Multiresolution Signal Decomposition Transforms, Subbands, and Wavelets phần 5 pps

208 CHAPTER 3. THEORY OF SUBBAND DECOMPOSITION Using the periodicity, Eq. (3.211), the last expression becomes Finally, (3) The vector of analysis filters h(z] is now of the form, The next step is to impose PR conditions on the polyphase matrix defined by h(z) = H p (z M )z_ M . To obtain this form, we partition C, G, and Z_2M m ^° where Co, Ci, go, g\ are each M x M matrices. Expanding Eq. (3.217) in terms of the partitional matrices leads to the desired form, 3.6. CASCADED LATTICE STRUCTURES 209 Figure 3.48: Structure of cosine-modulated filter bank. Each pair Gk and Gfc+M is implemented as a two-channel lattice. Equation (3.220) suggests that the polyphase components can be grouped into pairs, Gk and z~~ M Gk + M, as shown in Fig. 3.48. Moving the down-samplers to the left in Fig. 3.48 then gives us the structure of Fig. 3.49. Up to this point, the realization has been purely structural. By using the prop- erties of GO, Ci, go, gi in Eq. (3.220) and imposing H^\z~ l }H p (z) — /, it is shown (Koilpillai and Vaidyanathan, 1992) (see also Prob. 3.31) that the necessary and sufficient condition for paraunitary perfect reconstruction is that the polyphase component filters Gk(z) and Gfc+M^) be pairwise power complementary, i.e 210 CHAPTER 3. THEORY OF SUBBAND DECOMPOSITION Figure 3.49: Alternate representation of cosine modulated filter bank. For the case I — 1, all polyphase components are constant, Gk(z) = h(k). This last equation then becomes h 2 (k) + h 2 (k + M] = ^jg, which corresponds to Eq. (3.178). Therefore the filters in Fig. 3.49 {(?&(—£ 2 ), Gk+M(~z 2 )} can be realized by a two channel lossless lattice. We design Gk(z) and Gk+M(z) to be power complementary or lossless as in Section 3.6.1, Fig. 3.45 (with down-samplers shifted to the left), and then replace each delay z~ 1 by — z 2 in the realization. The actual design of each component lattice is described in Koilpillai and Vaidyanathan (1992). The process involves optimization of the lattice parameters. In Nguyen and Koilpillai (1996), these results were extended to the case where the filter length is arbitrary. It was shown that Eq. (3.222) remains necessary and sufficient for paraunitary perfect reconstruction. 3,7. IIR SUBBAND FILTER BANKS 211 3.7 IIR Subband Filter Banks Thus far, we have restricted our studies to FIR filter banks. The reason for this hesitancy is that it is extremely difficult to realize perfect reconstruction IIB analysis and synthesis banks. To appreciate the scope of this problem, consider the PR condition of Eq. (3.100): which requires Stability requires the poles of Q p (z] to lie within the unit circle of the Z-plane. Prom Eq. (3.223), we see that the poles of Q p (z) are the uncancelled poles of the elements of the adjoint of 'Hp(z) and the zeros of det('Hp(z)). Suppose 'H p (z) consists of stable, rational IIR filters (i.e., poles within the unit circle). Then adjHp(z) is also stable, since its common poles are poles of elements of H p (z}. Hence stability depends on the zeros of det('Hp(z))^ which must be minimum- phase—i.e., lie within the unit circle—a condition very difficult to ensure. Next suppose H p (z] is IIR lossless, so that H p (z~~ l }H p (z} — I. If H p (z) is stable with poles inside the unit circle, then "H p (z~ ) must have poles outside the unit circle, which cannot be stabilized by multiplication by z~~ n °. Therefore, we cannot choose Q p (z] = 'H^(z~ 1 ) as we did in the FIR case. Thus, we cannot obtain a stable causal IIR lossless analysis-synthesis PR structure. We will consider two alternatives to this impasse: (1) It is possible, however, to obtain PR IIR structures if we operate the synthe- sis filters in a noncausal way. In this case, the poles of Q p (z) outside the unit circle are the stable poles of an anticausal filter, and the filtering is performed in a non- causal fashion, which is quite acceptable for image processing. Two approaches for achieving this are described subsequently. In the first case, the signals are reversed in time and applied to causal IIR filters (Kronander, ASSP, Sept. 1988). In the second instance, the filters are run in both causal and anticausal modes (Smith, and Eddins, ASSP, Aug. 1990). (2) We can still use the concept of losslessness if we back off from the PR re- quirement and settle for no aliasing and no amplitude distortion, but tolerate some phase distortion. This is achieved by power complementary filters synthe- sized from all-pass structures. To see this (Vaidyanathan, Jan. 1990), consider a 212 CHAPTER 3. THEORY OF SUBBAND DECOMPOSITION lossless IIR polyphase analysis matrix expressed as where d(z) is the least common multiple of the denominators of the elements of 'H p (z), and F(z) is a matrix of adjusted numerator terms; i.e., just polynomials in 2" 1 . We assume that d(z) is stable. Now let Therefore, With this selection, P(z] is all-pass and diagonal, resulting in Hence \T(e^}\ — 1, but the phase response is not linear. The phase distortion implicit in Eq. (3.227) can be reduced by all-pass phase correction networks. A procedure for achieving this involves a modification of the product form of the M-band paraunitary lattice of Eq. (3.193). The substitution converts / H p (z) from a lossless FIR to a lossless IIR polyphase matrix. We can now select Q p (z) as in Eq. (3.225) to obtain the all-pass, stable T(z). To delve further into this subject, we pause to review the properties of all-pass filters. 3.7. IIR SUBBAND FILTER BANKS 213 3.7,1 All-Pass Filters and Mirror Image Polynomials An all-pass filter is an IIR structure defined by This can also be expressed as From this last expression, we see that if poles of A(z] are at (z\, z<2, • • •, %>), then the zeros are at reciprocal locations, (z^ , z^ , • • •, z" 1 ), as depicted in Fig. 3.50, Hence A(z] is a product of terms of the form (1 — az)/(l — az~ l ), each of which is all-pass. Therefore, and note that the zeros of A(z) are all non-minimum phase. Furthermore These all-pass filters provide building blocks for lattice-type low-pass and high- pass power complementary filters. These are defined as the sum and difference of all-pass structures, where AQ(Z), and A\(z] are all-pass networks with real coefficients. Two properties can be established immediately: (1) NQ(Z] is a mirror-image polynomial (even symmetric FIR), and NI(Z) is an antimirror image polynomial (odd symmetric FIR). (2) HQ(Z) and H\(z) are power complementary (Prob. 3.6): (3-233) 214 CHAPTER 3. THEORY OF SUBBAND DECOMPOSITION Figure 3.50: Pole-zero pattern of a typical all-pass filter. A mirror image polynomial (or FIR impulse response with even symmetry) is characterized by Eq. (3.196) as The proof of this property is left as an exercise for the reader (Prob. 3.21). Thus if z\ is a zero of F(z], then zf 1 is also a zero. Hence zeros occur in reciprocal pairs. Similarly, F(z) is an antimirror image polynomial (with odd symmetric impulse response), then To prove property (1), let AQ(Z}, A\(z) be all-pass of orders p 0 and pi, respectively. Then 3.7. IIR SUBBAND FILTER BANKS 215 arid Prom this, it follows that Similarly, we can combine the terms in H\(z) to obtain the numerator which is clearly an antimirror image polynomial. The power complementary property, Eq. (3.218), is established from the fol- lowing steps: Let Then But By direct expansion and cancellation of terms, we find and therefore, W(z) = 1, confirming the power complementary property. These filters have additional properties: (4) There exists a simple lattice realization as shown in Fig. 3.51, and we can write Observe that the lattice butterfly is simply a 2 x 2 Hadamard matrix. 216 CHAPTER 3. THEORY OF SUBBAND DECOMPOSITION Figure 3.51: Lattice realization of power complementary filters; AQ(Z), A\(z) are all-pass networks. 3.7.2 The Two-Band IIR QMF Structure Returning to the two-band filter structure of Fig. 3.20, we can eliminate aliasing from Eq. (3.36) by selecting GQ(Z) = HI(-Z) and GI(Z) - -H 0 (-z). This results in T(z) = H 0 (z)H 1 (-z) - H O (-Z)H!(Z). Now let HI(Z) — HQ(—Z], which ensures that H\(z] will be high-pass if H$(z] is low-pass. Thus T(z) = H%(z) - H 2 (-z) = H 2 (z) - H$(z). Finally, the selection of HQ(Z] and H\(z) by Eq. (3.232) results in Thus, T(z) is the product of two all-pass transfer functions and, therefore, is itself all-pass. Some insight into the nature of the all-pass is achieved by the polyphase representations of the analysis filters, The all-pass networks are therefore 3.7. IIR SUBBAND FILTER BANKS 217 These results suggest the two-band lattice of Fig. 3.52, where O,Q(Z) and a\(z) are both all-pass filters. We can summarize these results with In addition to the foregoing constraints, we also want the high-pass filter to have zero DC gain (and correspondingly, the low-pass filter gain to be zero at u? = ?r). It can be shown that if the filter length N is even (i.e., filter order JV — 1 is odd), then NQ(Z] has a zero at z — — 1 and NI(Z) has a zero at z = \. Pole-zero patterns for typical HQ(z}, H\(z] are shown in Fig. 3.53. Figure 3.52: Two-band power complementary all-pass IIR structure. [...]... in recording and reversing the signals We can account for these by multiplying $(z) by Z~(NI+N'^, where N[ and N% represent the delays in the first and second time-reversal operators Kronander (ASSP, Sept 88) employed this idea in his perfect reconstruction two-band structure shown in Fig 3 .55 Two time-reversals are used in each leg but these can be distributed as shown, and all analysis and synthesis... filters and time-reversal operators to obtain linear-phase filters, as suggested in Fig 3 .54 The design of {Ho(z), H\(z}} IIR pair can follow standard procedures, as outlined in the previous section We can implement HQ(Z) and H\(z) by the all-pass lattice structures as given by Eqs (3.241) and (3.242) and design the constituent all-pass filters using standard tables (Gazsi, 19 85) 3.7 IIR SUBBAND FILTER... section is Viscito and Allebach (1991) and supported also by Karlsson and Vetterli (1990) 3.9.1 2D Transforms and Notation A 2D signal x(ni,n . 19 85) . 3.7. IIR SUBBAND FILTER BANKS 221 Figure 3 .55 : Two-band perfect reconstruction IIR configuration; R denotes a time- reversal operator. Figure 3 .56 : Two-channel IIR subband configuration. . in Fig. 3 .56 as a two-band codec and in Fig. 3 .57 in the equivalent polyphase lattice form. The analysis section consists of low-pass and high-pass causal IIR filters, and the synthesis. implement HQ(Z) and H(z) by the all-pass lattice structures as given by Eqs. (3.241) and (3.242) and design the constituent all-pass filters using standard tables (Gazsi, 19 85) . 3.7.