M. Vetterli, A theory of multirate filter banks, IEEE Trans. Acoust. Speech Signal Processing 35 (1987) 356--372.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is that such decompositions exhibit statistical regularities that can be exploited (e.g. 20, 18, 2] Described below is one such decomposition, and a set of statistics collected from this decomposition. The decomposition employed here is based on separable quadrature mirror filters (QMFs) [23, 26, 22]. As illustrated in Fig. 1, this decomposition splits the frequency space into multiple scales and orientations. This is accomplished by applying separable lowpass and highpass filters along the image axes generating a vertical, horizontal, diagonal and lowpass subband. Subsequent scales are ....

M. Vetterli. A theory of multirate filter banks. IEEE Transactions on ASSP, 35(3):356--372, 1987.

....differentiates between natural and un natural images, and how it can be used in several digital forensic applications, specifically in detecting various types of digital tampering. 2. Image Statistics The image decomposition employed here is based on separable quadrature mirror filters (QMFs) [24, 25, 23]. As illustrated in Figure 1, this decomposition splits the Correspondence should be addressed to H. Farid (farid cs.dartmouth.edu) 6211 Sudikoff Lab, Department of Computer Science, Dartmouth College, Hanover, NH 03755 w x w y Figure 1: An idealized multi scale and orientation ....

M. Vetterli. A theory of multirate filter banks. IEEE Transactions on ASSP, 35(3):356--372, 1987.

....el ecom CNET DSE SGV, Issy lesMoulineaux, France (e mail: thierry.blu cnet.francetelecom.fr) Publisher Item Identifier S 1053 587X(98)03912 9. Conjugate quadrature filters (CQF s) 3] 4] made it clear that it was possible to have both FIR and perfect reconstruction. The research in this field [5] was then boosted, resulting in an algebraic description of a filter bank by a polyphase matrix, the determinant of which was shown to characterize perfect reconstruction with FIR filters. An essential achievement of this formulation was the mathematical expression of the CQF property known as ....

M. Vetterli, "A theory of multirate filter banks," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, pp. 356--372, May 1987.

....introduced in [218] CQF filters can be easily designed from Kaiser windows [151] The general formulation for M band perfect reconstruction filter banks was introduced in [261] using paraunitary polyphase matrices. A similar formulation, based on alias cancellation matrices, was presented in [277]. Cosine Modulated Banks One practical problem with the filter banks in Fig. 1 is that the computational complexity grows linearly with the number of subbands, M. If the analysis and synthesis filters have length L, it takes roughly 2L multiplies and additions per input sample to compute them. ....

M. Vetterli, "A theory of multirate filter banks," IEEE Trans. Acoust., Speech, Signal Processing, vol. 35, no. 3, pp. 365-372, Mar. 1987.

....16.35 Hz 32.70 Hz 65.41 Hz 130.8 Hz 261.6 Hz 523.3 Hz 1047 Hz 2093 Hz 4186 Hz 1 the F0 of a single tone since there may be several di#erent F0s within an STFT window. Because a single STFT provides bad time frequency resolution for a certain frequency range, we use a multirate filter bank [ Vetterli, 1987 ] Since the Wavelet transform providing the minimum uncertainty is hard to be performed in real time, we design an STFT based filter bank that provides an adequate time frequency resolution compromise under the real time constraint. Figure 3 shows an overview of our binary tree filter bank. At ....

Martin Vetterli. A theory of multirate filter banks. IEEE Trans. on ASSP, ASSP-35(3):356--372, 1987. -40-

....i#k (m) ffi i#k ffi m#m0 # i# k =0# 1#: #M ; 1# (3) where ffi i#k denotes the Kronecker symbol. For an ideal channel with impulse response c(m) ffi m#0 , 3) is the time domain version of the following condition on the polyphase matrices H p (z)andG p (z) of the analysis and synthesis filter bank [1]: H p (z) G p (z) z ;m0 I : 4) When PR holds for an ideal channel and when we have a non ideal channel in practice, at least intersymbol interference will arise. In addition, when a critically sampled system (N = M ) is used, the frequency bands have to be overlapping and we will have crosstalk ....

M. Vetterli, "A theory of multirate filter banks," IEEE Trans. Acoust., Speech, and Signal Processing, vol. 35, pp. 356--372, March 1987.

....perfect reconstruction property (PR) if X(z) X(z) It is critically sampled if p = q, and is oversampled if p q. The complementary filter problem, i.e. the problem of designing a PR filter bank with FIR analysis and synthesis filters, can be formulated in terms of polyphase decomposition ([1, 5]) In this method, we write H i (z) s and G i (z) s, l l l l l l i q# q# q# H0(z) H1(z) Hp 1(z) G0 (z) G1 (z) Gp 1 (z) X(z) q# q# q# X(z) Figure 1: Oversampled Filter Bank 0 # i # p 1, in the following form: H i (z) q 1 # j=0 z j H i,j (z q ) G i (z) q 1 ....

M. Vetterli, "A theory of multirate filter banks," IEEE Transactions on Acoustic, Speech and Signal Processing, vol. 10, pp. 219--244, April 1986.

....analysis and synthesis filter bank consists of fH i (z)g M Gamma1 i=0 and fF i (z)g M Gamma1 i=0 respectively, that are restricted to be causal and stable transfer functions. This filter bank system is important in subband coding of speech and digital audio applications; See, for example, [7, 9, 10, 11, 12, 14, 17, 19] and their corresponding references. We consider the design issue for synthesis filter banks that achieve zero aliasing error, and minimizes the H1 norm of the error system in Figure 1 with T id = z Gammad for some d 0. Such an H1 optimization is equivalent to minimization of the energy of the ....

M. Vetterli, "A theory of multirate filter banks", IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol.35, No.3, pp. 356-372, March 1987.

.... [5] 6] 7] Both orthogonal and biorthogonal wavelet bases correspond to perfect reconstruct filter banks (PRFBs) Research on wavelets mainly concerns real valued wavelet bases and filter banks as is evident by a large number of publications on the subject [5] 6] 7] 8] 9] 10] 11] [12], 13] 14] However, complex wavelet bases and filter banks have been seldom discussed. Both the continuous wavelet transform (CWT) and Fourier transform are defined for signals that are complex as well as real. The basis functions of the Fourier transform are the complex exponential functions ....

M. Vetterli, "A theory of multirate filter banks," IEEE Trans. ASSP, vol. 35, no. 3, pp. 356--372, Mar. 1987.

....Mux Mux II 1 2 Mux III 0 1 2 Mux IV FPM 1 y(n) x(n) BPM (P 1) P 1) 1 2 P 1 2 P FPM F . a) b) Fig. 1. N th Order FIR Filter (a) SFG representation with DH holdup delays (b) Base (folded) architecture with a BPM and F FPMs. like polyphase decomposition and noble identity [28] [29] 8 . The base architectures for these filters are folded architectures derived by exploiting the slowness of the sampling rate with respect to hardware speed (i.e. P = j Ts T crit k for decimation filters and P = j Ts LT crit k for interpolation filters) The folding sets S 1 ; Delta ....

M. Vetterli, "A theory of multirate filter banks," IEEE Trans. Acoustic Speech Signal Processing, vol. 35, no. 3, Mar. 1987, pp. 356-372.

....system, while visually, the reconstructed images look very similar. 1 Introduction Subband coding systems have been used for signal compression for more than a decade and the corresponding theory has progressed from initial, alias cancellation QMF solutions [1] to perfect reconstruction systems [2, 3, 4, 5]. However, all of these solutions were developed assuming that there is no coding loss. In reality, the system will possess a quantizer in the middle, and hence, information loss will occur. A typical approach in designing a subband coding system has been to find a perfect reconstruction (or an ....

....are drawn. 2 Review Material 2.1 A Glimpse at Perfect Reconstruction Filter Banks Here, we briefly recall some of the concepts from the theory of perfect reconstruction filter banks that are going to be used in the remainder of this paper. For a more extensive treatment of the subject, refer to [3, 4, 5, 14]. An analysis filter bank is a signal processing device that splits the input signal into M channel signals by means of filtering and downsampling by N (where N M ) In what follows, we will assume that N = M , i.e. the filter bank is critically sampled. The synthesis filter bank performs the ....

M. Vetterli, "A theory of multirate filter banks," IEEE Trans. Acoust., Speech, and Signal Proc., vol. 35, pp. 356--372, March 1987.

....2 (3.3) since PL is unitary and by a standard theorem on norms of convolution operators [8, 9] Here, T error = T Gamma T ideal . For our purposes, ess sup can be interpreted as sup , though ess sup is technically correct [9] The M modulation component (alias component) representation [2, 10] of an arbitrary x is given by Xmod (f) X(f) X(f 1 L ) X(f L Gamma1 L ) t : 3.4) Conversion between the modulation component form and the polyphase form is given by Xmod (f) FL S(f)XL (Lf) 3.5) where S(f) diag(1; e Gammaj 2 f ; e Gammaj 2 (L Gamma1)f ) ....

M. Vetterli, "A theory of multirate filter banks," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, pp. 356--372, Mar. 1987.

....disciplines with the advent of the Lapped Orthogonal Transform (LOT) The LOT is nothing but a filter bank featuring perfect reconstruction as well as having orthogonal unit pulse responses. Before this, it was well known that the transform coder can be seen as a special case of the subband coder [63]. The new concept in these papers was to design filter banks with short unit pulse responses and to use the AR(1) coding gain as criterion for goodness, instead of the channel separation properties previously mentioned. A substantial amount of effort has been put into the investigation of subband ....

Martin Vetterli. A theory of multirate filter banks. IEEE Trans. Acoust., Speech, Signal Processing, ASSP-35(3):356--372, March 1987.

....sequence h(n) H(z) will denote its Z transform. # M ] will denote the M fold downsampling defined by y(n) # M ] x(n) x(Mn) and [ M ] denotes the M fold upsampling operator defined by y(n) M ] x(n) x(M Gamma1 n) n 2 L(M) 0 otherwise where L(M) is the lattice generated by M [30, 11]. Finally, we will also adhere to the traditional abuse of notation and denote the Fourier transform H(e ) of the sequence h(n) by H( 2 M band Orthonormal Wavelet Bases M band compactly supported ON wavelet bases are characterized by a sequence known as the unitary scaling vector [15, ....

M. Vetterli. A theory of multirate filter banks. IEEE Trans. ASSP, 35(3):356--372, March 1987.

....unless stated otherwise. 2 A Glimpse at Perfect Reconstruction FIR Filter Banks Here, we briefly recall some of the concepts from the theory of perfect reconstruction filter banks that are going to be used in the remainder of this paper. For a more extensive treatment of the subject, refer to [3, 8, 9, 10]. An analysis filter bank is a signal processing device that splits the input signal into M channel signals by means of filtering and downsampling by N (where N M ) In what follows, we will assume that N = M , i.e. the filter bank is critically sampled. The synthesis filter bank performs the ....

....shift reversal) Let us also point out some facts on multirate filtering that are going to be used later. 1. Upsampling by p and downsampling by q can be interchanged if and only if p and q are relatively prime [9] 2. The output after filtering by H(z) and downsampling by N can be written as [8] Y (z) 1 N N Gamma1 X k=0 H(W k N z 1 N ) Delta X(W k N z 1 N ) 4) where WN denotes the N th root of unity, i.e. WN = e Gammaj 2 N . 3. A pair of useful identities known under the name of noble identities [9] gives conditions under which shift invariant filters can be ....

M. Vetterli, "A theory of multirate filter banks," IEEE Trans. Acoust., Speech, and Signal Proc., vol. 35, pp. 356--372, March 1987.

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M. Vetterli, A theory of multirate filter banks, IEEE Trans. Acoust. Speech Signal Processing 35 (1987) 356--372.

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M. Vetterli, "A theory of multirate filter banks," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, pp. 356--372, 1987.

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M. Vetterli, "A theory of multirate filter banks," IEEE Transactions on ASSP 35(3), pp. 356--372, 1987.

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M. Vetterli. A theory of multirate filter banks. IEEE Transactions on ASSP, 35(3):356--372, 1987.

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M. Vetterli. A theory of multirate filter banks. IEEE Transactions on ASSP, 35(3):356--372, 1987.

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M. Vetterli, "A theory of multirate filter banks," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, pp. 356--372, Mar. 1987.

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M. Vetterli, "A theory of multirate filter banks," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, pp. 356-372, Mar. 1987.

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M. Vetterli, "A theory of multirate filter banks," IEEE rans., vol.ASSP-35, no.3,p3,TJ]MM]T March 1987.

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M. Vetterli, "A theory of multirate filter banks," IEEE Trans. on Acoustics, Speech, and Signal Processing, vol. 35, pp. 356-372, March 1987.

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M.Vetterli: "A Theory of Multirate Filter Banks", Trans. ASSP, March, 1987, pp.356-372.

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