Multiresolution Signal Decomposition Transforms, Subbands, and Wavelets phần 8 potx

J.6. BLOCK TRANSFORM PACKETS 363 Figure 5.18: (a) Mixed tertiary/binary tree, (b) Tiling pattern. Scaling Function Wavelet Function Lowpass PR-QMF High-Pass PR-QMF OT 2 °fl 9 9 °T°to 0\ 4 2 2 a T<i> rt °t "I °l Daubechies 0.314 5.22 0.699 0.178 8.97 1.596 0.453 0.987 0.453 0.987 Mostregular 0.143 5.77 0.825 0.188 11.70 2.199 0.470 0.996 0.470 0.996 Coiflet 0.086 11.86 1.02 0.108 39.36 4.25 0.305 1.059 0.305 1.059 Table 5.7: Time-frequency localizations of six-tap wavelet niters and correspond- ing scaling and wavelet functions. 364 CHAPTER 5. TIME-FREQUENCY REPRESENTATIONS 5.6.1 Prom Tiling Pattern to Block Transform Packets The plots of several orthonormal 8x8 block transforms are shown in Fig. 2.7. These depict the basis sequences of each transform in both time- and frequency- domains. These plots demonstrate that the basis sequences are spread over all eight time slots whereas the frequency plots are concentrated over eight separate frequency bands. The variation in these from transform to transform is simply a matter of degree rather than of kind. The resulting time-frequency tiling then would have the general pattern shown in Fig. 5.19(a). There are eight time slots and eight frequency slots, and the energy concentration is in the frequency bands. The basis functions of these transforms are clearly frequency-selective and c«n be regarded as FIR approximations to a "brick wall" (i.e., ideal rectangular band-pas^ filter) frequency pattern which of course would necessitate infinite sine function time responses. Figure 5.19: (a) Tiling pattern for frequency-selective transform, (b) Tiling pattern for time-selective transform The other extreme is that of the shifted Kronecker delta sequences as basis functions as mentioned in Section 2.1. The time- and frequency-domain plots are shown in Fig. 5.20. This realizable block transform (i.e., the identity matrix) has perfect resolution in time but no resolution in frequency. Its tiling pattern is shown in Fig. 5.19(b) and can be regarded as a realizable brick-wall-in-time pattern, the dual of the rionrealizable brick-wall-in-frequency. 5.6. BLOCK TRANSFORM PACKETS 365 Figure 5.20: (a) Basis functions in time; (b) magnitude of Fourier transform of basis functions. The challenge here is to construct a specified but arbitrary tiling pattern while retaining the computational efficiencies inherent in certain block transforms, using the DFT, DOT. MET, and WH. Our objective then is to develop desired time- localized patterns starting from the frequency-selective pattern of Fig. 5.19(a), and conversely, to create frequency-localized tiling from the time-localized Kronecker delta pattern of Fig. 5.19(b). The first case is the time-localizable block transform, or TLBT (Kwak and Haddad, 1994), (Horng and Haddad, 1996). This is a unitary block transform which can concentrate the energy of its basis functions in desired time intervals— hence, time-localizable. We start with a frequency-selective block transform, viz., the DOT, DFT, WH. whose basis functions behave as band-pass sequences, from low-pass to high-pass. 366 CHAPTER 5. TIME-FREQ UENCY REPRESENTA TIONS Figure 5.21: Structure of the block transform packet. as in Fig. 2.7. Consider a subset of M& basis functions with contiguous frequency bands. We then construct a new set of M& time-localized basis functions as a linear combination of the original M& (frequency-selective) basis functions in such a way that each of the new M& basis functions is concentrated over a desired time interval but distributed over M& frequency bands. Hence we can swap frequency resolution for time resolution in any desired pattern. The construction of the TLBT system is shown in Fig. 5.21. Let 0i(n), i,n = 0,1, ,7V — 1, and ipi(n), «,n = 0,1, ,7V — 1, be the original set of orthonormal basis sequences, and the TLBT basis functions, respectively. These are partitioned into subsets by the TV x TV diagonal block matrix, where each coefficient matrix A^ is an M^xM^ unitary matrix, and Y^k-o ^k — N- Consider the kth partition indicated by 5.6. BLOCK TRANSFORM PACKETS and let a* be the iih row of A&, such that '0/c,i( n ) i g ^ le hirier product ^fc,i( n ) = a*$ fe (n). We want to find the coefficient vector a^ such that the TLBT basis sequence 'ipk,i( n ) maximally concentrates its energy in the interval Ii : [i(jf~), (i + l)(']g~) — 1]. We choose to minimize the energy of ^(n) outside the desired I t , : i.e., to minimize subject to orthonormality constraint on the rows of Af~, Q*QJ — 6i j. Hence, we minimize the objective function It can be shown (Prob. 5.5) that the optimal coefficient vector, a ? . is the eigenvector of a matrix E\ which depends only on J> fc , where v^ indicates the conjugate transpose. We now have a procedure for retiling the time-frequency plane so as to meet- any set of requirements. Figure 5.22: (a) Original tiling pattern for frequency concentrated 8x8 transform, (b) Tiling pattern for M fc = 1,1, 2,4. (c) Tiling pattern for M k = 4,1,1, 2. It is noted that the selection of the M^ values determines the time-frequency tiling patterns. The larger the value of Mfc, the more time resolution can be 368 CHAPTER 5. TIME-FREQUENCY REPRESENTATIONS obtained at the cost of sacrificing the resolution in frequency. For example, if we use an 8x8 block transform to construct the time-frequency tiling pattern in Fig. 5.22(b), the entire set of basis functions should be partitioned into four subsets with sizes ,M& = 1, 1,2,4. For the time-frequency tiling in Fig. 5.22(c), the values of Mjt are 4,1,1,2. Figure 5. 23 (a) shows a portion of a time- frequency tiling based on the 64 x 64 DCT transform. According to Fig. 5.23(a) the entire set of DCT basis functions is partitioned into several subsets: {<po(n), , ^>s(n)} for the subspace So with MQ = 4, {(j>4(n)} for Si with MI = 1, {$s(n)} for S 2 with M 2 = 1, {$e(n), $i 3 (n)} for 5,4 with A/4 = 8, • • • . For So, the subset of the TLBT basis functions is $o(n) = |0o( n )? , '03 (ra)}. In Fig. 5. 23 (a), the number on each cell represents the order of the TLBT basis functions. Cells ZQ and Z^ are the regions where tyo(n) and ^(ri) will concentrate their energies. The energy distributions in both the time- and frequency-domains for 0o(?0 and 02 (^) are illustrated in Figs. 5.23(b) and 5.23(c), respectively. These figures demonstrate that il'o(n) and "02 (^) concentrate their energies both around (1-5)(||) in the frequency-domain, but in the different time intervals [0 — 15] and [32 — 47], respectively. From these figures, we see that the TLBT basis functions concentrate most of their energies in the desired time interval and frequency band, specified in Fig. 5.23(a). The dual case is that of constructing a frequency-localized transform FLBT from the time-localized Kroriecker delta sequences. Here, we select M of these to be transformed into 0^,05 •••>'0fe.A/ i via the unitary M x M matrix B. where We define Jj as the energy concentration in the frequency domain in the iih frequency band, where ^^-(u;) is the Fourier transform of the basis function ^/^-(n), and the given frequency band is I Ui — {2m(N/M) < LJ < [27r(i + l)(AT/M)-l]}. J{ is maximized if we choose B to be the DFT matrix, i.e In other words, the DFT is the optimum sequence to transform the information from time-domain to the frequency-domain. Horng and Haddad (1998) describe a 5.6. BLOCK TRANSFORM PACKETS 369 Frequency Figure 5.23: (a) Portion of desired tiling pattern; (b) energy distribution of 0o( n ) in both time- and frequency-domain associated with cell 1 in (a); (c) energy dis- tribution of V'i(ri) in both time- and frequency-domain for cell 3 in (a). 370 CHAPTERS. TIME-FREQUENCY REPRESENTATIONS Figure 5.24: (a) Time localized Kronecker delta tiling pattern; (b) Intermediate pattern; (c) Desired tiling pattern. procedure for constructing a FLBT that matches a desired tiling pattern starting from the delta sequences. The process involves a succession of diagonal block DFT matrices separated by permutation matrices. The final transform can be expressed as the product of a sequence of matrices with DFT blocks along the diagonal and permutation matrices. In this procedure, no tree formulations are needed, and we are able to build a tiling pattern that cannot be realized by pruning a regular or, for that matter, an irregular tree in the manner suggested earlier, as in Figs. 5.16 and 5.17. The procedure is illustrated by the following example. Figure 5.24(c) is the desired pattern for an 8 x 8 transform. Note that this pattern is not realizable by pruning a binary tree, nor any uniformly structured tree as reported in the papers by Herley et. al. (1993). (1) We note that Fig. 5.24(c) is divided into two broad frequency bands, [0,7T/2] and [vr/2, TT]; therefore, we split the tiling pattern of Fig. 5.24(a) into two bands using a 2 x 2 DFT transform matrix. This results in the pattern of Fig. 5.24(b). The output coefficient vector y = [yQ,yi, yj] corresponding to the input data vector / is given by where AI — diag[<f>2, $2, $2» $2] and <!>& is a k x k DFT matrix. The 2x2 DFT matrix takes two successive time samples and transforms them into two frequency-domain coefficients. Thus $2 operating on the first two time- domain samples, /o and /]_, generates transform coefficients yo and y\, which rep- resent the frequency concentration over [0, Tr/2] and [?r/2, TT], respectively. These 5.6. BLOCK TRANSFORM PACKETS 371 are represented by cells yo? yi m Fig- 5.24(b), (2) We apply a permutation matrix P to regroup the coefficients yi into same frequency bands. In this case, P T — [<$Q , #2 > $F? $F? ^T? ^3 •> &§ •> ^i\i where 5k = [0 ,.,(). 1,0, ,0] is the Kronecker delta. (3) Next we observe that the lower frequency band in Fig. 5.24(c) consists of two groups: Group A has 3 narrow bands of width (?r/6) each and time duration 6 (from 0 to 5), and group B has one broad band of width (?r/2) and duration 2 (from 6 to 7). This is achieved by transformation matrix = diag\^^i] applied to the lower half of Fig. 5.24(b). The top half of Fig. 5.24(c) is obtained by splitting the high-frequency band of Fig. 5.24(b) from 7T/2 to TT into two bands of width 7T/4 and time duration 4. This is achieved by transformation matrix = diag\<&<2<&<2} applied to the top half of Fig. 5.24(b). Thus, where A% — diag[3>s, 3>i, $2, $2]- The final block transform is then where A = A$ P A 2 as given in Eq. (5.55) and C = e j27r / 3 . The basis sequences corresponding to cells ZI,ZQ are the corresponding rows of the A matrix. Concentration of these sequences in both time- and frequency- domains is shown in Figs. 5.25 and 5.26. zi(n) is concentrated over first 6 time slots as shown in the plot of zi(n)\ 2 . The associated frequency response \Zi(e^) 2 is shown in Fig. 5.25(b), which is concentrated in the frequency band (7T/6,27r/6). ZQ(H) is concentrated over last four time slots in Fig. 5.26(a) and |Ze(e :? ' u; )| 2 = sin(o;/2) — sin(2o;/2) 2 concentrates over (37r/4,7r) as in Fig. 5.26(b). 372 CHAPTER 5. TIME-FREQUENCY REPRESENTATIONS Figure 5.25: Energy distribution for cell Z\\ (a) time-domain; (b) frequency- domain. Figure 5.26: Energy distribution for cell Z 6 : (a) time-domain; (b) frequency- domain. [...]... BTP and DCT codecs The testing signal is a narrow band Gaussian signal Si with bandwidth^ 0.2 rad and central frequency 5?r/6 Because of the frequency-localized nature of this signal, BTP has only slight compaction improvement over the DCT The signal used in Fig 5. 28( b) is the narrow-band Gaussian signal Si plus time-localized white Gaussian noise $2 with 10% duty cycle and power ratio (81 /82 ) = —8dB... Communication and Image Processing, Vol 1360, pp 609-6 18, Oct 1990 J B Allen and L R R,abiner, "A Unified Approach to Short-Time Fourier Analysis and Synthesis," Proc IEEE, Vol 65, pp 15 58- 1564, 1977 M J Bastiaans, "Gabor's Signal Expansion and Degrees of Freedom of a Signal, " Proc, IEEE, Vol 68, pp 5 38- 539, 1 980 R B Blacknian and J W Tukey, The Measurement of Power Spectra Dover, 19 58 B Boashash and A Riley,... Combes, A Grossman, and P H Tchamitchian, Eds., Time-Frequency Methods and Phase Space Springer, 1 989 R E Crochiere and L R Rabiner, Multirate Digital Signal Processing Prentice-Hall, 1 983 I Daubechies, "Orthonormal Bases of Compactly Supported Wavelets, " Communications in Pure and Applied Math., Vol 41, pp 909-996, 1 988 I Daubechies, "The Wavelet Transform, Time-Frequency Localization and Signal Analysis,"... Younberg and S F Boll, "Constant-Q Signal Analysis and Synthesis." Proc IEEE ICASSP, pp 375-3 78, 19 78 Chapter 6 Wavelet Transform The wavelet transforms, particularly those for orthonormal and biorthogonal wavelets with finite support, have emerged as a new mathematical tool for multiresoiution decomposition of continuous-time signals with potential applications in computer vision, signal coding, and others... 1632, May 19 98 J Horng and R A Haddad, "Interference Excision in DSSS Communication System Using Time-Frequency Adaptive Block Transform," Proc IEEE Symp on Time-Frequency and Time-Scale Analysis, pp 385 - 388 , Oct 19 98 J Horng and R A Haddad, "Block Transform Packets - An Efficient Approach to Time-Frequency Decomposition" , IEEE Int'l Symp on Time-Frequency and Time-Scale Analysis, Oct 19 98 A, J E M Janssen,... S Lim and A V Oppenheim, Eds., Advanced Topics in Signal Processing, Prentice-Hall, 1 988 C H Page, "Instantaneous Power Spectra," J Appl Phys., Vol 23, pp 103 106, 1952 A Papoulis, Signal Analysis McGraw-Hill, 1977 F Peyrin and R Prost, "A Unified Definition for the Discrete-Time DiscreteFrequency, and Discrete-Time/Frequency Wigrier Distributions," IEEE Trans ASSP, Vol ASSP-34, pp 85 8 -86 7, 1 986 M R... Quadratic TimeFrequency Signal Representations," IEEE Signal Processing Magazine, pp 21 67, April 1992 J Horng and R A Haddad, "Variable Time-Frequency Tiling Using Block Transforms," Proc IEEE DSP Workshop, pp 25- 28, Norway, Sept 1996 J Horng and R A Haddad, "Signal Decomposition Using Adaptive Block Transform Packets", Proceedings of ICASSP' 98, Int'l Conf on Acoustics, Speech, and Signal Processing, pp... in frequency, are introduced to provide simple and easily understood examples of multiresolution wavelet concepts Upon establishing the link between wavelets and subbands, we then focus on the properties of wavelet filters and compare these with other multiresolution decomposition techniques developed in this book In particular the Daubechies filters (1 988 ) are shown to be identical to the Binomial QMFs... supported wavelet and its connection with subband techniques The two-band PR filter banks will play a major role in this linkage The multiresolution aspect becomes clear upon developing the one-to-one correspondence between the coefficients in the fast wavelet transform and the dyadic or octave-band subband tree The Haar wavelets, which are discontinuous in time, and the Shannon wavelets, discontinuous... SPIE Visual Communication and Image Processing, pp 723 734, Nov 1992 A N Akansu, R A Haddad and H Caglar, "The Binomial QMF-Wavelet Transform for Multiresolution Signal Decomposition, " IEEE Trans Signal Processing, Vol 41, No 1, pp 13-19, Jan 1993 A N Akansu and Y Liu, "On Signal Decomposition Techniques." Optical Engineering, pp 912-920, July 1991 A N Akansu, R A Haddad and H Caglar, "Perfect Reconstruction . 2 a T<i> rt °t "I °l Daubechies 0.314 5.22 0.699 0.1 78 8.97 1.596 0.453 0. 987 0.453 0. 987 Mostregular 0.143 5.77 0 .82 5 0. 188 11.70 2.199 0.470 0.996 0.470 0.996 Coiflet 0. 086 11 .86 1.02 0.1 08 39.36 4.25 0.305 1.059 0.305 1.059 Table . DCT. The signal used in Fig. 5. 28( b) is the narrow-band Gaussian signal Si plus time-localized white Gaussian noise $2 with 10% duty cycle and power ratio (81 /82 ) = —8dB. Basically, . testing signal is a narrow band Gaussian signal Si with bandwidth^ 0.2 rad and central frequency 5?r/6. Because of the frequency-localized nature of this signal, BTP has only slight