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 cancel- lation [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 desir- able, 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 chan- nel 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 complex modulation 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; how- ever, 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 fre- quency 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 intro- duced in a critically sampled SSB system with overlap- Reconstructed signal ped channel responses. --- In this paper a new critically sampledSSB analysidsyn- thesis 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.