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VIII Time Frequency and Multirate Signal Processing Cormac Herley Hewlett Packard Laboratories Kambiz Nayebi Sharif University 35 Wavelets and Filter Banks Cormac Herley Filter Banks and

VIII Time Frequency

and Multirate

Signal Processing

Cormac Herley

Hewlett Packard Laboratories

Kambiz Nayebi

Sharif University

35 Wavelets and Filter Banks Cormac Herley

Filter Banks and Wavelets

36 Filter Bank Design Joseph Arrowood, Tami Randolph, and Mark J.T Smith

Filter Bank Equations •Finite Field Filter Banks•Nonlinear Filter Banks

37 Time-Varying Analysis-Synthesis Filter Banks Iraj Sodagar

Introduction •Analysis of Time-Varying Filter Banks•Direct Switching of Filter Banks•

Time-Varying Filter Bank Design Techniques •Conclusion

38 Lapped Transforms Ricardo L de Queiroz

Introduction •Orthogonal Block Transforms•Useful Transforms•Remarks

A N IMPORTANT PROBLEM IN SIGNAL PROCESSING is the choice of how to represent a

signal It is for this reason that importance is attached to the choice of bases for the linear

expansion of signals That is, given a discrete-time signalx(n) how to find a i (n) and b i (n)

such that we can write

i

< x(n), a i (n) > b i (n) (VIII.1)

Ifb i (n) = a i (n), then (VIII.1) is the familiar orthonormal basis expansion formula [1] Otherwise, theb i (n) are a set of biorthogonal functions with the property

< b j (n), a i (n) > = δ i−j

The functionδ is defined such that δ i−j = 0, unless i = j, in which case δ0= 1 We shall consider cases where the summation in (VIII.1) is infinite, but restrict our attention to the case where it is

finite for the moment; that is, where we have a finite numberN 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 haveN 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), a0(n) >

< x(n), a1(n) >

< x(n), a N−2 (n) >

< x(n), a N−1 (n) >











(VIII.2)

and if biare the columns of B then

Clearly B = A−1; if B = A∗ then A is unitary, b i (n) = a i (n) and we have that (VIII.1) is the orthonormal basis expansion

Clearly the construction of bases is not difficult: any nonsingularN × N matrix will do for this

space Similarly, to get an orthonormal basis we need merely take the rows of any unitaryN × N

matrix, for example the identity IN 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 theith 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]

Since in (VIII.1) the signal is written as a superposition of the basis sequencesb i (n), we can say that

ifb i (n) has most of its energy concentrated around time n = n0, then the coefficient< x(n), a i (n) >

measures to some degree the concentration ofx(n) at time n = n0 Equally, taking the discrete Fourier transform of (VIII.1)

i

< x(n), a i (n) > B i (k).

Thus, ifB i (k) has most of its energy concentrated about frequency k = k0, then< x(n), a i (n) >

measures to some degree the concentration ofX(k) at k = k0 This basis function is mostly localized about the point(n0, k0) in the discrete-time discrete-frequency plane Similarly, for each of the basis

functionsb 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 Choosing a basis can then be loosely thought of as choosing some tiling of the time discrete-frequency plane For example, Fig.VIII.1shows the tiling corresponding to various orthonormal bases inC64 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

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 642in this discrete-time discrete-frequency plane The first is the identity basis, which has narrow vertical strips as tiles, since the basis sequencesδ(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 vectorse j2πin/N; these of course are perfectly localized at the frequenciesi = 0, 1, · · · N − 1,

but have equal energy at all times (i.e., 64 points wide, one point long) FigureVIII.1(c) shows the tiling corresponding to a discrete orthogonal wavelet transform (or logarithmic subband coder) operating over a finite length signal FigureVIII.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∗= BT and the

basis is orthonormal provided ATA = I = AAT Of the bases shown in Fig.VIII.1only 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-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 fromR Ntol2(R) are that an infinite number of basis

functions is required instead ofN, the matrices A and B become doubly infinite, and the tilings are in

the discrete-time continuous-frequency plane (the time axis ranges overZ, the frequency axis goes

from 0 toπ, 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 back together using (VIII.3) In general, for example, using the basis functions forR N, evaluating each

of the matrix multiplications in (VIII.2) and (VIII.3) will requireO(N2) 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 efficientO(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.