Tài liệu SEC 08 pptx

VIII TimeFrequency andMultirate SignalProcessing CormacHerley HewlettPackardLaboratories KambizNayebi SharifUniversity 35WaveletsandFilterBanks CormacHerley FilterBanksandWavelets 36FilterBankDesign JosephArrowood,TamiRandolph,andMarkJ.T.Smith FilterBankEquations • FiniteFieldFilterBanks • NonlinearFilterBanks 37Time-VaryingAnalysis-SynthesisFilterBanks IrajSodagar Introduction • AnalysisofTime-VaryingFilterBanks • DirectSwitchingofFilterBanks • Time- VaryingFilterBankDesignTechniques • Conclusion 38LappedTransforms RicardoL.deQueiroz Introduction • OrthogonalBlockTransforms • UsefulTransforms • Remarks A NIMPORTANTPROBLEMINSIGNALPROCESSINGisthechoiceofhowtorepresenta signal.Itisforthisreasonthatimportanceisattachedtothechoiceofbasesforthelinear expansionofsignals.Thatis,givenadiscrete-timesignalx(n)howtofinda i (n)andb i (n) suchthatwecanwrite x(n)=  i <x(n),a i (n)>b i (n). (VIII.1) Ifb i (n)=a i (n),then(VIII.1)isthefamiliarorthonormalbasisexpansionformula[1].Otherwise, theb i (n)areasetofbiorthogonalfunctionswiththeproperty <b j (n),a i (n)> =δ i−j . Thefunctionδisdefinedsuchthatδ i−j =0,unlessi=j,inwhichcaseδ 0 =1.Weshallconsider caseswherethesummationin(VIII.1)isinfinite,butrestrictourattentiontothecasewhereitis c  1999byCRCPressLLC finite for the moment; that is, where we have a finite number N of data samples, and so the space is finite dimensional. We next set up the basic notation used throughout the chapter. Assume that we are operating in C N , and that we have N basis vectors, the minimum number to span the space. Since the transform is linear, it can be written as a matrix. That is, if the a ∗ i are the rows of a matrix A, then A · x =        < x(n), a 0 (n) > < x(n), a 1 (n) > . . . < x(n), a N−2 (n) > < x(n), a N−1 (n) >        (VIII.2) and if b i are the columns of B then x = B · A · x. (VIII.3) Clearly B = A −1 ;ifB = A ∗ then A is unitary, b i (n) = a i (n) and we have that (VIII.1)isthe orthonormal basis expansion. Clearly the construction of bases is not difficult: any nonsingular N × N matrix will do for this space. Similarly, to get an orthonormal basis we need merely take the rows of any unitary N × N matrix, for example the identity I N . There are many reasons for desiring to carry out such an expansion. Much as Taylor or Fourier series are used in mathematics to simplify solutions to certain problems, the underlying goal is that a cleverly chosen expansion may make a given signal processing task simpler. A major application is signal compression, where we wish to quantize the input signal in order to transmit it with as few bits as possible, while minimizing the distortion introduced. If the input vector comprises samples of a real signal, then the samples are probably highly correlated, and the identity basis (where the ith vector contains 1 in the ith position and is zero elsewhere) with scalar quantization will end up using many of its bits to transmit information which does not vary much from sample to sample. If we can choose a matrix A such that the elements of A · x are much less correlated than those of x, then the job of efficient quantization becomes a great deal simpler [2]. In fact, the Karhunen-Lo ` eve transform, which produces uncorrelated coefficients, is known to be optimal in a mean squared error sense [2]. Sincein(VIII.1) the signal is written as a superposition of the basis sequences b i (n), we can say that if b i (n) has most of its energy concentrated around time n = n 0 , then the coefficient < x(n), a i (n) > measurestosomedegree the concentration of x(n)at time n = n 0 . Equally, takingthediscreteFourier transformof(VIII.1) X(k) =  i < x(n), a i (n)>B i (k). Thus, if B i (k) has most of its energy concentrated about frequency k = k 0 , then < x(n), a i (n) > measures to some degree the concentration of X(k)at k = k 0 . This basis function is mostly localized about the point (n 0 ,k 0 ) in the discrete-time discrete-frequency plane. Similarly, for each of the basis functions b i (n) we can find the area of the discrete-time discrete-frequency plane where most of their energy lies. All of the basis functions together will effectively cover the plane, because if any part were not covered there would be a “hole” in the basis, and we would not be able to completely represent all sequences in the space. Similarly the localization areas, or tiles, corresponding to distinct basis functions should not overlap by too much, since this would represent a redundancy in the system. Choosingabasis canthenbe looselythought ofaschoosingsome tilingofthe discrete-timediscrete- frequency plane. For example, Fig. VIII.1 shows the tiling corresponding to various orthonormal bases in C 64 . The horizontal axis represents discrete-time, and the vertical axis discrete-frequency. Naturally, each of the diagrams contains 64 tiles, since this is the number of vectors required for a c  1999 by CRC Press LLC FIGURE VIII.1: Examples of tilings of the discrete-time discrete-frequency plane; time is the hori- zontal axis, frequency the vertical. (a) The identity transform. (b) Discrete Fourier transform. (c) Finite length discrete wavelet transform. (d) Arbitrary finite length transform. basis, and each tile can be thought of as containing 64 points out of the total of 64 2 in this discrete- time discrete-frequency plane. The first is the identity basis, which has narrow vertical strips as tiles, since the basis sequences δ(n + k) are perfectly localized in time, but have energy spread equally at all discrete frequencies. That is, the tile is one discrete-time point wide and 64 discrete-frequency points long. The second, shown in Fig. VIII.1(b), corresponds to the discrete Fourier transform basis vectors e j2πin/N ; these of course are perfectly localized at the frequencies i = 0, 1, ···N − 1, but have equal energy at all times (i.e., 64 points wide, one point long). Figure VIII.1(c) shows the tiling corresponding to a discrete orthogonal wavelet transform (or logarithmic subband coder) operating over a finite length signal. Figure VIII.1(d) shows the tiling corresponding to a discrete orthogonal wavelet packet transform operating over a finite length signal, with arbitrary splits in time and frequency; construction of such schemes is discussed in Section 7.1. In Fig. VIII.1(c) and (d), the tiles have varying shapes but still contain 64 points each. It should be emphasized that the localization of the energy of a basis function to the area covered by one of the tiles is only approximate. In practice, of course, we will always deal with real signals, and in general we will restrict the basis functions to be real also. When this is so, B ∗ = B T and the basis is orthonormal provided A T A = I = AA T . Of the bases shown in Fig. VIII.1 only the discrete Fourier transform will be excluded with this restriction. One can, however, consider a real transform which has many properties in common with the DFT, for example the discrete Hartley transform [3]. While the above description was given in terms of finite-dimensional signal spaces, the interpre- c  1999 by CRC Press LLC tation of the linear transform as a matrix operation, and the tiling approach remains essentially unchanged in the case of infinite length discrete-time signals. In fact, for bases with the structure we desire, construction in the infinite-dimensional case is easier than in the finite-dimensional case. The modifications necessary for the transition from R N to l 2 (R) are that an infinite number of basis functions is required instead of N, the matrices A and B become doubly infinite, and the tilings are in the discrete-time continuous-frequency plane (the time axis ranges over Z, the frequency axis goes from 0 to π, assuming real signals). Good decorrelation is one of the important factors in the construction of bases. If this were the only requirement, we would always use the Karhunen-Lo ` eve transform, which is an orthogonal data- dependent transform which produces uncorrelated samples. This is not used in practice, because estimating the coefficients of the matrix A can be very difficult. Very significant also, however, is the complexity of calculating the coefficients of the transform using (VIII.2), and of putting the signal backtogetherusing (VIII.3). Ingeneral, for example,using thebasisfunctionsfor R N , evaluatingeach of the matrix multiplications in (VIII.2) and (VIII.3) will require O(N 2 ) floating point operations, unless the matrices have some special structure. If, however, A is sparse, or can be factored into matrices that are sparse, then the complexity required can be dramatically reduced. This is the case, for example, with the discrete Fourier transform, where there is an efficient O(N log N) algorithm to do the computations, which has been responsible for its popularity in practice. This will also be the case with the transforms that we consider, A and B will always have special structure to allow efficient implementation. References [1] Gohberg, I. and Goldberg, S., Basic Operator Theory, Birkh ¨ auser, Boston, MA, 1981. [2] Gersho, A. and Gray, R.M., Vector Quantization and Signal Compression, Kluwer Academic, Nor- well, MA, 1992. [3] Bracewell, R., The Fourier Transform and its Applications, 2nd ed., McGraw-Hill, New York, 1986. c  1999 by CRC Press LLC . is one discrete-time point wide and 64 discrete-frequency points long. The second, shown in Fig. VIII.1(b), corresponds to the discrete Fourier transform. splits in time and frequency; construction of such schemes is discussed in Section 7.1. In Fig. VIII.1(c) and (d), the tiles have varying shapes but still