T. Saramäki and R. Bregovi, ultirate Systems and Filter Banks, Chater in ultirate Systems: Design T. Saramäki and R. Bregovi, ultirate Systems and Filter Banks, Chater in ultirate Systems: Design ULTIRATE SYSTES AND FILTER BANKS Taio Saramäki and Robert Bregovi Signal Processing Laboratory, Tamere University of Technology P.O. Box 553, FIN-33 Tamere, Finland v [n] H () v [n] H () v [n] H () v -- [n] H -- () Analysis bank Processing Unit w [n] F () w [n] F () w [n] F () w -- [n] F -- () Synthesis bank Introduction During the last two decades, multirate filter banks have found various alications in many different areas, such as seech coding, scrambling, adative signal rocessing, image comression, signal and image rocessing alications as well as transmission of several signals through the same channel (alvar, 99a; Vaidyanathan, 993; Vetterli amd Kova evi, 995; Fliege, 994; isiti, isiti, Oenheim, and Poggi, 996). The main idea of using multirate filter banks is the ability of the system to searate in the frequency domain the signal under consideration into two or more signals or to comose two or more different signals into a single signal. When slitting the signal into two or more signals an analysis-synthesis system is used. The analysis-synthesis systems under consideration in this chater are critically samled multichannel or -channel uniform filter banks and octave filter banks as shown in Figures and, resectively. In the analysis bank of the uniform bank, the signal is slit with the aid of filters H k () for k =,,, into bands of the same bandwidth and each sub-signal is decimated by a factor of. In the case of octave filter banks, the overall signal is first slit into two bands of the same bandwidth and both sub-signals are decimated by a factor of two. After that, the decimated lowass filtered signal is slit into two bands and so on. Doing this three times gives rise to a three-level octave filter bank corresonding to the structure shown in Figure. In this case, H () is a highass filter with bandwidth equal to half the baseband and the decimation factor is, H () and H () are bandass filters with bandwidths equal to one fourth and one eighth of the baseband, resectively, and the corresonding decimation factors are 4 and, whereas H 3 () is a lowass filter with the same bandwidth and decimation factor as H (). In many alications, the rocessing unit corresonds to storing the signal into the memory or transferring it through the channel. The main goal is to significantly reduce, with the aid of roer coding schemes, the number of bits reresenting the original signal for storing or transferring uroses. When slitting the signal into various frequency bands with the aid of the analysis filter bank, the signal characteristics are different in each band and various numbers of bits can be used for coding and decoding the sub-signals. In some alications, the rocessing unit is used for treating the sub-signal in order to obtain the desired oeration for the outut signal of the overall system. A tyical examle is the use of the overall system for making adative signal rocessing more efficient. Another examle is the de-noising of a signal erformed with the aid of a secial octave filter bank, called a discrete-time wavelet bank (Vetterli and Kova evi, 995; isiti et al., 996). H F / H 3 F 3 / f s /() H () H () H () f s /6 H 3 () H F / 4 Analysis bank H F / f s / f s /() v [n] v [n] v [n] v 3 [n] H F / H F /4 H -- F -- / H -- F -- / 3f s /() (--)f s /() (--)f s /() Processing Unit w [n] w [n] w [n] w 3 [n] f s /4 4 F () F () F () F 3 () Synthesis bank H F / f s / Figure. Analysis-synthesis filter bank. -channel uniform filter bank. Three-level octave filter bank. Note that in the case of interolation by a given factor, the corresonding filter should aroximate this factor in the assband in order to reserve the signal energy. The role of the filters in the synthesis art is to aroximately reconstruct the original signal. This is erformed in two stes. First, for the uniform filter bank, the sub-signals at the outut of the rocessing unit are interolated by a factor of and filtered by synthesis filters F k () for k =,,,, whereas for the octave filter bank, the interolation factors for the subsignals are the same as for the analysis art. Second, the oututs of these filters are added. In the transferring and storing alications, the ultimate goal is to design the overall system such that, desite of a significantly reduced number of bits used in the rocessing unit, the reconstructed signal is either a delayed version of the original signal or suffers from a negligible lost of information carried by the sub-signals. There are two tyes of coding techniques, namely, lossy and lossless codings. For the lossless coding, it is desired to design the overall system such that the outut signal is si mly a delayed version of the inut signal or suffers from some hase distortion being tolerable in some alications. For the lossy coding, it is beneficial to design the analysis-synthesis filter bank such that some distortions, including amlitude distortion and aliasing errors, being less than those caused by coding distortions are allowed. This increases the overall filter bank erformance or, alternatively, the same erformance can be achieved by shorter filter orders or a shorter delay caused for the inut signal by the overall filter bank. These facts are very crucial for seech, f s /
T. Saramäki and R. Bregovi, ultirate Systems and Filter Banks, Chater in ultirate Systems: Design 3 T. Saramäki and R. Bregovi, ultirate Systems and Filter Banks, Chater in ultirate Systems: Design 4 audio, and communication alications. The coding techniques are not considered at all in this chater and we concentrate on the case where the rocessing unit does not cause any errors to the sub-signals. In the case of audio or seech signals, the goal is to design the overall system together with coding such that our ears are not able to notice the errors caused by reducing the number of bits used for storing or transferring uroses. In the case of images our eyes serve as referees, that is, the urose is to reduce the number of bits to reresent the image to the limit that is still satisfactory to our eyes. Deending on how many channels are used for the signal searation, there are two grous of uniform filter banks, namely, multi-channel or -channel filter banks ( > ) and two-channel filter banks ( = ). In the first grou, the signal is searated into different channels and in the second grou into two channels. Using a tree-structure, two-channel filter banks can be used for building -channel filter banks in the case where is a ower of two. A more effective way of building -channel filter banks is to first design a rototye filter in a roer manner. The filters in the analysis and synthesis banks are then generated with the aid of this rototye filter by using a cosine-modulation or a modified discrete Fourier transform (DFT) technique (alvar, 99b; Vaidyanathan, 993; Fliege, 993; Heller, Kar, and Nguyen 999; Kar, ertins, and Schuller ). Two-channel filter banks are very useful in generating octave filter banks. In this case, the overall signal is first slit with the aid of a two-channel filter bank into two bands. After that, the decimated lowass filtered signal is slit into two bands using the same two-channel filter bank and so on. There are two basic tyes of octave filter banks, namely, frequency-selective filter banks mostly used for audio and telecommunications alications and discrete-time wavelet banks used in alications where the signal waveform is desired to be reserved, like in the case of images. For discrete-time wavelet banks, the frequency selectivity of the filters in the octave analysis-synthesis filter banks is not so imortant due to their different alications. There are other roerties that are more imortant, as will be discussed in Subsection 4. and in more details in Chater 3. In these cases, the main goal is to reserve the waveform of the inut signal after treating it in an aroriate manner in the rocessing unit. When two or more different signals are comosed into a single signal, then a uniform synthesis-analysis system is used, as shown in Figure. This system is also called a transmultilexer. In this system, all the signals are interolated by a factor of and filtered by synthesis filters F k () for k =,,,. Then, the oututs are added to give a single signal with samling rate being times that of the inut signals. The next ste is to transfer the signal through a channel. Finally, in the analysis bank the original signals are reconstructed with the aid of analysis filters H k () for k =,,,. These signals have the original samling rates due to the decimation by a factor of. If the outut signal in the analysis-synthesis system is just a delayed version of the inut signal, then for the corresonding transmultilexer the outut signals in the case of the ideal channel are delayed versions of the inuts. Therefore, the design of a transmultilexer can be converted to the design of an analysis-synthesis filter bank. Based on this fact, this chater does not consider the design of transmultilexers. An interested reader is referred to the textbook written by Fliege (993). x [n] x [n] x [n] x -- [n] F () F () F () F -- () Synthesis bank Channel H () H () H () H -- () y [n] y [n} y [n] y -- [n] Analysis bank Figure. Synthesis-analysis filter bank: Transmultilexer. The outline of this chater is as follows. Section reviews various tyes of existing finite imulse resonse (FIR) and infinite imulse resonse (IIR) two-channel filter banks. The basic oerations of these filter banks are considered and the requirements are stated for alias-free, erfect-reconstruction (PR), and nearly erfect-reconstruction (NPR) filter banks. Also some efficient synthesis techniques are referred to. Furthermore, examles are included to comare various two-channel filter banks with each other. Section 3 concentrates on the design of multichannel (-channel) uniform filter banks. The main emhasis is laid on designing these banks using tree-structured filter banks with the aid of two-channel filter banks and on generating the overall bank with the aid of a single rototye filter and a roer cosine-modulation or DFT technique. In Section 4, it is shown how octave filter banks can be generated using a single twochannel filter bank as the basic building block. Also, the relations between the frequencyselective octave filter banks and discrete-time wavelet banks are briefly discussed. Finally, concluding remarks are given in Section 5. Two-Channel Filter Banks This section considers the synthesis of two-channel filter banks based on the use of FIR and IIR filters. First, basic oeration rinciles are discussed and the necessary requirements for alias-free, erfect-reconstruction (PR), and nearly PR (NPR) filter banks are stated. Second, an overview of the most imortant filter bank tyes and references to some existing synthesis schemes are given.. Basic Oeration of a Two-Channel Filter Bank The block diagram of a two-channel filter bank is shown in Figure 3. This system consists of an analysis and a synthesis bank as well as a rocessing unit between these two banks. H () H () x [n] x [n] Analysis bank v [n] v [n] Processing unit w [n] w [n] u [n] u [n] F () F () Synthesis bank Figure 3. Two-channel filter bank... Oeration of the analysis bank The role of the analysis bank is to slit the inut signal into lowass and highass filtered channel signals, denoted by x [n] and x [n] in Figure 3, using a lowass highass filter air with transfer functions H () and H (). Hence, the -transforms of these signals are exressible as X k = H k X for k =,. () y [n] y [n]
T. Saramäki and R. Bregovi, ultirate Systems and Filter Banks, Chater in ultirate Systems: Design 5 T. Saramäki and R. Bregovi, ultirate Systems and Filter Banks, Chater in ultirate Systems: Design 6 After filtering, the signals in both channels are down-samled by a factor of two by icking u every second samle, resulting in two subband signal comonents, denoted by v [n] and v [n] in Figure 3. If the inut samling rate is F s, then the samling rates of v [n] and v [n] are F s /. The -transforms of these comonents are given by / / / / Vk = [ Hk ( ) X ( ) Hk ( ) X ( )] for k =,. () Tyically, H () and H () have the same transition band region with the band edges being located around f = F s /4 at f = ( ρ )F s /4 and f = (ρ )F s /4 with ρ > and ρ >, as shown in Figure 4. In order to give a ictorial viewoint of what is haening in the frequency domain, Figure 4 shows the Fourier transforms of an inut signal, whereas Figure 5 shows those of signals x [n], x [n], v [n], and v [n]. These transforms are obtained from the corresonding - transform by simly using the substitution = e jπf/fs. It is seen that after decimation V k () for k =, contain two overlaing comonents X k ( / ) and X k ( / ). This overlaing can, however, be eliminated in the overall system of Figure 3 by roerly designing the transfer functions H (), H (), F (), and F (), as will be seen later on. X(e jπf/fs ) Fs/4 Fs/ 3Fs/4 V(e jπf/(fs) ) X(e jπf/fs ) X(e jπ(ffs/)/fs ) / F s/4 F s/ 3F s/4 U(e jπf/fs ) X(e jπf/fs ) X(e jπ(ffs/)/fs ) / (c) Low-ass channel Fs/4 Fs/ 3Fs/4 Fs F s Fs X(e jπf/fs ) (f) V(e jπf/(fs) ) Fs/4 Fs/ / X(e jπ(ffs/)/fs ) (g) X(e jπf/fs ) U(e jπf/fs ) / (h) High-ass channel 3Fs/4 F s/4 F s/ 3F s/4 Fs/4 X(e jπ(ffs/)/fs ) X(e jπf/fs ) Fs/ 3Fs/4 Fs F s Fs X(e jπf/fs ) H(e jπf/fs ) H(e jπf/fs ) F(e jπf/fs ) X(e jπf/fs ) (d) F(e jπf/fs ) X(e jπf/fs ) (i) Fs/ Fs Fs/4 Fs/ 3Fs/4 ( ρ ) Fs/4 (ρ ) Fs/4 F(e jπf/fs ) F(e jπf/fs ) Fs F(e jπf/fs ) X(e jπ(ffs/)/fs ) F s/4 F s/ 3F s/4 F s F(e jπf/fs ) X(e jπ(ffs/)/fs ) F s/4 F s/ 3F s/4 F s (c) (e) (j) Fs/4 Fs/ ( ρ ) Fs/4 (ρ ) Fs/4 Figure 4. agnitude of the Fourier transform of an inut signal. Amlitude resonses for H () and H (). (c) Amlitude resonses for F () and F ()... Oeration of the rocessing unit In the rocessing unit, the signals v [n] and v [n] are comressed and coded suitably for either transmission or storage uroses. Before using the synthesis art, signals in both channels are decoded. The resulting signals denoted by w [n] and w [n] in Figure 3 may differ from the original signals v [n] and v [n] due to ossible distortions caused by coding and quantiation errors as well as channel imairments. In the sequel, it is suosed, for simlicity, that there are no coding, quantiation, or channel degradations, that is, w [n] v [n] and w [n] v [n]. 3Fs/4 Fs Fs/4 Fs/ 3Fs/4 Fs Fs/4 Figure 5. agnitudes of the Fourier transforms of the signals in the two-channel filter bank of Figure 3., (f) Transforms of x [n] and x [n]., (g) Transforms of v [n] and v [n]. (c), (h) Transforms of u [n] and u [n]. (d), (i) Transforms of unaliased comonents of y [n] and y [n]. (e), (j) Transforms of aliased comonents of y [n] and y [n]...3 Oeration of the synthesis bank The role of the synthesis bank is to aroximately reconstruct in three stes the delayed version of the original signal from the signal comonents w [n] and w [n]. In the first ste, these signals are u-samled by a factor of two by inserting ero-valued samles between the existing samles yielding two comonents, denoted by u [n] and u [n] in Figure 3. In the w [n] v [n] and w [n] v [n] case, the -transforms of these signals are exressible as U = k k k k V ( ) = X X ( ) Fs/ 3Fs/4 [ ] for =, k. (3) Simultaneously, the samling rate is increased from F s / to F s and the baseband from [, F s /4] to [, F s /]. Therefore, u [n] and u [n] contain in their basebands, in addition to the frequency comonents of v [n] and v [n] in their baseband [, F s /4], the comonents in [F s /4, F s /], as illustrated in Figure 5. In Equation (3), X k () is the -transform of the desired unaliased signal comonent, whereas X k ( ) is the -transform of the unwanted aliased signal comonent that should be eliminated. Fs
T. Saramäki and R. Bregovi, ultirate Systems and Filter Banks, Chater in ultirate Systems: Design 7 T. Saramäki and R. Bregovi, ultirate Systems and Filter Banks, Chater in ultirate Systems: Design The second ste involves rocessing u [n] and u [n] by a lowass highass filter air with transfer functions F () and F (), whereas the third ste is to add the filtered signals, denoted by y [n] and y [n] in Figure 3, to yield the overall outut. The -transform of is given by Y = Y Y, (4) where Y = k k k k k k k [ ] for =, F V ( ) = F X F X ( ) k. (5) The role of the synthesis filters with transfer functions F () and F () is twofold. First, it is desired that Y() does not contain the terms X ( ) and X ( ). This is achieved by requiring that F ()X ( ) = F ()X ( ). Second, if it is desired that is aroximately a delayed version of, that is, x[n K], then F ()X () F ()X () K X() should be satisfied. In order to satisfy these requirements, F () and F () should generate a lowass highass filter air in a manner similar to H () and H (). There exist two main differences. First, due to the alias-free conditions to be considered in the next subsection, the lower and uer edges of this filter air are located at f = ( ρ )F s /4 and f = (ρ )F s /4, as shown in Figure 4(c). Second, because of interolation, the amlitude resonses should aroximate two in the assbands. The last subfigures on the left and right sides of Figure 5 show how the above conditions can be satisfied in the frequency domain. The exact simultaneous conditions for H (), H (), F (), and F () to satisfy the above-mentioned two conditions will be given in the following subsections.. Alias-Free Filter Banks Combining Equations (), (4), and (5) the relation between the inut and the outut signals of Figure 3 is exressible as Y = T X A X ( ), (6) where T [ H F H F ] = (7) is the overall distortion transfer function and A [ H ( ) F H ( ) F ] = () is the aliasing transfer function. In order to generate an alias-free filter bank, this term has to be canceled, that is, A(). The most straightforward way of achieving this is to select F () and F () as follows: F = H( ) (9) F = H ( ). () In the sequel, these conditions are used excet for Subsection.5. where the filter bank is constructed with the aid of causal and anti-causal IIR filters. After fixing F () and F () in the above manner, the inut-outut relation of Equation (6) takes the following simlified form: T X Y =, () where T = H H( ) H ( ) H( ). () There are two imortant reasons for concentrating on the synthesis of two-channel filter banks in such a manner that they are alias-free. First, relating F () and F () to H () and H () according to Equations (9) and () makes both the design and imlementation of the overall system more efficient comared to the nearly alias-free case. Second, it has been observed that when synthesiing the bank without the exact alias-free condition leads to a system that is ractically alias-free. This fact can be seen, for instance, from the results given by Nayebi, Barnwell III, and Smith in (99)..3 Perfect-Reconstruction (PR) and Nearly Perfect-Reconstruction (NPR) Filter Banks The following subsection states the necessary and sufficient conditions for an FIR twochannel filter bank to satisfy the PR conditions and describes their connections to the linearhase half-band FIR filters. Furthermore, these conditions are extended to their IIR counterarts..3. Theorem for the PR Proerty The PR roerty, that is, = x[n K], is the ability of a system to roduce an outut signal that is a delayed relica of the inut signal. The necessary conditions for the PR roerty are given for a two-channel FIR filter bank by the following theorem: Theorem for the PR roerty: Consider the alias-free two-channel filter bank shown in Figure 3 with w [n] v [n] and w [n] v [n] and let H () and H () be the transfer functions of FIR N n N filters given by H ( = = h and H ( h ) n ) n = =. Then, = x[n K] with K n being an odd integer, that is, T() = K, is met rovided that the imulse-resonse coefficients of N N n E = H H( ) = e (3) satisfy / for = for n = K e (4) n is odd and n K. In order to rove this theorem, Equation () is rewritten as N N N N n n T = t = E( ) [ E( ) ] = ( e eˆ ), (5) where ˆ = e e for for n n even e (6) odd. The imulse-resonse coefficients of this T() satisfy = e eˆ = for for n = K t (7) yielding T K = E ) E( ) = n K, (. () This imlies that the outut signal is the relica of the inut signal delayed by K samles, as is desired.
evi T. Saramäki and R. Bregovi, ultirate Systems and Filter Banks, Chater in ultirate Systems: Design 9 T. Saramäki and R. Bregovi, ultirate Systems and Filter Banks, Chater in ultirate Systems: Design It is well known that the PR roerty can be satisfied only when K is an odd integer and N N is two times an odd integer (see, e.g., (Vaidyanathan, 993)). There are two basic alternatives to achieve the PR roerty, namely, K = (N N )/ and K < (N N )/, as illustrated in Figures 6 and 6, resectively. In the first case, E() is an FIR filter transfer function with a symmetric imulse resonse and the imulse-resonse value occurring at the odd central oint n = K being equal to /, whereas the other values occurring at odd values of n are ero. Hence, E() is the transfer function of a linear-hase FIR half-band filter (see, e.g., (Saramäki, 993)). In the second case, the imulse-resonse values at odd values of n are also ero excet for one odd value n = K, where the imulse resonse takes on the value of /. The K = (N N )/ case is attractive when the overall delay of K samles is tolerable, whereas the K < (N N )/ case is used for reducing the delay caused by the filter bank to the overall signal. imulse resonse imulse resonse imulse resonse.5 Imulse resonse for E(): K=.5 5 5 n in samles Imulse resonse for E( ).5.5 5 5 n in samles Imulse resonse for T()=E() E( ).5 5 5 n in samles imulse resonse imulse resonse imulse resonse.5 Imulse resonse for E(): K=7.5 5 5 n in samles Imulse resonse for E( ).5.5 5 5 n in samles Imulse resonse for T()=E() E( ).5 5 5 n in samles Figure 6. Imulse resonses for E(), E( ), and T() for PR filter banks with N N =. K = = (N N )/. K = 7 < (N N )/. In the PR case, there is no amlitude or hase distortion. This is due to the fact that t[n] is nonero only at n = K achieving the value of unity. In the NPR case, the imulse resonse values t[n] differ slightly from ero for n K and slightly from for n = K so that there exists some amlitude and/or hase distortions. These distortions are tolerable in many ractical alications (lossy channel coding and quantiation) rovided that they are smaller than the errors introduced in the rocessing unit. oreover, by slightly releasing the PR condition, filter banks with better selectivities can be synthesied, as will be seen later on. The above criteria for the PR and NPR roerty are also valid for IIR filter banks to be considered in Subsection.5 (excet for the banks to be described in Subsection.5.). The main difference is that the imulse resonses of IIR analysis filters are of infinite length. Therefore, the imulse resonse e[n] of E() and the resonse t[n] of T() are also of infinite length. For a PR system e[n] and t[n] must satisfy the conditions of Equations (4) and (7) for n <, whereas in the NPR case these conditions should be aroximately satisfied..3. PR Filter Bank Design and the Theorem for the PR Reconstruction Consider N N N N n = e = S ( ) E k (9) k = with the imulse-resonse coefficients satisfying the conditions of Equations (4). The factoriation of E() as E() = H () H ( ) can be erformed as follows. First, the eros k of E() for k =,,, N N are divided into two grous α k for k =,,, N and β k for k =,,, N in such a manner that the eros in both grous are either real or occur in comlex-conjugate airs. Second, the constant S of E() is factoried as S = S S. Then, according to the theorem of the revious subsection, the transfer functions given by N N n = h = S ( ) H α () k n = k = N N ˆ ( ) = ˆ n H h = S ( k ) k = can be used for generating a PR two-channel filter bank. In the above, H () is directly the transfer function of the lowass analysis filter. The corresonding highass filter transfer function H () is obtained from H ˆ by selecting the imulse-resonse coefficients to be n h = ( ) hˆ for n =,,, N. The amlitude resonses of these two filters are related j( π ω ) through ( e ) = H e ˆ and H () form a lowass highass filter air H β () H ˆ so that H with the amlitude resonses obtained from each other using a lowass-to-highass transformation ω π ω. According to the above discussion, the synthesis of a PR two-channel filter bank can be stated as follows: Given an odd integer K and integers N and N such that their sum is two times an odd integer, find E() of order N N such that its imulse-resonse coefficients satisfy the conditions of Equation (4) and H () and H () generated using the above factoriation scheme form a lowass highass filter air with the desired roerties. In general, this roblem statement cannot be exloited in a straightforward manner for designing two-channel FIR filter banks. However, there are two excetional cases. In the first case, K = (N N )/ and this roblem statement is widely used for designing start-u two-channel filter banks for generating discrete-time wavelet banks (Vetterli and Herley, 99; Vetterli and Kova 995; isiti et al., 996). In this case, half-band linear-hase FIR filter transfer functions E() with the maximum numbers of eros at = are of great imortance. As a curiosity, these filters are secial cases of the maximally flat FIR filters introduced by Herrmann (97). In the second excetional case, the imulse-resonse coefficients of H () and H ˆ are time-reversed version of each other, that is, hˆ = h[ N n] for n =,,, N (see Figure 7). Furthermore N = N = K. These conditions imly the following (see, e.g., (Herrmann and Schussler, 97; Saramäki, 993)). If H () has a real ero or a comlex-conjugate ero air inside (outside) the unit circle, then H ˆ has a recirocal real ero or a comlex-conjugate ero air outside (inside) the unit circle. ost imortantly, if H () has eros on the unit circle, then = H Hˆ is a linear-hase ˆ has the same eros on the unit circle meaning that H E
T. Saramäki and R. Bregovi, ultirate Systems and Filter Banks, Chater in ultirate Systems: Design T. Saramäki and R. Bregovi, ultirate Systems and Filter Banks, Chater in ultirate Systems: Design half-band FIR filter with the restriction that the eros occurring on the unit circle are double eros (see Figure 7). Furthermore, ˆ E() Im Re H() Im E e = H e = H e. Re e[n] h[n] h[n] 7 n n n 4 7 7 H() Im Figure 7. Factoriation of a half-band E() having double eros on the unit circle into H () and ˆ having time-reversed imulse resonses. H This alternative for constructing PR FIR two-channel filter banks, called later on in this chater as orthogonal banks, was indeendently observed by inter (inter, 95) and Smith and Barnwell (Smith, 96). In their design schemes, E() of order N having double eros on the unit circle is designed to exhibit an equirile amlitude resonse in the stoband [ω s, π] with ω s > π/. The actual synthesis is erformed by first designing, by means of the Reme algorithm (cclellan, Parks, and Rabiner 973), a conventional half-band filter of the same order to have a minimax behavior in the same stoband. Then, the coefficients of this filter are modified to give a rise to the desired half-band filter. After knowing E(), all what is needed is to share the eros between H () and H ˆ as described above (see Figure 7) as well as to factorie S as S = S S so that the resulting imulse resonses are time-reversed versions of each other. This technique results in analysis transfer functions H () and H () such that the maximum amlitude value of H () [H ()] in the stoband [ω s, π] ([, π ω s ]) is δ s with δ s being the stoband rile of E()..4 FIR Filter Banks and Their Design This subsection reviews the synthesis of various two-channel filter banks where both H () and H () are transfer functions of FIR filters as given by N = n H h () N = n H h. (3) oreover, in order to obtain an alias-free system, F () and F () are also FIR filters satisfying Equations (9) and (). Re.4. FIR Filter Bank Classification Alias-free two-channel FIR filter banks can be classified into several filter bank tyes according to Table I. The tye deends first on the relation between the analysis transfer functions H () and H (), second on whether these transfer functions are linear-hase FIR filters or not, and third on whether the PR or NPR roerty is desired to be achieved. In addition to these roerties, Table I shows for each tye the number of unknowns to be otimied, whether the order(s) must be odd or even, and the relation of the overall filter bank delay to the filter orders. Filter banks with the filter bank elay less than N (for QF banks) and (N N )/ (for biorthogonal banks) are later on referred to as low-delay two-channel filter banks. Table I Classification of two-channel FIR filter banks Filter bank tye QF Orthogonal Biorthogonal Filter relation Phase H (), H () Filter order odd Number of unknowns Filter bank delay (N )/ N PR N <N NPR odd N N PR NPR linearhase H () = H ( ) nonlinear -hase H () = nonlinear N H ( ) -hase linearhase even-even (N N )/ N )/ odd-odd (N N )/ (N nonlinear odd-odd N -hase even-even N <(N N )/ In the following subsections the definition of each filter bank tye is given in more details along with their roerties and a short review of the existing synthesis schemes. Examles comaring the various filter tyes with each other will be given in Subsection.4.7. We start by stating a general otimiation roblem including all FIR filter banks of Table I..4. General FIR Filter Bank Design Problem It has turned out to be beneficial to state a general otimiation roblem including all the filter tyes of Table I as follows (Bregovi and Saramäki, 999, b): Given the tye of twochannel filter bank, the filter orders N and N, the assband and stoband band frequencies () () () () ω < π/ and ω s > π/ for H () as well as ω > π/ and ω s < π/ for H (), the reconstruction error δ a, and the assband rile δ as well as the filter bank delay K, find the adjustable coefficients of H () and H (), as given by Equations () and (3), to minimie () π ωs = ε max H e dω, H e dω, (4) ω s subject to max H( e ) δ and max H( e ) δ, (5) ω ω () (, ω ) () () ( ω, ω ) ( e ) ω () ( ω, π ) max H δ and max H s ω () () ( ω, ω ) s ( e ) δ PR NPR, (6)
T. Saramäki and R. Bregovi, ultirate Systems and Filter Banks, Chater in ultirate Systems: Design 3 T. Saramäki and R. Bregovi, ultirate Systems and Filter Banks, Chater in ultirate Systems: Design 4 and jkω max T( e ) e δ a, (7) ω [, π ] where j( ω π ) j( ω π ) T( e ) = H ( e ) H( e ) H ( e ) H( e ). () The main objective is to minimie the maximum of the stoband energies of H () and H () subject to some constraints, as illustrated in Figure. First, the amlitude resonses of both H () and H () have to stay in the assband within the given limits ±δ. Second, the maximum allowable value for these amlitude resonses in the transition bands is δ. Third, the maximum of the absolute value of the deviation between the overall frequency resonse and the constant delay of K samles has to be in the overall frequency range less than or equal to δ a. For the PR filter banks, this deviation is ero. As will be seen later, some of the constraints are automatically satisfied by some of the above-mentioned tyes of two-channel filter banks. The above otimiation roblem has been stated in terms of the angular frequency ω that is related to the real frequency f and the samling rate F s trough ω = πf/f s. The roblem formulation is very general. For instance, the band edges for H () and H () can be selected arbitrarily unlike in the case of Figure 4, where the edges for H () and H () are the same. The secial feature of the roblem statement is that the maximum of the stoband energies of the two filters is minimied, allowing us to treat both filters in a similar manner. In most other existing statements the sum of these energies is minimied. The above roblem has been stated in such a manner that the otimum solution can be found using a two ste-rocedure roosed in Bregovi further otimiation is generated using a simle design scheme for the selected filter bank tye. The second ste involves otimiing the filter bank with the aid of an efficient constrained nonlinear otimiation algorithm (Dutta, 977; Saramäki, 99). δ δ and Saramäki (999, b). In the first ste, a good starting-oint filter bank for H(e ) H(e ) () ω s ( ) ω ) ω ( π/ Figure. Secifications for H () and H (). () ω s.4.3 NPR Quadrature irror Filter (QF) Banks Quadrature mirror filter (QF) banks were the first tye of filter banks used in signal rocessing alications for searating signals into subbands and for reconstructing them from individual subbands (Esteban and Galand, 977). For a QF bank, H () = H ( ) so that H (), as given by Equation (), is the only transfer function to be otimied. In this case, T(), as given by Equation (), becomes π ω N n T = t = [ H ] [ ] ( ) H( ). (9) This transfer function is desired to aroximate the delay K with K being an odd integer satisfying K N. It is well-known that in this case the PR roerty cannot be satisfied excet for the trivial case with H () = ( )/ that does not rovide good attenuation characteristics (Vaidyanathan, 993). Using the decomosition H () = G ( ) G ( ), the overall bank can be effectively imlemented as shown in Figure 9 (alvar, 99a). According to Table I, there are two tyes of QF banks to be considered next. G() G() G() G() Analysis bank Processing unit Synthesis bank Figure 9. Efficient imlementation for a QF two-channel bank..4.3. QF banks with linear-hase subfilters. For these banks, the imulse-resonse coefficients of H () ossess an even symmetry, that is, h [N n] = h [n] for n =,,, (N )/ and K = N. Hence, for the overall filter bank, there are only (N )/ unknowns. Based on the linear-hase roerty of H (), the transfer function between the outut and inut of the overall system has also an imulse resonse of an even symmetry, that is, t[n n] = t[n] for n =,,, N in Equation (9). Hence, it suffers only from the amlitude distortion. The first systematic aroach for synthesiing QF banks was roosed by Johnston (Johnston, 9). In his synthesis technique, the weighted sum of the filter stoband energy and [ ] the integral of ( e ) T over the band [, π/] is minimied. The resulting reconstruction error is not equirile. An iterative technique resulting in an equirile reconstruction error has been roosed by Chen and Lee (99). Further generaliations of this method described by Lim, Yang, and Koh (993), and Goh, Lim, and Ng (999a) enable us to obtain also equirile behaviors in the filter stobands. It has turned out that in many alications it is beneficial to design filter banks in such a way that the reconstruction error exhibits an equirile (minimax) behavior, whereas the stoband energies of the filters are minimied (Bregovi and Saramäki 999, b). This roblem can be solved conveniently using the general roblem statement described in Subsection.4. as well as the two-ste otimiation scheme mentioned in the same subsection. In this case, the roblem takes a simlified form since the stoband energies of both filters are the same and there is no need to control the assband and transition band behaviors. All what is needed is to find a good start-u solution. For this urose, efficient iterative methods described by Xu, Lu, and Antoniou (99) and Lu, Xu, and Antoniou (99) can be used. It should be emhasied that the roer use of constrained otimiation algorithms guarantees the convergence at least to a good local otimum solution. However, if such an algorithm together with a good initial solution is not available, then iterative methods are of a ractical value. This
T. Saramäki and R. Bregovi, ultirate Systems and Filter Banks, Chater in ultirate Systems: Design 5 T. Saramäki and R. Bregovi, ultirate Systems and Filter Banks, Chater in ultirate Systems: Design 6 is because they give in many cases with a small comutation workload a satisfactory sub-otimal solution..4.3. Low-delay QF banks with nonlinear-hase subfilters. In this case, H () is a nonlinear-hase filter. Hence, all the imulse-resonse values h [n] for n =,,, N are unknowns. The filter bank delay K is no more equal to the filter order N. This enables us to increase the filter orders to imrove the filter banks erformance without increasing the overall filter bank delay. The reconstruction error is now given by jkω j( ω ) ω [ ] [ ( π jk T e e = H e H e )] e, (3) that is desired to be made small in the overall baseband [, π]. Due to the nonlinear-hase characteristics, the erformance of H () in the assband must also be controlled unlike for the linear-hase case, where a good stoband characteristics together with a small overall amlitude distortion automatically guarantees that the assband amlitude resonse of H () aroximates unity with a small tolerance. Similar to the linear-hase QF banks, iterative methods described by Xu et al. (99) and Lu et al. (99) exist for designing low-delay QF banks. These filter banks can be used as start-u solutions for the general otimiation roblem stated in Subsection.4.. For this roblem, the stoband energy of only one filter has to be minimied, but all the remaining constraints must be included..4.4 PR Orthogonal Filter Banks These filter banks were considered in Subsection.3.. According to this discussion, the relation between the analysis filters is defined as N H = H( ), (3) where H () is given by Equation () and N, the filter order, is an odd integer. This makes the imulse resonse coefficients of H () and H ( ) time-reversed versions of each other, that is, h [n] = ( ) n h [N n] for n =,,,N. Furthermore, the PR roerty imlies that in this case the following conditions are satisfied:. E() = H ()H ( ) is a linear-hase half-band FIR filter of order N.. The eros of E() occurring on the unit circle are double eros. As was ointed out in Subsection.3., this enables us to factorie E() into two transfer functions H () and H ( ) of order N in such a manner that their imulse resonse are timereversed versions of each other and the amlitude resonse of the corresonding H () is related j( π ω ) to that of H () trough H ( e ) = H ( e ). In Subsection.3., a synthesis scheme was briefly described for designing H () and H () to exhibit minimax amlitude behaviors in the stobands. For designing corresonding filters with least-mean-square error behaviors in the stobands, the most efficient way is to use iterative algorithms described by Blu (99) and Bregovi solution is indeendent of the initial solution. Besides of the direct form imlementation of Figure 3, orthogonal filter banks can be also realied using a lattice form as shown in Figure. This structure is a slightly modified version of those roosed by Vaidyanathan and Hoang (9). For a given odd order N, there are (N )/ unknown lattice coefficients α k as well as the scaling constants β and β. The role of the scaling constants is to make the amlitude resonses of the analysis and synthesis filters to and Saramäki. These algorithms are fast and the convergence to the otimum aroximate one and two in their assband regions, resectively. The advantage of using a lattice structure is that the PR roerty is satisfied for any combination of the lattice coefficients that is very attractive in the case of coefficient quantiation. For the conversion between the lattice coefficients and the original coefficients, see Vaidyanathan and Hoang (9) and Goh and Lim (999b). β α(n )/ α(n )/ α α α α Analysis bank α α Synthesis bank α α Figure. Lattice structure for orthogonal filter banks. α(n )/.4.5 PR Biorthogonal Filter Banks For biorthogonal filter banks, H () and H () are different transfer functions being related to each other through the PR roerty. According to the linear-hase roerty of the filters, they are divided into two tyes (see Table I), namely, filter banks with linear-hase subfilters and filter banks with nonlinear-hase subfilters. These two tyes will be considered next..4.5. PR biorthogonal filter banks with linear-hase subfilters. For PR biorthogonal filter banks with linear-hase subfilters, H () and H () satisfy the following conditions:. The imulse resonses of H () and H ( ) ossess an even symmetry, meaning that the coefficients of H () and H () satisfy h [N n] = h [n] for n =,,, N and h [N n] = ( ) N h [n] for n =,,, N, resectively (for N even (odd), the imulse resonse of H () ossesses an even (odd) symmetry).. The sum of the filter orders N and N is two times an odd integer, that is, N N = K with K being an odd integer. 3. E() = H ()H ( ) is a half-band linear-hase FIR filter of order N N. There exist only the following two cases to meet these conditions: Case A: N and N are odd integers and their sum is two times an odd integer K. Case B: N and N are even integers and their sum is two times an odd integer K. In both cases, the filter bank delay is K = (N N )/. The first effective iterative algorithm for designing biorthogonal filter banks has been roosed by Horng and Willson (99). An iterative algorithm giving rise to analysis filters exhibiting equirile amlitude erformances in their stobands has been roosed by Yang, Lee, and Chieu (99). The results of the above-mentioned algorithms can be used as start-u solutions for the general otimiation roblem stated in Subsection.4.. In this case, all the constraints are needed. It has turned out that, instead of using δ a = in Equation (7), δ a = 3 gives a very accurate solution. α(n )/ β
T. Saramäki and R. Bregovi, ultirate Systems and Filter Banks, Chater in ultirate Systems: Design 7 T. Saramäki and R. Bregovi, ultirate Systems and Filter Banks, Chater in ultirate Systems: Design For imlementing biorthogonal filter banks, like in the case of the orthogonal banks, there exists also a lattice structure as shown in Figure (Nguyen and Vaidyanathan, 99) that can be used in some cases (Vaidyanathan, 993). The coefficient values for the lattice structure can be found by direct otimiation (Vaidyanathan, 993) or by transforming the otimied directform coefficients to those used in the lattice structure (Nguyen, 99b, 995). A very useful alternative for designing and imlementing biorthogonal two-channel filter banks is to use the lifting scheme that was introduced by Sweldens (996) and Daubechies and Sweldens (99) for designing biorthogonal two-channel banks for generating discrete-time wavelet banks. As shown by Daubechies and Sweldens (99), any PR biorthogonal and orthogonal two-channel filter bank can be imlemented using this technique. The main advantage of the resulting structure is that the PR roerty is still remaining after the quantiation of the lifting coefficients into very simle reresentation forms (without any extra scaling). When alying the lifting scheme for designing biorthogonal two-channel filter banks, a PR filter bank with short analysis and synthesis filters is used as a starting oint. After that, PR filter banks with higher filter orders are successively generated alying the so-called lifting and dual lifting stes. For more details, see Sweldens (996) and Daubechies and Sweldens (99). The lifting scheme has been also used for designing cosine-modulated low-delay biorthogonal filter banks by Kar and ertins (997). β α l α l α α α α Analysis bank α α Synthesis bank α Figure. Lattice structure for biorthogonal filter banks..4.5. Low-delay PR biorthogonal filter banks with nonlinear-hase subfilters. For low-delay PR biorthogonal filter banks with nonlinear-hase subfilters, H () and H () satisfy the following conditions:. The imulse resonses of H () and H () are not symmetric. N N. The imulse resonse of E( ) = H H ( ) = e / for = for n = K n is odd and n K, α α l α l β n satisfies e (3) where K is an odd integer with K < (N N )/. An examle for an imulse resonse of E() is shown on Figure 6. The second condition imlies that the overall transfer function between the outut and inut is T() = K with K less than (N N )/. The sum of the filter orders must be two times an odd integer. Since the overall system delay is less than half the sum of the filter orders, the imulse resonses of H () and H () cannot ossess symmetries. Due to the nonlinearity, all the imulse resonse values are unknowns. The high number of unknowns (altogether N N ) and the PR condition with the delay less than half the sum of the filter orders makes the synthesis of the overall system very nonlinear and comlicated. Low-delay PR biorthogonal filter banks have been first introduced by Nayebi, Barnwell, and Smith (99, 994). The filter banks obtained by using their design scheme are subotimal, as has been shown in some later aers. However, they have made several imortant observations concerning the roerties of low-delay filter banks. First, it is not advisable to design filter banks with a very small delay comared to the filter orders. The efficiency of such systems is low in the sense that after certain filter orders for the same overall delay, the use of larger filter orders result only in a negligible imrovement in the erformance of the filter bank. Second, additional constraints are necessary in the transition bands of the filters due to the artifacts often occurring in these bands. Similar to the linear-hase case for designing this tye of filter banks, to obtain a good result, the roblem formulation according to Subsection.4. together with a two-ste design method described there is recommended. In this case, a good initial solution is obtained using an iterative method described by Abdel-Raheem, El-Guibaly, and Antoniou in (996). Other methods for designing low-delay PR biorthogonal filter banks have been roosed by Schuller and Smith (995, 996)..4.6 Generalied NPR Filter Banks As mentioned earlier, it is beneficial in many cases to release the PR condition until the errors caused by the non-idealities of the filter bank to the signal are lower than those caused by the rocessing unit. The ultimate goal is to achieve better filter bank roerties. According to Table I, there are two tyes of NPR banks that will be considered next..4.6. NPR filter banks with linear-hase subfilters. For an NPR filter bank with linear-hase subfilters, H () and H () satisfy the same conditions as for the corresonding PR bank (see Subsection.4.5.) with the excetion that now n E is nearly a half-band linear-hase FIR N N in Condition 3 = H H ( ) = e filter of order N N, that is, its imulse resonse coefficients satisfy / for for n = K e (33) n is odd and n K, where K = (N N )/. Consequently, the overall transfer function T() aroximates the delay term K. Like for the QF banks with linear-hase subfilters, the imulse-resonse coefficients of T() ossess an even symmetry so that the reconstruction error consists only of an amlitude error. The actual design of these filters can be accomlished by first stating the otimiation roblem according to Subsection.4. and then solving the roblem with the aid of the two-ste design technique mentioned in the same subsection. As an initial solution for the second ste, the corresonding PR filter bank can be used..4.6. Low-delay NPR filter banks with nonlinear-hase subfilters. For a low-delay NPR filter bank with nonlinear-hase subfilters, H () and H () satisfy the following conditions:. The imulse resonses of H () and H () are not symmetric. N N. The imulse resonse of E( ) = H H = e n satisfies
T. Saramäki and R. Bregovi, ultirate Systems and Filter Banks, Chater in ultirate Systems: Design 9 T. Saramäki and R. Bregovi, ultirate Systems and Filter Banks, Chater in ultirate Systems: Design / for for n = K e (34) n is odd and n K, where K is an odd integer with K (N N )/. The otimiation of these banks can be erformed like for the corresonding banks with linear-hase subfilters. The difference is that now the reconstruction error consists of both amlitude and hase errors..4.7 FIR filter bank examles This subsection comares, in terms of examles, various FIR filter banks considered in this section. An overview of all designs under consideration is given in Table II. For all filter banks, the filter bank delay is K = 3 and the assband and stoband edges are located at () () () () ω = ω s =. 44π and ω s = ω =. 56π. For the banks with linear-hase subfilters and for the orthogonal banks, the orders of both H () and H () are N = N = 3. For the low-delay banks, N = N = 63. Additionally, in the low-delay NPR biorthogonal case, the solution for N = N = 33 is also included. For the biorthogonal banks, Table II shows δ, the allowable assband rile of H () and H () as well the maximum allowable overshoot in the transition band, and δ a, the allowable reconstruction error for NPR banks. Furthermore, the table shows the figure where amlitude resonses of the analysis filters of the corresonding filter bank are given. For the NPR banks, the reconstruction error is also included in the figures. Figure shows how the amlitude resonses of the analysis filters of a QF bank can be imroved comared to the Johnston 3D design (Johnston, 9) given in Figure by minimiing the stoband energies of the filters subject to the maximum reconstruction error of the Johnston design. As seen from Figure, this otimiation aroach results in a minimax reconstruction error. As illustrated in Figure 3, the channel selectivity can further be imroved for the same filter bank delay by using, instead of linear-hase filters, nonlinear-hase filters of higher orders. Figures 4 and 4 comare orthogonal filter banks where the stoband behaviors of the analysis filters have been otimied in the least-mean-square and minimax senses, resectively. As can be exected, the attenuations rovided by the filters designed in the least-mean-square sense are lower near the stoband edges, but become higher for frequencies further away from the edges. Table II FIR filter bank examles Filter bank tye PR Phase N δ δ a Figure Design QF NPR linear-hase 3-3.3 3 minimax Figure Johnston 3D low-delay 63 Figure 3 low-delay Orthogonal PR nonlinear 3 - least-squares Figure 4 minimax PR linear-hase 3. linear-hase Figure 5 low-delay 63. low-delay 3 δ a = 3 Biorthogonal linear-hase 3. 5 Figure 6 δa = NPR 5 33 N = 33 low-delay 63. 5 Figure 7 N = 63 Figures 5 and 5 rovide a comarison between a linear-hase and a low-delay PR biorthogonal filter bank, resectively. For designing these filters, the overall roblem has been stated according to Subsection.4. and the two-ste otimiation scheme mentioned there has been used. As can be exected, the stoband attenuations of the analysis filters in the low-delay filter bank are higher due to their higher orders. Figure 6 shows the amlitude resonses of the analysis filters as well as the reconstruction errors for two NPR biorthogonal two-channel filter banks with linear-hase subfilters. It is clearly seen that a larger allowable reconstruction error results in higher stoband attenuations. NPR low-delay biorthogonal two-channel filter banks with nonlinear-hase subfilters rovide higher attenuations even with a smaller assband rile (see Table II) as shown in Figure 7. This figure shows the amlitude characteristics of the analysis filters and the reconstruction errors for filter banks with two different filter orders. Reconstruction error 4 6 x 3 5 minimax 5...3.4.5...3.4.5 Normalied frequency (ω/(π)) 5...3.4.5 4 6 x 3 5 Figure. QF banks with linear-hase filters. Reconstruction error 5...3.4.5 4 6 x 3 5...3.4.5 Normalied frequency (ω/(π)) Figure 3. Low-delay QF bank. 4 6 least mean square...3.4.5 Normalied frequency (ω/(π)) 4 6 Figure 4. PR orthogonal filter banks. Johnston 3D...3.4.5 Normalied frequency (ω/(π)) minimax...3.4.5 Normalied frequency (ω/(π))