P. P. Vaidyanathan. Mutirate Systems and Filter Banks. Prentice Hall, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... 92 UNITARY FIR FILTER BANKS AND SYMMETRY R.A.Gopinath and C.S. Burrus Department of Electrical and Computer Engineering, Rice University, Houston, TX 77251 CML TR 92 17 20th June 92 Abstract In the last decade a number of perfect reconstruction filter bank structures have been proposed [11]. Of these the modulated filter banks are the easiest to design and implement [7, 6, 4] However, the filters in a modulated filter bank cannot be linear phase. Recently, unitary FIR filter banks with linear phase have been completely parameterized [10] In that paper, the eigenstructure of the ....

....symmetry we will consider is, h i (n) Sigmah i (N Gamma 1 Gamma n) H i (z) SigmaH R i (z) 3) where the filters are all of fixed length N , and the symmetry is about N Gamma1 2 . For these three symmetries, the next section studies the form of the (first orthant or Type 1 [11, 1]) polyphase component matrix, H p (z) of the analysis bank. 2 Characterization of Filter Banks with Symmetry We will only consider the case of even M . For even M , if the filters are linear phase half the filters are symmetric, while the other half are anti symmetric [10] Let J denote the ....

P. P. Vaidyanathan. Mutirate Systems and Filter Banks. Prentice Hall, 1992.

....neatly [3] Fact 1 L(M 1 ) is a subset of L(M 2 ) iff there exists an integer matrix K such that M 1 = M 2 K. Fact 1 implies that L(M) L = L(I) iff M is integral (see Fig. 3) In the example above, the number of lattice points per unit volume 1=36 = 1= jM j. This is true for all lattices [3, 18]. The generator of a lattice is not unique. In Fig. 3 M = 8 2 6 6 # = 6 2 0 6 # 1 0 1 1 # is also a generator of same lattice. The generator is unique only up to right multiplication by a unimodular matrix. If M is unimodular, then the number of lattice points per unit volume is ....

....2, we get Theorem 3. 2 6 Commuting Filters and Upsamplers Downsamplers If we can analyze cascades of filters and up downsamplers a complete set of tools will be in place. Filter upsampler and downsampler filter cascades are easily analyzed duals (these results are known as the Noble identities [18, 15]) Filter downsampler and upsampler filter cascades involve important concepts that play a major role in filter bank theory. This section introduces the new concept, generalized polyphase representations, giving more general results than are available in the literature. An important new identity ....

P. P. Vaidyanathan. Mutirate Systems and Filter Banks. Prentice Hall, 1992.

....problem is given in Fig. 1. The filter bank problem involves the design of the real coefficient realizable (i.e. FIR or causal stable IIR) filters h i (n) and g i (n) with the following goals: Perfect Reconstruction (i.e. y(n) x(n) and approximation of ideal frequency responses (see Fig. 2) [37, 41, 42, 45, 49, 46, 31, 44]. Closely x(n) hM Gamma1 (n) h 1 (n) h 0 (n) # M # M # M dM Gamma1 (n) d 0 (n) d 1 (n) M : M M g M Gamma1 (n) g 1 (n) g 0 (n) j i ....

.... : hM Gamma1 (n) h 1 (n) h 0 (n) # M # M # M xM Gamma1 (n) x 0 (n) x 1 (n) Figure 3: An M channel Transmultiplexer For PR the filters h i and g i have to satisfy a set of algebraic conditions [44]. Let L(M) f: Gamma2M; GammaM; 0; M; 2M; g and R(M) f0; 1 : M Gamma 1g. L(M) is the lattice generated by M and R(M) is the set of representatives of L(M ) For any sequence x(n) one can define the polyphase representation (see [44, 2, 15] with respect to R(M) as follows: ....

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P. P. Vaidyanathan. Mutirate Systems and Filter Banks. Prentice Hall, 1992.

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P. P. Vaidyanathan. Mutirate Systems and Filter Banks. Prentice Hall, 1992.

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