Tài liệu 37 Time-Varying Analysis-Synthesis Filter Banks docx

Iraj Sodagar “Time-Varying Analysis-Synthesis Filter Banks” 2000 CRC Press LLC Time-Varying Analysis-Synthesis Filter Banks 37.1 37.2 37.3 37.4 Iraj Sodagar DavidSar noffResearch Center 37.1 Introduction Analysis of Time-Varying Filter Banks Direct Switching of Filter Banks Time-Varying Filter Bank Design Techniques Approach I: Intermediate Analysis-Synthesis (IAS) • Approach II: Instantaneous Transform Switching (ITS) 37.5 Conclusion References Introduction Time-frequency representations (TFR) combine the time-domain and frequency-domain representations into a single framework to obtain the notion of time-frequency TFR offer the time localization vs frequency localization tradeoff between two extreme cases of time-domain and frequency-domain representations The short-time Fourier transform (STFT) [1, 2, 3, 4, 5] and the Gabor transform [6] are the classical examples of linear time-frequency transforms which use time-shifted and frequencyshifted basis functions In conventional time-frequency transforms, the underlying basis functions are fixed in time and define a specific tiling of the time-frequency plane The term time-frequency tile of a particular basis function is meant to designate the region in the plane that contains most of that function’s energy The short-time Fourier transform and the wavelet transform are just two of many possible tilings of the time-frequency plane These two are illustrated in Fig 37.1(a) and (b), respectively In these figures, the rectangular representation for a tile is purely symbolic, since no function can have compact support in both time and frequency Other arbitrary tilings of the time-frequency plane are possible such as the example shown in Fig 37.1(c) In the discrete domain, linear time-frequency transforms can be implemented in the form of filter bank structures It is well known that the time-frequency energy distribution of signals often changes with time Thus, in this sense, the conventional linear time-frequency transform paradigm is fundamentally mismatched to many signals of interest A more flexible and accurate approach is obtained if the basis functions of the transform are allowed to adapt to the signal properties An example of such a time-varying tiling is shown in Figure 37.1(d) In this scenario, the time-frequency tiling of the transform can be changed from good frequency localization to good time localization and vice versa Time-varying filter banks provide such flexible and adaptive time-frequency tilings c 1999 by CRC Press LLC FIGURE 37.1: The time-frequency tiling for different time-frequency transforms: (a) The STFT, (b) the wavelet transform, (c) an example of general tiling, and (d) an example of the time-varying tiling The concept of time varying (or adaptive) filter banks was originally introduced in [7] by Nayebi et al The ideas underlying their method were later developed and extended to a more general case in which it was also shown that the number of frequency bands could also be made adaptive [8, 9, 10, 11] De Queiroz and Rao [12] reported time-varying extended lapped transforms and Herley et al [13, 14, 15] introduced another time-domain approach for designing time-varying lossless filter banks Arrowood and Smith [16] demonstrated a method for switching between filter banks using lattice structures In [17], the authors presented yet another formulation for designing time-varying filter banks using a different factorization of the paraunitary transform Chen and Vaidyanathan [18] reported a noncausal approach to time-varying filter banks by using time-reversed filters Phoong and Vaidyanathan [19] studied time-varying paraunitary filter banks using polyphase approach In [11, 20, 21, 22], the post filtering technique for designing time-varying filter bank was reported The design of multidimensional time-varying filter bank was addressed in [23, 24] In this article, we introduce the notion of the time-varying filter banks and briefly discuss some design methods 37.2 Analysis of Time-Varying Filter Banks Time-varying filter banks are analysis-synthesis systems in which the analysis filters, the synthesis filters, the number of bands, the decimation rates, and the frequency coverage of the bands are changed (in part or in total) in time, as is shown in Fig 37.2 By carefully adapting the analysis section to the temporal properties of the input signal, better performance can be achieved in processing the signal In the absence of processing errors, the reconstructed output x(n) should closely approximate ˆ a delayed version of the original signal x(n) When x(n − ) = x(n) for some integer constant, , ˆ then we say that the filter bank is perfectly reconstructing (PR) The intent of the design is to choose the time-varying analysis and synthesis filters along with the time-varying down/up samplers so that the system requirements are met subject to the constraint that the analysis-synthesis filter bank be PR at all times c 1999 by CRC Press LLC FIGURE 37.2: The time-varying filter bank structure with time-varying filters and time-dependent down/up samplers One general method for analysis of time-varying filter banks is the time-domain formulation reported in [10, 22] In this method, the time-varying impulse response of the entire filter bank is derived in terms of the analysis and synthesis filter coefficients Figure (37.3) shows the diagram of a time-varying filter bank In this figure, the filter bank is divided into three stages: the analysis filters, the down/up samplers, and the synthesis filters The signals x(n) and x(n) are the filter bank input and output at time n, respectively The outputs ˆ of the analysis filters are shown by v(n) = [v0 (n), v1 (n), , vM(n)−1 (n)]T , where vi (n) is the output of the ith analysis filter at time n The outputs of the down/up samplers at time n is called w(n) = [w0 (n), w1 (n), , wM(n)−1 (n)]T FIGURE 37.3: Time-varying filter bank as a cascade of analysis filters, down/up samplers, and synthesis filters The input/output relation of the analysis filters can be expressed by v(n) = P(n)xN (n) (37.1) P(n) is an M(n) × N(n) matrix whose mth row is comprised of the coefficients of the mth analysis filter at time n and xN (n) is the input vector of length N (n) at time n: xN (n) = [x(n), x(n − 1), x(n − 2), , x(n − N (n) + 1)]T (37.2) The input/output function of down/up samplers can be expressed in the form w(n) = (n)v(n) (37.3) where (n) is a diagonal matrix of size M(n) × M(n) The mth diagonal element of (n), at time n, is if the input and output of the mth down/up sampler are identical, otherwise it is zero c 1999 by CRC Press LLC To write the input/output relationship of the synthesis filters, Q(n) is defined as  g0 (n, 1) g0 (n, 2) g0 (n, N (n) − 1) g0 (n, 0)  g1 (n, 1) g1 (n, 2) g1 (n, N (n) − 1) g1 (n, 0)   g2 (n, 1) g2 (n, 2) g2 (n, N (n) − 1) g2 (n, 0) Q(n) =    gM(n)−1 (n, 0) = q0 (n) q1 (n)        gM(n)−1 (n, 1) gM(n)−1 (n, 2) gM(n)−1 (n, N (n) − 1) q2 (n) qN (n)−1 (n) (37.4) where qi (n) = [g0 (n, i), g1 (n, i), g2 (n, i), , gM(n)−1 (n, i)]T , is a vector of length M(n) and gi (n, j ) denotes the j th coefficient of the ith synthesis filter At time n, the mth synthesis filter is convolved with vector [wm (n), wm (n − 1), , wm (n − N (n) + 1)]T and all outputs are added together Using Eq (37.4), the output of the filter bank at time n can be written as: N (n)−1 x(n) = ˆ i=0 T qi (n) w(n − i) (37.5) If s(n) and w(n) are defined as ˆ T T T T s(n) = q0 (n), q1 (n), q2 , , qN (n)−1 (n) T w(n) = w T (n), w T (n − 1), w T (n − 2), , w T (n − N (n) + 1) ˆ (37.6) T , (37.7) then Eq (37.5) can be written in the form of one inner product, ˆ x(n) = sT (n)w(n) ˆ (37.8) where s(n) and w(n) are vectors of length N (n)M(n) Using Eqs (37.1), (37.3), (37.7), and (37.8), ˆ the input/output function of the filter bank can be written as:   (n) P(n) xN (n)   (n − 1) P(n − 1) xN (n − 1)     (n − 2) P(n − 2) xN (n − 2) (37.9) x(n) = sT (n)  ˆ      (n − N(n) + 1) P(n − N (n) + 1) xN (n − N (n) + 1) As the last N(n) − elements of vector xN (n − i) are identical to the first N (n) − elements of vector xN (n − i − 1), the latter equation can be expressed by   x(n) ˆ =   sT (n)  (n) P(n) O .O (n − 1) P(n − 1) O O (n − 2) P(n − 2) O O (n − N (n) + 1) P(n − N (n) + 1) O O         O O O x(n) x(n − 1) x(n − 2) x(n − 2N (n) + 1) c 1999 by CRC Press LLC             (37.10) where O is the zero column vector with length M(n) Thus, the input/output function of a timevarying filter bank can be expressed in the form of x(n) = zT (n)xI (n) ˆ (37.11) where xI (n) = [x(n), x(n − 1), , x(n − I + 1)]T and I (n) = 2N (n) − and z(n) is the timevarying impulse response vector of the filter bank at time n: z(n) = A(n) s(n) (37.12) The matrix A(n) is the [2N(n) − 1] × [N (n) M(n)] matrix      A(n) =        P(n)T OT OT OT (n)   OT  P(n − 1)T  (n − 1)  OT OT   OT OT  P(n − N (n) + 1)T  (n − N (n) + 1)          (37.13) For a perfect reconstruction filter bank with a delay of , it is necessary and sufficient that all elements but the ( + 1)th in z(n) be equal to zero at all times The ( + 1)th entry of z(n) must be equal to one If the ideal impulse response is b(n), the filter bank is PR if and only if A(n) s(n) = b(n) 37.3 for all n (37.14) Direct Switching of Filter Banks Changing from one arbitrary filter bank to another independently designed filter bank without using any intermediate filters is called direct switching Direct switching is the simplest switching scheme and does not require additional steps in switching between two filter banks But such switching will result in a substantial amount of reconstruction distortion during the transition period This is because during the transition, none of the synthesis filters satisfies the exact reconstruction conditions Figure (37.4) shows an example of a direct switching filter bank Figure (37.5) shows the time-varying impulse response of the above system around the transition periods In this figure, z(n, m) is the response of the system at time n to the unit input at time m For a PR system, z(n, m) has a height of along the diagonal and everywhere else in the (m, n)-plane As is shown, the time-varying filter bank is PR before and after but not during the transition periods In this case, each switching operation generates a distortion with an 8-sample duration One way to reduce the distortion is to switch the synthesis filters with an appropriate delay with respect to the analysis switching time This delay may reduce the output distortion, but it can not eliminate it 37.4 Time-Varying Filter Bank Design Techniques The basic time-varying filter bank design methods are summarized in Table 37.1 These techniques can be divided into two major approaches which are briefly described in the following sections c 1999 by CRC Press LLC FIGURE 37.4: Block diagram of a time-varying analysis/synthesis filter bank that switches between a two- and three-band decomposition TABLE 37.1 Comparison of Time-Varying Filter Bank Different Designing Methods Intermediate analysis Changing freq resolution Filter bank requirement Computational complexity Arrowood Smith Yes Indirect Lattice structures Low de Queiroz Rao Yes Indirect ELT Low Intermediate analysis Gopinath Burrus Yes Indirect Paraunitary Low synthesis Herley et al Yes Direct Paraunitary Low (IAS) Chen Vaidyanathan Direct Noncausal synthesis Low Low Yes Instantaneous No Direct General (not PR) transform switching Redesigning analysis No Direct General High (ITS) 37.4.1 Least square synthesis Post filtering No Direct General Low Approach I: Intermediate Analysis-Synthesis (IAS) In the first approach, both analysis and synthesis filters are allowed to change during the transition period to maintain perfect reconstruction We refer to this approach as the intermediate analysissynthesis (IAS) approach In [16], the authors have chosen to start with the lattice implementation of time-invariant twoband filter banks, originally proposed by Vaidyanathan [25] for time-invariant case Consider the lattice structure shown in Fig 37.6 Figure 37.6(a) represents a lossless two-band analysis filter bank, consisting of J + lattice stages The corresponding synthesis filter bank is shown in Fig 37.6(b) As is shown, for each stage in the analysis filter bank, there exists a corresponding stage in the synthesis filter bank with similar, but inverse functionality As long as each two corresponding lattice stages in the analysis and synthesis sections are PR, the overall system is PR To switch one filter bank to another, the lattice stages of the analysis section are changed from one set to another If the corresponding lattice stages of the synthesis section are also changed according to the changes of the analysis section, the PR property will hold during transition Due to the existence of delay elements, any change in the analysis section must be followed with the corresponding change in the synthesis section, but with an appropriate delay For example, the parameter αj of the analysis and synthesis filter banks can c 1999 by CRC Press LLC FIGURE 37.5: The time-varying impulse response for direct switching between the two- and the three-band system The filter bank is switched from the two-band to the three-band at time n = and switched back at time n = 13 (a) Surface plot, (b) contour plot be changed instantaneously But any change in parameter αj −1 in the analysis filter bank must be followed with the similar change in the synthesis filter bank after one sample delay Because of such delays, switching between two PR filter banks can occur only by going through a transition period in which both analysis and synthesis filter banks are changing in time In [12, 26], the design of time-varying extended lapped transform (ELT) [27, 28] was reported The extended lapped transform is a cosine-modulated filter bank with an additional constraint on the filter lengths Here, the design procedure is based on factorization of the time-domain transform matrix into permutation and rotation matrices As the ELT is paraunitary, the inverse transform can be obtained by reversing the order of the matrix multiplication Since any orthogonal transform is a succession of plane rotations, any changes in these rotation angles result in changing the filter bank without losing the orthogonality property The authors derived a general frame work for Mband ELT transforms compared to the two-band case approach in [16] This method parallels the lattice technique [16] except with the mild modification of imposing the additional ELT constraints In [17], the authors presented yet another formulation for designing time-varying filter banks In this paper, a different factorization of the paraunitary transform has been shown which is not based on plane rotations unlike the ones in [12, 26] Using this factorization, a paraunitary filter bank can be implemented in the form of some cascade structures Again, to switch one filter bank to c 1999 by CRC Press LLC FIGURE 37.6: The block diagram of a two-band paraunitary filter bank in lattice form: (a) analysis lattice, (b) synthesis lattice another, the corresponding structures in the analysis and synthesis filter bank are changed similarly but with an appropriate delay If the orthogonality property in each cascade structure is maintained, the time-varying filter bank remains PR This formulation is very similar to the ones in [12, 16, 26], but represent a more general form of factorization In fact, all above procedures consider similar frameworks of structures that inherently guarantee the exact reconstruction Herley et al [13, 14, 15, 29] introduced a time-domain method for designing time-varying paraunitary filter banks In this approach, the time-invariant analysis transforms not overlap As a simple example, consider the case of switching between two paraunitary time-invariant filter banks The analysis transform around the transition period can be written as          T=      P1 P T  P2            (37.15) The matrices P1 and P2 represent paraunitray transforms and therefore are unitary matrices Their nonzero columns also not overlap with each other The matrix PT represents the analysis filter bank during the transition period In order to find this filter bank, the matrix PT is initially replaced with a zero matrix Then, the null space of the transform T is found Any matrix that spans this subspace can be a candidate vector for PT By choosing enough independent vectors of this null space and applying the Gram-Schimidt procedure to them, an orthogonal transform can be selected for PT This method has also been applied to time-varying modulated lapped transforms [24] and two-dimensional time-varying paraunitary filter banks [30] The basic property of all above procedures is the use of intermediate analysis transforms in the transition period The characteristics of these analysis transforms are not easy to control and typically the intermediate filters are not well-behaved c 1999 by CRC Press LLC 37.4.2 Approach II: Instantaneous Transform Switching (ITS) In the second approach, the analysis filters are switched instantaneously and time-varying synthesis filters are used in the transition period We refer to this approach as the instantaneous transform switching (ITS) approach In the ITS approach, the analysis filter bank may be switched to another set of analysis filters arbitrarily This means that the basis vectors and the tiling of the time-frequency plane can be changed instantaneously To achieve PR at each time in the transition period, a new synthesis section is designed to ensure proper reconstruction In the least squares (LS) method [10], for any given set of analysis filters, a LS solution of Eq (37.14) can be used to obtain the “best” synthesis filters of the corresponding system (in L2 norm): s (n)LS = A (n)T A(n) −1 A (n)T b(n) (37.16) The advantage of the LS approach is that there is no limitation on the number of analysis filter banks that can be used in the system The disadvantage of the LS method is that it does not achieve PR However, experiments have shown that the reconstruction is significantly improved in this method compared to direct switching [10] In the LS solution, b(n) is projected onto the column space of A(n) For PR, the projection error should be zero Thus, to obtain time-varying PR filter banks, the reconstruction error, ||A(n)s(n) − b(n)||2 , can be brought to zero with an optimization procedure The optimization operates on the analysis filter coefficients and modifies the range space of A(n) until b(n) ∈ range(A(n)) Although the s(n)’s at different states are independent of each other, since the A(n)’s have some common elements, optimization procedures should be applied to all analysis sections at the same time This method is referred to as “redesigning analysis” [10] The last ITS method, post filtering, uses conventional filter banks with time-varying coefficients followed by a time-varying post filter The post filter provides exact reconstruction during transition periods, while it operates as a constant delay elsewhere Assume at time n0 the time-varying filter bank is switched from the first filter bank to the second If the length of the transition period is L samples, the output of the filter bank in the interval [n0 , n0 + L − 1] is distorted because of switching The post filter removes this distortion The block diagram of such a system is shown in Fig (37.7) In this figure, z(n) and y(n) are the analysis/synthesis filter bank and post filter impulse responses, FIGURE 37.7: The block diagram of time-varying filter bank and post filter respectively If the delays of the filter bank and the post filter are denoted can write and , respectively, we Distorted if n0 ≤ n < n0 + L x(n − ) otherwise (37.17) x(n) = x(n − ˜ x(n) = ˆ (37.18) The desired output of the post filter is c 1999 by CRC Press LLC − ) The input/output relation of the time-varying filter bank during the transition period can be written as (37.19) x(n) = zT (n) xI (n) ˆ where xI (n) is the input vector at time n: xI (n) = [x(n), x(n − 1), x(n − 2), , x(n − I + 1)]T z(n) is a vector of length I and represents the time-varying impulse response of the filter bank at time n If the transition impulse response matrix is defined to be     Z=   O z(n0 + L − 2) O O z(n0 + L − 1) O O O O O O z(n0 )   ,  (37.20) then the input/output relation of the filter bank in the transition period can be described as xL (n0 + L − 1) = ZT xK (n0 + L − 1) ˆ (37.21) where Z is a K × L matrix and K = I + L − In Eq (37.21), the I − − samples before and samples after the transition period are used to evaluate the output The above intervals are called the tail and head of the transition period, respectively Since the first and second filter banks are PR, the tail and head samples are exactly reconstructed We write xK (n0 + L − 1) as the concatenation of three vectors: xa xt xb xK (n0 + L − 1) = , (37.22) where xa and xb are the input signals in the head and tail regions while xt represents the input samples which are distorted during the transition period Using this notation, Eq (37.21) can be written as T T xL (n0 + L − 1) = Za xa + ZtT xt + Zb xb ˆ (37.23) where Z= Za Zt Zb (37.24) By replacing vectors xb and xa with their corresponding output vectors xa and xb , xt of Eq (37.23) ˆ ˆ can be written as xt = = T T (ZtT )−1 (xt − Za xa − Zb xb ) ˆ ˆ ˆ ˆ Y T xK (37.25) Equation (37.25) describes the post filter input-output relationship during the transition region In this equation, Y is the time-varying post filter impulse response which is defined as   −Za Zt−1 (37.26) Y =  Zt−1  −Zb Zt−1 From Eq (37.25), it is obvious that the condition for causal post filtering is ≥L+ c 1999 by CRC Press LLC −1 (37.27) The post filter exists if Zt has an inverse It can be shown that the transition response matrix Zt , can be described by a matrix, product of the form Zt = L S (37.28) where L is the analysis transform applied to those input samples that are distorted during the transition period and S contains the synthesis filters during the transition period In order for Zt to be invertible, it is necessary (but not sufficient) that L and S be full rank matrices The analysis sections are defined by the required properties of the first and second filter banks and L is fixed Therefore, a filter bank is switchable to another filter bank if the corresponding L is a full rank matrix In this case, by proper design of the synthesis section, both S and Zt will be full rank Two methods to obtain proper synthesis filters are shown in [20, 22] 37.5 Conclusion In this article, we briefly review some analysis and design methods of time-varying filter banks Time-varying filter banks can provide a more flexible and accurate approach in which the basis functions of the time-frequency transform are allowed to adapt to the signal properties A simple form of time-varying filter bank is achieved by changing the filters of an analysis-synthesis system among a number of choices Even if all the analysis and synthesis filters are PR sets, exact reconstruction will not normally be achieved during the transition periods To eliminate all distortion during a transition period, new time-varying analysis and/or synthesis sections are required for the transition periods Two different approaches for the design were discussed here In the first approach, both analysis and synthesis filters are allowed to change during the transition period to maintain PR and so it is called the intermediate analysis-synthesis (IAS) approach In the second approach, the analysis filters are switched instantaneously and time-varying synthesis filters are used in the transition period This approach is known as the instantaneous transform switching (ITS) approach In the IAS approach, both analysis and synthesis filters can change during the transitions rather than only the synthesis filters in ITS approach That implies that maintaining PR conditions is easier in the IAS approach Note that the analysis filters in the transition periods are designed only to satisfy PR conditions and they not usually meet the desired time and frequency characteristics In the ITS approach, only synthesis filters are allowed to be time-varying in the transition periods These methods have the advantage of providing instantaneous switching between the analysis transforms compared to IAS methods But they have different drawbacks: the LS method does not satisfy PR conditions at all times, the redesigning analysis method requires jointly optimization of the time-invariant analysis section, and finally the post filtering method has the drawback of additional computational complexity required for post filtering The analysis and design methods of the time-varying filter bank have been developed to design adaptive time-frequency transforms These adaptive transforms have many potential applications in areas such as time-frequency representation, subband image and video coding, and speech and audio coding But since the developments of the time-varying filter bank theory is very new, its applications have not been investigated yet References [1] Allen, J.B., Short-term spectral analysis, synthesis, and modification by discrete fourier transform, IEEE Trans Acoustics, Speech, Signal Processing, 25, 235–238, June 1977 [2] Allen, J.B and Rabiner, L.R., A unified approach to STFT analysis and synthesis, Proc IEEE, 65, 1558–1564, Nov 1977 c 1999 by CRC Press LLC [3] Rabiner, L.R and Schafer, R.W., Digital Processing of Speech Signals, Prentice-Hall, Englewood Cliffs, NJ, 1978 [4] Portnoff, M.R., Time-frequency representation of digital signals and systems based on shorttime fourier analysis, IEEE Trans Acoustics, Speech, Signal Processing, 55–69, Feb 1980 [5] Nawab, S.N and Quatieri, T.F., Short-Time Fourier Transform, Chapter in Advanced Topics in Signal Processing, Prentice-Hall, Englewood Cliffs, NJ, 1988 [6] Gabor, D., Theory of communication, J IEE (London), 93(III), 429–457, Nov 1946 [7] Nayebi, K., Barnwell, T.P., and Smith, M.J.T., Analysis-synthesis systems with time-varying filter bank structures, Proc Intl Conf Acoustics, Speech, Signal Processing, Mar 1991 [8] Nayebi, K., Sodagar, I., and Barnwell, T.P., The wavelet transform and time-varying tiling of the time-frequency plane, IEEE-SP Intl Symp Time-Frequency and Time-Scale Analysis, Oct 1992 [9] Sodagar, I., Nayebi, K., and Barnwell, T.P., A class of time-varying wavelet transforms, Proc Intl Conf Acoustics, Speech, Signal Processing, April 1993 [10] Sodagar, I., Nayebi, K., Barnwell, T.P., and Smith, M.J.T., Time-varying filter banks and wavelets, IEEE Trans Signal Processing, Nov 1994 [11] Sodagar, I., Analysis and Design of Time-Varying Filter Banks, Ph.D thesis, Georgia Institute of Technology, Atlanta, GA, Dec 1994 [12] de Queiroz, R.L and Rao, K.R., Adaptive extended lapped transforms, Proc Intl Conf Acoustics, Speech, Signal Processing, April 1993 [13] Herley, C., Kovacevic, J., Ramchandran, K., and Vetterli, M., Arbitrary orthogonal tilings of the time-frequency plane, IEEE-SP Intl Symp Time-Frequency and Time-Scale Analysis, Oct 1992 [14] Herley, C and Vetterli, M., Orthogonal time-varying filter banks and wavelets, Proc Intl Symp Circuits Syst., Apr 1993 [15] Herley, C., Wavelets and Filter Banks, Ph.D thesis, Columbia University, New York, 1993 [16] Arrowood, J.L and Smith, M.J.T., Exact reconstruction analysis/synthesis filter banks with time-varying filters, Proc Intl Conf Acoustics, Speech, Signal Processing, Apr 1993 [17] Gopinath, R.A., Factorization approach to time-varying filter banks and wavelets, Proc Intl Conf Acoustics, Speech, Signal Processing, Apr 1994 [18] Chen, T and Vaidyanathan, P.P., Time-reversed inversion for time-varying filter banks, Proc 27th Asilomar Conf on Signals, Systems, and Computers, 1993 [19] Phoong, S and Vaidyanathan, P.P., On the study of lossless time-varying filter banks, Proc 29th Asilomar Conf on Signals, Systems, and Computers, 1995 [20] Sodagar, I., Nayebi, K., Barnwell, T.P., and Smith, M.J.T., A new approach to time-varying FIR filter banks, Proc 27th Asilomar Conf on Signals, Systems, and Computers, 1993 [21] Sodagar, I., Nayebi, K., Barnwell, T.P., and Smith, M.J.T., A novel structure for time-varying FIR filter banks, Proc Intl Conf Acoustics, Speech, and Signal Processing, 1994 [22] Sodagar, I., Nayebi, K., and Barnwell, T.P., Time-varying analysis-synthesis systems based on filter banks and post filtering, IEEE Trans Signal Processing, Oct 1995 [23] Sodagar, I., Nayebi, K., Barnwell, T.P., and Smith, M.J.T., Perfect reconstruction multidimensional filter banks with time-varying basis functions, Proc 27th Asilomar Conf on Signals, Systems, and Computers, 1993 [24] Kovacevic, J and Vetterli, 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banks, Proc 27th Asilomar Conf on Signals, Systems, and Computers, 1993 c 1999 by CRC Press LLC .. .Time-Varying Analysis-Synthesis Filter Banks 37. 1 37. 2 37. 3 37. 4 Iraj Sodagar DavidSar noffResearch Center 37. 1 Introduction Analysis of Time-Varying Filter Banks Direct Switching of Filter Banks. .. time-varying filter banks and briefly discuss some design methods 37. 2 Analysis of Time-Varying Filter Banks Time-varying filter banks are analysis-synthesis systems in which the analysis filters, the... (37. 6) T , (37. 7) then Eq (37. 5) can be written in the form of one inner product, ˆ x(n) = sT (n)w(n) ˆ (37. 8) where s(n) and w(n) are vectors of length N (n)M(n) Using Eqs (37. 1), (37. 3), (37. 7),