + no)). (16) K k=O Equations (15) and (16) can be recognized as the trans- r = 0 *..M - 1. (13) form operations required,when the block time is even, for the-analysis and synthesis, respectively (ignoring the Assuming that P = K, i.e., maximum overlap, makes modifications ( - l)mk>. the analysis straightforward. The results for shorter-length Substitution of (15) into (16) and use of simple trigo- windows will follow by assuming that the length K win- nometry gives dow hasan appropriate number of zero-valued coeffi- cients at either end. This means that the channel signals are the modified(X (- l)mk)DCT or DST of the sequence [see (31 K- 1

gm(r> = h(K - 1 - r) xm(r), r = 0 * * K - 1. k=O (14) Now Also, y”,(r) is the inverse DCT or DST of the modified K- 1 channel signals, Le., (-- l)”kk(,). Since (-- 1)2mk = 1, k=O ym(r)is the sequence obtained from the forward and then the reverse DCT or DSTof (14). Synthesis can be imple- n = sK, s integer mented using the block transform approach shownin Fig. = other integer n, 2. Thechannel signals are modified by (- andan inverse DCT or DST transform is applied to form the se- hence, quence y,(r). This sequence is multiplied by the synthesis f‘(n) = f(n)/2 + R(K - n - 2no)/2, 2no integer window f(n) and overlapped and added to shifted output from the previous block time. In most applications, the (1 8) analysis/synthesis procedure can be simplified by remov- where the time indexes areinterpreted as reduced modulo ing the modifications (- l)mk[3]. K. In order to determine the relationship between f(n)and Equation (18) shows that the resulting sequence is equal x(n), it is necessary to investigate the properties of a se- to the original sequence plus an aliasing term which is a quence recovered from the cosine and sine transforms of shifted replica of the original sequence, time reversed. a rea1 sequence. The relative shift between the two sequences can be controlled by the phase factor no. 111. PROPERTIESOF SINEAND COSINETRANSFORMS Similarly, for the DST let If we have a K-point real sequence, it is possible to represent it by K unique points in the frequency domain and recover the original sequence from these frequency domain points. However, if a K-point sequence is repre- sented by either the sine or cosine transforms used in the SSB analysis/synthesis procedures, fewer than K unique and frequency domain points are determined, and hence, the origipal sequence cannot be recovered from the frequency domain samples. The inverse transform yields adistorted replica of the original sequence. In fact, thedistortion can Equations (19) and (20) are the transform operations re- 1158 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND ‘SIGNAL PROCESSING, VOL. ASSP-34,NO. 5, OCTOBER 1986

quired for analysis and synthesis when the block time is The first term in (24) is the desired signal component, odd. while the second term is due to time domain aliasing. It can be shown that For reconstruction we require f’(n) = f(n)/2 - 2 (K - n - 2no)/2, 2no integer. f(r + M)&(M - 1 - r) +f(r) (2 1) *h(2M-l-r)=2, r=O*-*M-l Again, time indexes are interpreted as reduced moduloK. (254 In this case the aliasing term is equal in magnitude to and that in the cosine transform; however,it has opposite sign. It can be seen that for a critically sampledSSB analysis/ f(r) h(r + 2no - 1) - f(r + M) synthesis system, time domain aliasing is introduced. The next section shows that careful designof the windows h (n) -K(M+~+~~,-I)=o, ~=O...M-I. andf(n) and selection of a particular value of no allows (25b) this distortion to be cancelled in the OLA synthesis. Equation (25a) is the well-known condition that adjacent analysis/synthesis windows must add so that the result is Iv. PERFECT RECONSTRUCTIONFROM CRITICALLY SAMPLEDANALYSIS SIGNALS flat [2], [3]. This is the time domain dualof the condition that the frequency response of the filter bank be flat. As discussed, jjm(r)is the sequence obtained by a for- Equation (25b) must be satisfied so that time domain ward and then inverse DCT or DST of the sequence (14). aliasing introduced by the frequency domain representa- Using the reconstruction equations developed in the pre- tion is cancelled between adjacent time segments. This vious section, i.e., (18) and (21), and substituting (14), condition is the dual of the condition for adjacent band jjmo(r)can be expressed in termsof the windowed se- frequency domain aliasing cancellation in the frequency quence as domain approaches [7], [8]. To satisfy both these conditions, choose jj%(r) = gm0(r)/2+ (- l)mogm,(K - Y - 2no)/2. (22) f(r) = h (r). The recovered sequence is the sum of the original windowed sequence and an alias term. The sign of the alias Note that term is dependent on the block time mo (whether a DCT h(r) = K(Y), Y = 0 * * . K - 1, or DST is applied). For the case where mo is even, (22) and (14) can be and if used in (13) to give the recovered signal segment at n = .h(K- 1 -r)=h(r), r=O.-.K- 1, QM in terms of the input signal. then (25a) becomes imo(r)= f(r + M) {K(K - 1 - r - M) f2(r + M) + f2(r) = 2, r = 0 - - M - 1 (26a) * 2%-1(r + M) - K(r + M + 2no - 1) and (25b) becomes

* 3%- 1(K - r - 2no)}/2 f(r) f(-2M - 1 + Y + 2no) - f(r + M)J;(-M - 1 + r + 2no) = 0, r=O*..M- 1. (26b) + K (r + 2no - 1) Zm0(K- r - 2n0)}/2. If we let (23) 2no = M + 1, (27) Noting that M = K/2 and then

Tm0-](r+ M) = i&,(r), r = 0 + - * M - 1, f(--~-1+r+2%)=f(r), ~=o.-.M-I, (23) becomes and

imo(r)= Zm(r) (f(r + M) K(M - 1 - r) f(-2M - 1 + I- + 2no) = f(r + M),

+ h(2M - r - 1)f(r)}/2 r=O-*.M- 1, + fm(K - r -2no) (f(r) h(r + 2no - 1) satisfying (26b). A similar development is possible for the casewhen mo - f(r + M) K(M + r + 2no - 1)}/2, is odd. Equation (25a) would remain the same, and the signs on the aliasing components of (25b) would be in- r=O..+M- 1. (24) verted. PRINCEN AND BRADLEY:ANALYSISISYNTHESIS FILTER BANKDESIGN 1159

cally sampled analysis signals by applying time domain constraints. The technique is the time domain dual of the frequency domain design techniques based on cancellation of frequency domain aliasing between adjacent channels. The analysis bank described is even and providesK/ 2 + 1 unique outputs which are critically sampled.

Forwardllnverie Forwardllnverre cosine transform sine transform V. IMPLEMENTATIONAND WINDOW DESIGN Implementation of thetechnique can be efficiently v v achieved using the OLA structure shown in Fig. 2. Anal- ysis involves windowing and then a DCT or DST as defined in Section 11. Synthesis requires an inverse DCT or X X DST, windowing the recoveredsequence, and overlap-

Synthesis window ping and adding to a buffer. The data in the input and output buffers are then shifted relative to the analysis and synthesis windows, and the process is repeated for the U next block time. In terms of efficiency, the method has a characteristic common to most time domaindesigns in that it is more efficient for a given number of bands than cor- respondingfrequency domain designs. Forexample, a critically sampled design based on the methods described by Rothweiler .[7] and Chu [SI, which has greater than 40 dB stopband attenuation and less than 0.2 dB of passband ripple, would require an 80-sample window [8] to implement a 32-band system (16 or 17 unique bands). The new n = rnoM " technique, with maximum overlap, uses 32-sample win- Fig. 5. The mechanism of aliasing cancellation. dows.The transform operations required are similar in both cases. It should be noted that theadjacent band over- The mechanism for time domain aliasing cancellation lap conditions, which are a requirement for aliasing can- can be made obvious by the useof some simple diagrams, cellation in the frequency domain techniques, can onlybe shown in Fig. 5. Assume that the input signal is flat in the approximatedusing finite impulseresponse filters. One time domain. Windowing results in a finite signal. Con- could, of course, use shorter-length filters to implement sider first that the block time m is even. The recovered the frequencydomain techniques; however, this would sequence after a forward and inverse DCT contains alias- increase the residual frequency domain aliasing and mag- ing distortion.The dashed line represents the aliased nitude distortion, which exists even in the absence of cod- terms, which arereversed in time. Notethat the alias terms ing. The timedomain designs presented here haverestric- are shown separated, but of course the actual signal is the tions on the window length and shape which can be met sum of the original windowed sequence and the aliased exactly and, in the absence of coding, will give perfect sequence. The sequence can be interpreted asperiodic reconstruction. The frequency domain designs presented with period K. The synthesis window extracts a portion in [7] and [8] are more efficient than the designs presented of the sequence of length K. In the next block time, the here for a given channel bandwidth. For example, the de- window is shifted by M = Kt2 and a forward and an in- signs presented here do not closely satisfy the adjacent verse DST are performed on the windowed sequence and band overlap condition required in the frequency domain give the periodic sequence shown. Note that the time-re- designs. versed aliased sequence has opposite sign to that in the The proposed technique is, in some ways,more flexible previous block time. The synthesis window extracts a K- than existing transform-typeanalysis/synthesis systems point sequence, which is added to the sequence obtained used in coding schemes. Using overlapped windows with in theprevious block time.The aliasing terms at the upper conventional techniques implies that the systems must be edge of the segment from block timem, and at the lower noncritically sampled. This generally means that the win- edge of the segment from block timem + 1, are equal in dows used in transform coders are limited to have a small magnitude but have opposite sign and will cancel when amount of overlap, and hence, the systems have a neces- the time segments are overlapped and added. These dia- sarily broad 'channel frequency response. Using the tech- grams may be compared to those in [8] for even and odd nique presented in this paper, significant overlap can be channel frequency responses of the dual frequency do- introduced,allowing narrower channel frequency remain design. sponseand critical sampling tobe achieved simulta- Hence, it is possibleto design an analysis/synthesis neously. system based on an SSB structure which allows perfect Two possible window designs are shownin Fig. 6, and reconstruction of the original signal from a set of criti- the coefficient values are given in Table I. Both are for 1160 IEEETRANSACTIONS ON ACOUSTICS. SPEECH,AND SIGNAL PROCESSING, VOL. ASSP-34, NO. 5, OCTOBER 1986

1.6

W 0.8 20 I 'i.

0 0 : TIME 1 .I 0 -n -n -3n n 4 2 4 : FREQUENCY (a) 'I \ \ 1. I 0 c 16 32

TIME Fig. 6. (a) A 20-point window for a 32-band system. (b) A 32-point window for a 32-band system.

TABLE I WINDOWCOEFFICIENTS FOR Two POSSIBLE DESIGNSWHICH GIVEPERFECT RECONSTRUCTIONIN A 32-BAND SYSTEM

0 -n -7l -3n n 4 2 4 WINDOW IESIGNS r= FREQUENCY [w) r DESIGN 1 DESIGN2 - (b) 0 0 0.18332 Fig. 7. (a) The frequency responseof an analysis (synthesis) channel using 1 0 0.26722 the 20-point window. (b) The frequency response for the 32-point win- 2 0 0.36390 dow. 3 0 0.47162 4 0 0.58798 5 0 0.70847 32-bandL' evenly stacked (i.e., 17 unique bands) systems. 6 0.50000 0.82932 The designs represent two extremes. The first is similar 7 0.86603 0.94533 to designs used in current transform coders andgives only 8 1.11803 1.05202 a small amount of overlap between adjacent windows. The 9 1.32287 1.14558 correspondingchannel response is shown in Fig.7(a). 10 1.41421 1.22396 Fig. 6(b) shows a design which exhibits maximum over- 11 1 ,41421 1.28610 12 1 .41421 1.33307 lap. This design represents an analysis/synthesis system 13 1.41421 1.36655 with properties in between those currently used in sub- 14 1.41421 1 .38866 band coding and those used in transform coders. The fre- 15 1.41421 1.40212 quency response is shown in Fig. 7(b). It is not suggested that these window designs are opti- Windowssymmetric, ie, h(K-I-r) = h(r) mum in any sense for use in coding systems. An investi- - gation of the performance of the new technique in a cod- PR INCEN AND BRADLEY: ANALYSISISYNTHESIS FILTER BANK DESIGN BANK ANALYSISISYNTHESISFILTER BRADLEY: AND PRINCEN 1161

ing systemand appropriate window designs forthese [3] R. E. Crochiere, “A weighted overlap-add method of short time Fou- rieranalysis/synthesis,” ZEEE Trans. Acoust., Speech, Signal Pro- systems is a subject for future research. We can say, how- cessing, vol. ASSP-28, pp. 99-102, Feb. 1980. ever, that the techniquepresented allows a number of de- [4] J. B. Allen and L. R. Rabiner, “A unified approach to short-time Fou- sign choices not possible with other critically sampled rieranalysis and synthesis,” Proc. IEEE, vol. 65,pp. 1558-1564, Nov.1977. analysidsynthesis techniques. [5] M. R. Portnoff, “Implementation of the digital phase vocoder using the fast Fourier transform,” IEEE Trans. Acoust., Speech, Signal Pro- VI. CONCLUSION cessing, vol. ASSP-24, pp. 243-248, June 1976. [6] A. Croisier et al., “Perfect channel splitting by use of interpolation/ In this papera new critically sampledSSB analydsyn- decimationltree decomposition techniques,” presented at the Int.’Conf. thesis system which provides perfect reconstruction has Inform. Sci./Syst., Patras, Greece, Aug. 1976. been developed. The technique allows overlap between [7] J. Rothweiler, “Polyphase quadrature filters-A new subband coding technique,” presented at the ICASSP, Boston, MA, 1983. adjacent analysis and synthesis windows, and hence, time [8] P. L. Chu, “Quadrature mirror filter design for an arbitrary numberof domain aliasing is introduced in the analysis. However, equal bandwidth channels,” IEEE Trcms. Acoust., Speech,Signal Pro- this aliasing is cancelled in thesynthesis process. Achiev- cessing, vol. ASSP-33, pp. 203-218, Feb. 1985. ing aliasing cancellation places time domain constraints on the analysis and synthesis windows. The proposed system was developed using the OLA method of analysis/ synthesis, which also provides an efficient implementa- John P. Princen (S’83) was born in Melbourne, Australia, on September 12, 1961. Hereceived the tion. The technique is the dual ofrecently developed crit- B.Eng.degree in communications engineering ically sampled frequency domain designs based on alias- with distinction from the Royal Melbourne Insti- ing canceilation.It adds considerable flexibility in the tute of Technology (R.M.I.T.), Melbourne, in 1984 and has completed the M.Eng. program in design of transform coders, can provide reasonably band- thearea of analogspeech scrambling, also at limited channel responses, and is more efficient than cor- R.M.I.T. responding frequency domain designs for a given number Hisresearch interests include digital signal of bands. processing and digital communciations. ACKNOWLEDGMENT The authors wouldlike to thank A. Johnson for his helpful comments, and also the reviewers, whose sugges- Alan Bernard Bradley was born in Melbourne, Australia, in 1946. He received the B.E. degree tions contributed significantly to the quality of the final in electrical engineering from Melbourne Univer- manuscript. sity in 1968 and the M.E.Sc. degree in electrical engineering from Monash University in 1972. REFERENCES Since1973 he hasbeen working as Lecturer and Senior Lecturer in the Department of Com- [I] J. M.Tribolet and R. E. Crochiere, “Frequencydomain coding of municationand Electronic Engineering at the speech,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP: Royal Melbourne Institute of Technology, Mel- 27, pp. 512-533, Oct. 1979. bourne. His research interests are in the field of [2] R. E. Crochiere and L. R. Rabiner, Multirate Digital Signal Process- digital signal processing, with particular emphasis ing. EnglewoodCliffs, NJ: Prentice-Hall,1983. on speech analysis and coding.