SDP APPROXIMATION OF THE HALF DELAY AND THE DESIGN OF HILBERT PAIRS. Bogdan Dumitrescu
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1 SDP APPROXIMATION OF THE HALF DELAY AND THE DESIGN OF HILBERT PAIRS Bogdan Dumitrescu Tampere International Center for Signal Processing Tampere University of Technology P.O.Box 553, 3311 Tampere, FINLAND ABSTRACT This paper presents a method for designing an orthogonal Hilbert pair of wavelets. The wavelets are generated each by an orthogonal (CQF) filter bank. The scaling filter defining the first filter bank and wavelet is given. The second filter is optimized using an exact semidefinite programming (SDP) description of a special H error measure of the half-sample delay. The result of the SDP problem is then refined to meet the orthogonality conditions. Experimental results show that the proposed method can bring significant improvement to Hilbert pairs designed with other methods. 1. INTRODUCTION The dual-tree complex wavelet transform (DT-CWT) [1] is a structure using two real wavelet transforms that emulate a complex one. One of the wavelets is an approximate Hilbert transform of the other; we say that the wavelets form a Hilbert pair. The DT-CWT has several properties that make it appealing in certain applications; its more important features are the nearly shift invariance and the directional selectivity (in two dimensions); a detailed review of the DT-CWT can be found in []. Let us denote H(z) and G(z) the scaling filters that determine the wavelets forming the DT-CWT. Each of them can be seen as the filter on the first channel of the analysis part of an orthogonal (CQF) filter bank. (We discuss here only the case of orthogonal wavelets; the techniques we propose can be extended to biorthogonal wavelets, with relatively minor modifications.) So, the filters must satisfy the conditions: H(z)H(z 1 ) + H( z)h( z 1 ) =, G(z)G(z 1 ) + G( z)g( z 1 ) =. Let ψ h (t) and ψ g (t) be the wavelets determined by the two filters. They form a Hilbert pair if their Fourier transforms obey to { jψ h () for > Ψ g () = () jψ h () for < (1) Work supported by Academy of Finland, project No. 1346, Finnish Centre of Excellence program (6-11). The author is on leave from the Department of Automatic Control and Computers, Politehnica University of Bucharest, Romania. Equivalently, the complex wavelet ψ h (t) + jψ g (t) must have a Fourier transform that is analytic, i.e. Ψ h () + jψ g () =, for <. (3) It was suggested in [3] and proved in [4] that the only condition for the filters to generate a DT-CWT is that their magnitude response is the same and that their phase response differ by ω/, i.e. G(ω) = H(ω)e jω/. (4) In other words, the ratio of the transfer functions of the two filters is a half-sample delay. The condition (4) cannot be met exactly with practical filters. We will work with FIR filters and consider the following optimization problem, introduced in [5]. We assume that one of the filters, say H(z), is given; of course, this filter satisfies the orthogonality and perfect reconstruction condition (1). We want to find G(z) such that G(z) z 1/ H(z) (5) is minimum. It is also desired that the filter G(z) has a preset number L of zeros in 1 (the corresponding wavelet has L vanishing moments), typically the same number of zeros that H(z) has in 1. This optimization problem has been solved approximately in [5], by transformation into a (continuous-time) sampled-data problem, solved using standard H control tools. In this paper, we show that the minimization of (5) is equivalent to a linear matrix inequality (LMI) and can be performed exactly. If G(z) z 1/ H(z) and H(z) satisfies (1), then G(z) satisfies approximately the condition (1). However, there is no control on the error in satisfying (1) and this aspect is not mentioned in [5]. Unfortunately, unlike (5), the constraint (1) is not convex. So, it cannot be simply added to the optimization problem. To deal with this issue, we propose to modify the G(z) minimizing (5), by finding the nearest filter that satisfies (5). Although this two-step approach cannot guarantee optimal results, it proves efficient in practice. Section presents our method, the main original contribution being the exact minimization of (5). Section 3 contains two examples of design, showing better performance than those obtained with previous methods.
2 . BOUNDED REAL LEMMA APPROACH The minimization of (5) is clearly related to the inequality G(ω) e jω/ H(ω) γ, ω [, ω ], (6) where γ is a positive real (to be minimized, but for now constant) and ω [, π]; we take ω = π to minimize (5), but an arbitrary value makes the problem more general and may be useful in practice. The inequality (6) has a form that suggests a trigonometric polynomial Bounded Real Lemma (BRL) approach; only the term e jω/ does not fit into the usual BRL framework. This difficulty is fixed easily, by noticing that (6) is equivalent to G(ω) e jω H(ω) γ, ω [, ω /]. (7) Let us assume that H(z) and G(z) are FIR filters, written as H(z) = N N h k z k, G(z) = g k z k. (8) k= k= The filters may have different degrees in what follows, with modifications that are obvious. An LMI that is a sufficient condition for inequalities like (7) has been presented in [6], for multivariate polynomials. In the univariate case, the LMI is equivalent to inequality. We discuss here the particular form of the LMI that corresponds to (7). We define F(z) = G(z ) z 1 H(z ). (9) Let M be an even integer and denote R(z) = M r k z k, r k = r k, (1) M a symmetric trigonometric polynomial. This polynomial is nonnegative on the interval [, ω /] if and only if (see e.g. [7, Th. 1.17]) there exist globally nonnegative (symmetric) trigonometric polynomials A(z) and B(z) of degrees M and M, respectively, such that R(z) = A(z) + ( z+z 1 cos ω ) ( 1 z+z 1 ) B(z). (11) The coefficients of the polynomials A(z), B(z) can be expressed as linear functions of positive semidefinite matrices Q a and Q b, using the trace parameterization [8, 9, 1], by a k = tr[θ k Q a ], b k = tr[θ k Q b ], (1) where Θ k is the elementary Toeplitz matrix with ones on diagonal k and zeros elsewhere. Note that the sizes of matrices Q a and Q b are (M + 1) (M + 1) and (M 1) (M 1), respectively. Denoting α = 1 ω cos, β = 1 ω (1 + cos ), χ = 1 4 (13) the coefficients of the polynomial multiplying B(z) in (11) and combining (1) and (11), we obtain r k = a k + αb k + β(b k 1 + b k+1 ) + χ(b k + b k+ ) = tr[θ k Q a ] + tr[αθ k + β(θ k 1 + Θ k+1 ) + χ(θ k + Θ k+ )]Q b. (14) In this context, Q a and Q b are named Gram matrices associated with R(z). The BRL for FIR systems is formulated as follows (a general proof can be found in [7, Ch. 4]). Let f be a vector containing the coefficients of F(z) from (9); since the degree of F(z) is N + 1, i.e. odd, we raise it to M = (N + 1) by adding a zero coefficient; this coefficient comes on the last position and so the expression of the vector f R N+3 is f = [g h g 1 h 1... g N h N ] T. (15) The inequality F(ω) γ holds for all ω [, ω /] if and only if there exist Q a, Q b such that γ δ k = tr[θ k Q a ] + tr[αθ k + β(θ k 1 + Θ k+1 ) + χ(θ k + Θ k+ )]Q b (16) and [ Qa f f H 1 ]. (17) (These relations come from a majorization relation between the Gram matrices associated with the nonnegative polynomials γ and F(ω).) Hilbert pair optimization. We return now to our original problem. Given an FIR filter H(z) of order N and the frequency bound ω, find G(z) of order N such that the H norm γ from (6) is minimum. Using the BRL described above, this problem can be formulated as min γ γ,g,q a,q b subject to (16), (17) Q a, Q b (18) Due to (15), the coefficients of the variable G(z) appear linearly, and so it results that (18) is an SDP problem. Moreover, the regularity (number of vanishing moments) condition can be imposed on G(z) without changing the nature of the problem. Indeed, if G(z) has L roots in 1, then it can be written as G(z) = G(z)(1 + z 1 ) L. Denoting g R N L+1 the vector containing the coefficients of the new variable G(z), it results that f = A g+b, i.e. the new variable appears affinely in (17); the constant matrix A and vector b can be derived easily and so we do not give explicit relations for them. Orthogonality conditions. The conditions (1) cannot be imposed on G(z) in conjunction with (18) because they are not convex. We have added a single condition to (18); since G(ω) ω= = 1, it results that N g k = 1, (19) k=
3 which is a linear constraint and so it fits into the SDP framework. Expressed on the coefficients of G(z), the conditions (1) are equivalent to N l k= g k g k+l = δ l, for l = : (N 1)/. () (This shows that it is natural to take N odd, otherwise the last coefficient of G(z) is zero.) These conditions are quadratic equalities and so they are not convex. Solution refinement. The given filter H(z) satisfies the orthogonality conditions. As it is expected that the optimal γ given by (18) is small, it results that we have a good approximation in G (z) z 1/ H(z), where G (z) is the solution of (18). So, we expect that the error E o = (N 1)/ l= ( δ l N l k= g k g k+l ) 1/ (1) is relatively small. So, there must be a filter G(z) satisfying (), relatively near from G (z), for which the value γ from (6) is relatively near from γ. We thus propose to solve the quadratic problem min g g g s.t. () () To this goal, we use a general purpose constrained optimization routine, initialized with g. Although () has many local minima, we expect that the detected solution is near g. 3. EXPERIMENTAL RESULTS We have solved the SDP problem (18) using the library SeDuMi [11]. For solving (), we have used the Matlab function fmincon. Since the filters orders are relatively small, the problems are solved in few seconds. Example 1. We consider first the filters from [3], Example, with N = 9, L = 3. The first two columns of Table 1 contain the coefficients of the filters designed in [3] (using algebraic methods), which form an approximate Hilbert pair. Using the first filter, H(z), as input for our algorithm, we obtain a filter G (z) after solving (18) with the additional constraint (19) and a filter G(z) after solving (). The coefficients of this later filter are shown in the third column of Table 1. We name G (z) brute filter (since it does not respect the orthogonality constraints) and G(z) refined filter. The orthogonality error (1) is E o = for the brute filter and E o =. 1 1 for the refined filter. Figure 1 shows the error E(ω) = G(ω) e jω/ H(ω). (3) The dash-dot curve for the brute filter has the typical equiripple aspect of a H solution. The solid curve corresponds to the refined filter and is clearly better than the dashed curve for the original filters from [3]. However, it is more important to look at the analyticity error of the complex wavelet ψ(t) = ψ h (t)+jψ g (t). The magnitude of its Fourier transform Ψ() is shown in Figure, for the refined (solid curve) and original (dashed) filters. The detailed region from Figure 3 shows that the complex wavelets designed with the current method are nearer from zero for <. Although the visual information is relevant, we can use also the analyticity measures proposed in [1], for the peak error, and E 1 = max < Ψ() max > Ψ() E = Ψ() d Ψ() d (4) (5) for the energy error. In percents, the errors for the original filters are E 1 =.14, E =.59. For our method, the errors are E 1 = 1.16, E =.1, i.e. much better. Example. We look now at the filters from [13], Example 1A, with N = 11, L = 4, designed using an allpass approximation of the half delay. The information we present is structured as in the previous example. Table gives the coefficients of the filters. Figure 4 presents the error (3) and Figures 5 6 the spectrum of the complex wavelet. It is again clear that the proposed method produces wavelets with better analyticity measures. Indeed, the errors for the original filters are E 1 = 1.6, E =.37, while for our method the errors are E 1 = 1.1, E =.176. We also note that the orthogonality error (1) is E o = for the brute filter and E o = for the refined filter. For all the other few examples from the cited papers, our method was able to improve the Hilbert pair designed there with other means. 4. CONCLUSION We have presented a method that, given a filter defining an orthogonal filter bank, designs a second such filter such that the wavelets generated by the two scaling filters form a Hilbert pair. The design method has two steps. The first consists of an SDP optimization problem and find the second filter such that its ratio with the first one is (approximately) a half delay. The second step is a nonconvex optimization that make the filter obey to the orthogonality constraints, without deteriorating significantly the quality of the Hilbert pair. The numerical examples show that our method gives better results, in terms of diverse analyticity measures of the Hilbert pair, than previous designs. Future work will be directed to study the effect of ω on the solution of (6). Another way of improving the solution may be the insertion of a weighting filter in (6); if this filter is FIR, then the optimization problem can still be transformed into SDP and thus solved efficiently.
4 Table 1. Coefficients of Hilbert pair filters for Example 1. h k g o,k [3] g k Table. Coefficients of Hilbert pair filters for Example. h k g o,k [13] g k
5 .6.5 Brute Brute E.3 E Normalized frequency Normalized frequency Figure 1. Errors E(ω) for Example 1. Figure 4. Errors E(ω) for Example Figure. Spectrum of complex wavelets for Example 1. Figure 5. Spectrum of complex wavelets for Example Figure 3. Spectrum of complex wavelets for Example 1 (detail). Figure 6. Spectrum of complex wavelets for Example (detail).
6 5. REFERENCES [1] N.G. Kingsbury, Complex wavelets for shift invariant analysis and filtering of signals, J. Applied and Computational Harmonic Analysis, vol. 1, no. 3, pp , June 1. [] I.W. Selesnick, R.G. Baraniuk, and N.G. Kingsbury, The Dual-Tree Complex Wavelet Transform, IEEE Signal Proc. Magazine, vol. 6, pp , Nov. 5. [3] I.W. Selesnick, Hilbert Transform Pairs of Wavelet Bases, IEEE Signal Proc. Letters, vol. 8, no. 6, pp , June 1. [4] R. Yu and H. Ozkaramanli, Hilbert Transform Pairs of Orthogonal Wavelet Bases: Necessary and Sufficient Conditions, IEEE Trans. Signal Processing, vol. 5, no. 1, pp , Dec. 5. [5] R. Yu and A. Baradarani, Design of Hilbert transform pairs of wavelet bases via LMI optimization, in 17th Int. Symp. on Math. Theory of Networks and Systems, Kyoto, Japan, 6, pp [6] B. Dumitrescu, Trigonometric Polynomials Positive on Frequency Domains and Applications to -D FIR Filter Design, IEEE Trans. Signal Proc., vol. 54, no. 11, pp , Nov. 6. [7] B. Dumitrescu, Positive trigonometric polynomials and signal processing applications, Springer, 7. [8] B. Alkire and L. Vandenberghe, Convex optimization problems involving finite autocorrelation sequences, Math. Progr. ser. A, vol. 93, no. 3, pp ,. [9] B. Dumitrescu, I. Tăbuş, and P. Stoica, On the Parameterization of Positive Real Sequences and MA Parameter Estimation, IEEE Trans. Signal Proc., vol. 49, no. 11, pp , Nov. 1. [1] Y. Genin, Y. Hachez, Yu. Nesterov, and P. Van Dooren, Optimization Problems over Positive Pseudopolynomial Matrices, SIAM J. Matrix Anal. Appl., vol. 5, no. 1, pp , 3. [11] J.F. Sturm, Using SeDuMi, a Matlab Toolbox for Optimization over Symmetric Cones, Optimization Methods and Software, vol. 11-1, pp , 1999, [1] D.B.H. Tay, N.G. Kingsbury, and M. Palaniswami, Orthonormal Hilbert-Pair of Wavelets With (Almost) Maximum Vanishing Moments, IEEE Signal Proc. Letters, vol. 13, no. 9, pp , Sept. 6. [13] I.W. Selesnick, The Design of Approximate Hilbert Transform Pairs of Wavelet Bases, IEEE Trans. Signal Processing, vol. 5, no. 5, pp , May.
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