P. P. Vaidyanathan and Phuong-Quan Hoang. Lattice structures for optimal design and robust implementation of twochannel perfect reconstruction QMF banks. IEEE Trans. ASSP, 36(1):81--93, January 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....structure enables flexible and fast biorthogonal transform, and it also allows lossless transform, making it a powerful building tool for wavelet transforms. It has been proven that any orthogonal filterbank can be decomposed into delay elements and plane rotations by lattice factorizations [33]. It is easy to show that any plane rotation can be represented by lifting steps. Therefore, it follows that the DCT a simple orthogonal filterbank can be constructed from the lifting scheme if we start from any plane rotation based factorization of the DCT matrix, such as those in [12] 15] ....

P. Vaidyanathan and P. Hoang, "Lattice structures for optimal design and robust implementation of two-band perfect reconstruction QMF banks," IEEE Trans. Acoust., Speech, Signal Processing, vol. 36, pp. 81--94, 1988.

....means that any filter obtained will meet our requirements. IV. DESIGN RESULTS AND COMPARISONS This section provides some design examples. We compare our filters with those obtained by other existing methods. Quite a few design algorithms are now classical in the dyadic orthonormal case [3] 4] [13], 14] 31] as well as in the dyadic biorthonormal case [9] 32] In contrast, much less [16] 22, orthonormal] 32, biorthonormal] has been done in the true rational case. We shall first compare our new algorithm with others in the dyadic case. Then, for , we shall see that our designed ....

P. P. Vaidyanathan and P. Q. Hoang, "Lattice structures for optimal design and robust implementation of two-channel perfect reconstruction QMF banks," IEEE Trans. Acoust., Speech, Signal Processing, vol. 36, pp. 81--94, Jan. 1988.

....power of in and . The order comes directly from the filter lengths. Our objective is to develop complete and minimal factorization structures for paraunitary matrices of given order with or without further constraints. Many excellent works on this topic have been reported by other researchers [8] [19] For general paraunitary filterbanks, Vaidyanathan et al. propose a complete and minimal structure in [9] and [10] which shows that any paraunitary matrix of McMillan degree can be factorized into a product of degree one paraunitary building blocks and an additional unitary matrix. The ....

....delay property that has been found to be valuable in object based audiovisual signal compression. It is interesting that all the reported results on factorizing a special class of a paraunitary system take the stage order one form. These special systems also include the twochannel filterbanks [8] and cosine modulated paraunitary filterbanks [20] 25] Does a generalized factorization for paraunitary system exist It is reasonable to handle the factorization problem of causal FIR paraunitary systems with or without constraints in one framework. Section II shows that if the degree one ....

[Article contains additional citation context not shown here]

P. P. Vaidyanathan and P.-Q. Hoang, "Lattice structure for optimal design and robust implementation of two- channel perfect reconstruction QMF banks," IEEE Trans. Acoust., Speech, Signal Processing, vol. 36, pp. 81--94, Jan. 1988.

.... we have (1) which appears at the bottom of the next page, where N is the length of , r) It is shown in [15] that the 2M polyphase components of the prototype filter H(z) can be grouped into M power complementary pairs, where each pair is implemented as a two channel lossless lattice filter bank [13], 15] The lattice coefficients are optimized to minimize the stopband attenuation of the prototype filter. The design example in [15] for a 17 channel PR cosine modulated filter bank with stopband attenuation of 40 dB has been presented. In [151, the length of the prototype filter H(z) is ....

.... length (as in (2) It tums out that the necessary and sufficient conditions for the arbitrary length case, in terms of the polyphase functions of the prototype filter, are the same as those in [15] Moreover, these filter banks can be parameterized by pairs of two channel paraunitary filter banks [13], 15] Section IV presents the corresponding lattice structure for the paraunitary cosine modulated filter bank with arbitrary length. The design problem is formulated using the quadratic constrained least squares method [25] and PR cosine modulated filter banks with high stopband attenuation ....

P. P. Vaidyanathan and P. Q. Hoang, "Lattice structures for optimal design and robust implementation of two-chamel perfect-reconstruction QMF banks," IEEE Trans. Acoust., Speech. Signal Processing, vol. 36, pp. 81-94, Jan. 1988.

....rise to overlapping DWT type orthonormal frequency decompositions. Independently of her efforts, in the theory of Filter Banks (where sequences satisfying Eqn. 90 are called Conjugate Quadrature Filters (CQFs) or Quadrature Mirror Filters (QMFs) all such sequences had also been parameterized [46]. We will now see how the DWT can be computed using a filter bank . Thinking of the channel W i as the orthogonal complement of V i in will be useful. Let ci, denote expansion coemcients of f(t) ci, i, f(t) 35) Then ci, is related to ci , and bi, in terms of h0 and h as, bi , h( ....

....discussed in Section. 59 is based on Householder reflections. From a computational perspective, in the two channel case, we find that the Given s rotation based factorizations gives a (moderately) more efficient structure to implement the filter bank. The structure is called the lattice structure [46]. In the general M channel case, the Householder reflection method is much more efficient. Factorization approaches are useful only for IIR and FIR rational, stable unitary matrices. Consider an FIR filter bank with filters hi(n) all of length NM. For simplicity let us further assume that they ....

P. P. Vaidyanathan and Phuong-Quan Hoang. Lattice Structures for Optimal Design and Robust Implementation of Two-Channel Perfect Reconstruction Qmf banks. IEEE T'ats. it ASSP, 36(1):81 93, 1988.

....Coiflets [5] ffl Symmetric biorthogonal wavelets [5, p. 271] ffl Family of wavelets with optimal Sobolev regularity constructed by the author [9, 10] the n z = 1 family from [10] ffl Beylkin 18 , see [13, p. 444] ffl A filter used in speech coding constructed by Vaidyanathan and Huong [11]. The function wavecoef takes as arguments a string describing the family and a number that specifies the order of the wavelet (usually the number of coeffients in the filter) Selwavlt is menu based, instead. WAVECOEF returns some wavelet filter coefficients [h,g] wavecoef(selection,n) ....

P. P. Vaidyanathan and P.-Q. Huong. Lattice structures for optimal design and robust implementation of two-channel perfect-reconstruction QMF banks. IEEE Trans. on Acoustics, Speech, and Signal Processing, 36(1):81--94, 1988.

....1 in E(z) and R(z) The order comes directly from the lter lengths. Our objective is to develop complete and minimal factorization structures for paraunitary matrices of given order with or without further constraints. Many excellent works on this topic have been reported by earlier researchers [8] [19] For general paraunitary lter banks, Vaidyanathan et.al. propose a complete and minimal structure in [9] 10] which shows that any paraunitary matrix of McMillan degree N can be factorized into a product of N degree one paraunitary building blocks and an additional unitary matrix. With ....

....to M [19] However, the condition is too strict to cover all paraunitary systems of a given order. It is interesting that all the reported results on factorizing a special class of paraunitary system take the (K 1) stage order one form. These special systems also include the 2 channel lter banks [8] and cosine modulated paraunitary lter banks [20] 25] Does a generalized factorization for paraunitary system exist It is reasonable to handle the factorization problem of causal FIR paraunitary systems with or without constraints in one framework. Section 2 shows that if the degree one ....

[Article contains additional citation context not shown here]

P. P. Vaidyanathan and P.-Q. Hoang, \Lattice structure for optimal design and robust implementation of two- channel perfect reconstruction QMF banks," IEEE Trans. Acoustics, Speech, and Signal Processing, Vol.36, No.1, pp. 81-94, 1988.

....A framework for the design of wavelet lters was presented, which can be generalized to higher dimensions. There are very few di erent approaches to direct multidimensional orthogonal lter design. The most important among these is the paraunitary polyphase decomposition due to Vaidyanathan ([VH88]) But since his building matrices do not commute in general, the a priori ordering of these matrices is not clear and thus there is no unique representation of all possible orthogonal lters of a given shape, which can be obtained by the proposed method. However, numerical experiments lead to the ....

P.P. Vaidyanathan and P.-Q. Hoang. Lattice Structures for Optimal Design and Robust Implementation of Two-Channel Perfect Reconstruction Filter Banks. IEEE Trans. Acoust., Speech and Signal Proc., 36(1):81-94, 1988.

....structure enables exible and fast biorthogonal transform, and it also allows lossless transform, making it a powerful building tool for wavelet transforms. It has been proven that any orthogonal lter bank can be decomposed into delay elements and plane rotations by lattice factorizations [33]. It is easy to see that any plane rotation can be represented by lifting steps. Therefore, it follows that the DCT a simple orthogonal lter bank can be constructed from the lifting scheme, if we start from any plane rotation based factorization of the DCT, such as those in [12] 13] 14] ....

P. Vaidyanathan and P. Hoang, \Lattice structures for optimal design and robust implementation of two-abdn perfect reconstruction qmf banks," IEEE Trans. ASSP, vol. 36, pp. 81-94, 1988.

....of compactly supported wavelets originated both from mathematical analysis and the signal processing community. The roots of critically sampled wavelet transforms are actually older than the word wavelet and go back to the context of subband filters, or more precisely quadrature mirror filters [36, 37, 41, 51, 52, 53, 54, 58, 56, 59]. In mathematical analysis, wavelets were defined as translates and dilates of one fixed function and were used to both analyze and represent general functions. 15, 20, 25, 35, 24] In the late eighties the introduction of multiresolution analysis and the fast wavelet transform by Mallat and ....

....ff 6= 0) cos ff Gamma sin ff sin ff cos ff # = 1 (cos ff Gamma 1) sin ff 0 1 # 1 0 sin ff 1 # 1 (cos ff Gamma 1) sin ff 0 1 # This corresponds to the well known fact in geometry that a rotation can always be written as three shears. The lattice factorization of [52] allows the decomposition of any orthonormal filter pair into shifts and Givens rotations. It follows any orthonormal filter can be written as lifting steps, by 13 first writing the lattice factorization and then using the example above. This provides a different proof of Theorem 7 in the ....

P. P. Vaidyanathan and P.-Q. Hoang. Lattice structures for optimal design and robust implementation of two-band perfect reconstruction QMF banks. IEEE Trans. Acoust. Speech Signal Process., 36:81--94, 1988.

....filter banks by Mintzer [30] Smith and Barnwell [35, 36] and Vetterli [41] Parks and McClellan [31] developed a method, based on Chebyshev approximation, that can be used to construct finitely supported CQF s whose modulus approximates the modulus of a specified CQF. Vaidyanathan and Hoang [37] described an efficient lattice filterbank structure that is equivalent to factoring elements in SU(2; A 5 ) into products of factors [z; 0] and [fi; fl] fi; fl 2 C: Daubechies [7, 8] used certain finitely supported CQF s, obtained as minimal phase spectral factors of filters constructed by ....

P. P. Vaidyanathan and P. Q. Hoang, Lattice structures for optimal design and robust implementation of two--channel perfect reconstruction QMF banks, IEEE Transactions on Acoustics, Speech, and Signal Processing, Volume ASSP-36, pages 81-94, January 1988.

.... Perfect Reconstruction filter bank (PR) this bank can be implemented using lattice structures, which in the compactsupport orthonormal DWT case implies a finite number R 1 of stages (fig.1) where R = L 2 with L being the length of the original filters, and the km parameters can be found as in [3]. These lattices correspond with the polyphase matrix of those filters, so the downsampling and upsampling operators are translated to the input (analysis) or output (synthesis) reducing in this way the number of adds and multiplications to nearly half of the direct implementations. z 1 A j 1 A ....

P.Vaidyanathan and P.Q.Hoang, "Lattice structures for optimal design and robust implementation of two-channel perfect-reconstruction QMF banks", IEEE Trans. on Acoustic, Speech and Signal Proc., vol. 36, pp. 81-94, 1988.

....the system s PTM is paraunitary than the corresponding LT (H) is orthogonal and vice versa. 0. 4 Factorization of lapped transforms There is an important result for paraunitary PTM which states that any paraunitary E(z) can be decomposed into a series of orthogonal matrices and delay stages [8] [63]. In this decomposition there are N z delay stages and N z 1 orthogonal matrices, where N z is the McMillan degree of E(z) the degree of the determinant of E(z) Then, E(z) B 0 Nz Y i=1 ( Upsilon(z)B i ) 47) where Upsilon(z) diagfz Gamma1 ; 1; 1; 1g, and B i are ....

....1g, and B i are orthogonal matrices. It is well known that an M Theta M orthogonal matrix can be expressed as a product of M(M Gamma 1) 2 plane rotations. However, in this case, only B 0 is a general orthogonal matrix, while the matrices B 1 through BNz have only M Gamma 1 degrees of freedom [63]. This result states that it is possible to implement an orthogonal lapped transform using a sequence of delays and orthogonal matrices. It also defines the total number of degrees of freedom in a lapped transform, i.e. if one changes arbitrarily any of the plane rotations composing the ....

P. P. Vaidyanathan and P. Hoang, "Lattice structures for optimal design and robust implementation of 2-channel PR-QMF banks," IEEE Trans. Acoust., Speech, Signal Processing, ASSP-36, pp. 81--94, Jan. 1988.

....Achieving PR through the use of the Lattice Structure From the results of the previous section it is seen that the bank still has some distortion (though very small) This distortion occurs mainly due the numerical error in computation of the zero, or the cepstrum inv. cepstrum . It is shown in [2] that any pair of filter H o (z) and H 1 (z) that satisfy jH o (e jw )j 2 jH 1 (e jw )j 2 = constant (1:1) can be represented by the lattice shown in Fig 1.9. Furthermore, if we impose the additional constraint that H 1 (z) z GammaN f H o ( Gammaz) 9 0 500 1000 1500 2000 2500 ....

....1.8: Reconstruction of the test signal an an a1 a1 ao ao Z 1 Z Z 1 1 x[n] Ho(z) H1(z) Figure 1.9: Lattice structure for filters satisfying Eq. 1.1 10 then the lattice coeff s must satisfy the following constraint ff 2m = 0 i.e, the even lattice coefficients must be zero. In [2] the author proposes an optimisation based on the above property to design the lattice coeff s and hence the filters to achieve PR. The advantage of design via the lattice structure is that the system is always PR regardless of the coeff s . Hence the technique is not prone to any numerical errors ....

Vaidyanathan, P.P., Hoang P.Q. , Lattice Structures for Optimal Design and Robust Implementation of Two-Channel Perfect Reconstruction QMF Banks, IEEE Trans. on Acoustics, speech and signal processing. Vol. 36, No. 1, pp. 81-94,Jan 1988.

....arbitrary precision by algebraic wavelet QMF lters that have the conjugacy property. Moreover, the minimal necessary eld for these lters can be chosen of degree two. We need a handsome description of all wavelet QMFs to prove this theorem. There exist several parametrizations of QMFs [6] 7] [8], 9] It turns out that the parametrization due to D. Pollen [7] is well suited for our purposes. We review the necessary facts of this parametrization in the next subsection. A. Pollen s Parametrization Pollen associates to every scaling lter a 2 2 matrix with Laurent polynomials in R[z; ....

P. P. Vaidyanathan and P. Q. Hoang, \Lattice structures for optimal design and robust implementation of two-channel perfect reconstruction lter banks", IEEE Trans. Acoust., Speech, Signal Processing, vol. 36, pp. 81-94, 1988.

....power complementarity, other Perfect Reconstruction schemes have sacrificed either linear phase or power complementarity, and thus we divide them into two categories accordingly. The first category consists of schemes such those proposed by Smith and Barnwell [SI84, SI86] and Vaidyanathan et al. [VH88] that use FIR filters that are not linear phase. The second category consists of schemes such as those reported in [ABMD90, VH88, VD89, NV89, VH90, VG89, RV91, ABMD90] that use FIR linear phase filters that give rise to a non orthogonal wavelet expansion, because they do not satisfy power ....

....and thus we divide them into two categories accordingly. The first category consists of schemes such those proposed by Smith and Barnwell [SI84, SI86] and Vaidyanathan et al. [VH88] that use FIR filters that are not linear phase. The second category consists of schemes such as those reported in [ABMD90, VH88, VD89, NV89, VH90, VG89, RV91, ABMD90] that use FIR linear phase filters that give rise to a non orthogonal wavelet expansion, because they do not satisfy power complementarity. Note that approximate power complementarity implies approximate orthonormality of the underlying wavelet basis. It is not clear that the filters described ....

[Article contains additional citation context not shown here]

P. P. Vaidyanathan and P. Q. Hoang. Lattice structures for optimal design and robust implementation of two-band perfect reconstruction QMF banks. IEEE Transactions ASSP, ASSP-36-1:pp. 81--94, January 1988.

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P. P. Vaidyanathan and Phuong-Quan Hoang. Lattice structures for optimal design and robust implementation of twochannel perfect reconstruction QMF banks. IEEE Trans. ASSP, 36(1):81--93, January 1988.

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P P Vaidyanathan and P-Q Hoang, "Lattice structures for optimal design and robust implementation of two-channel perfect reconstruction QMF banks", IEEE Trans. on ASSP, Jan 1988, pp 81-94.

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Vaidyanathan P. P. And Hoang P, "Lattice structures for optimal design and robust implementation of twochannel perfect reconstruction QMF banks", IEEE Transaction on Acoustic, Speech and Signal Processing vo. ASSP-36, pp. 81-94, 1988.

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P. P. Vaidyanathan and P. Q. Hoang, #Lattice structures for optimal design and robust implementation of two-channel perfect reconstruction #lter banks," IEEE Trans. Acoust., Speech, Signal Processing 36, pp. 81#94, 1988.

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P.P. Vaidyanathan and P. Q. Hoang, "Lattice Structures for Optimal Design and Robust Implementation of TwoChannel Perfect-Reconstruction QMF Banks", IEEE Transactions on ASSP, Vol. 36 No. 1, Jan. 1988, pp.8194.

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P. P. Vaidyanathan and P.-Q. Hoang, "Lattice structures for optimal design and robust implementation of twochannel perfect-reconstruction QMF banks," IEEE Trans. Acoust., Speech, Signal Processing 36, pp. 81-94, Jan. 1988.

No context found.

P. Vaidyanathan and P. Hoang, \Lattice structures for optimal design and robust implementation of two-abdn perfect reconstruction qmf banks," IEEE Trans. Accoust. Speech and Signal Process. Vol. 36, pp. 81-94, 1988.

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P. P. Vaidyanathan and P. Q. Hoang. Lattice structures for optimal design and robust implementation of two-channel perfect reconstruction filter banks. IEEE Trans. Acoust., Speech, Signal Processing, 36:81--94, 1988.

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P.P. Vaidyanathan and P.Q. Hoang, "Lattice structures for optimal design and robust implementation of two-channel perfect reconstruction QMF banks", IEEE Transactions on Acoustics, Speech and Signal Processing, Vol.36, pp.81-94, January 1988.

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