Variable Fractional Delay FIR Filters with Sparse Coefficients

Transcription

1 Variable Fractional Delay FIR Filters with Sparse Coefficients W.-S. Lu T. Hinamoto Dept. of Electrical & Computer Engineering Graduate School of Engineering University of Victoria Hiroshima University Victoria, Canada Hiroshima, Japan

2 Outline Design Problem and Related Work Significance Design Method at a Glance A Design Example Concluding Remarks

3 . Design Problem and Related Work Design Problem Find a VFD FIR digital filter of order (N, K) N K n n n nk n= k= H (, zp) = a( pz ), a( p) = a p k with p and sparse A = {a ij } ( N ) ( K ) R + + that optimally approximates a desired frequency response H ( ω, p) = e ω in a weighted least-squares sense. π d 3 j ( D+ p) J ( A) = W( ω, p) H( ω, p) H ( ω, p) dpdω subject to: sparsity( A) = N z d

4 Related Work A. Tarczynski, G. D. Cain, E. Hermanowicz and M. Rojewki, 997 W.-S. Lu and T.-B. Deng, 999. T.-B. Deng,. C.-C. Tseng, 4. T.-B. Deng and Y. Lian, 6. Linear programming algorithms for sparse FIR filters (T. Baran, D. Wei, and A.V. Oppenheim, March ) l -minimization algorithms for sparse -D and -D FIR filters (W.-S. Lu and T. Hinamoto,, ) 4

5 . Significance Implementing a VFD filter in Farrow model is costly as each coefficient of a VFD filter is a polynomial rather than a scalar. Hence VFD digital filters with sparse coefficients are of interest because the sparsity implies reduced implementation complexity (cost), hence real-time application potential. VFD filters are not sparse in general. 5

6 3.Design Method at a Glance Define T jω jnω K ω = e e and p = p p and assume a separable weighting function W( ω, p) = W( ω) W( p), then up to a constant one can write where π J ( A) = W( ω, p) H( ω, p) H ( ω, p) dpdω T = tr( PA ΩA) tr( SA) d T P S π T T = W( p) dp, Re W( ω) dω pp Ω = ωω π T T T jω ( D+ p) = W( p) pdp, p Re W( ω) e dω pω ω = ω 6

7 To deal with large condition numbers of P and Ω, Cholesky T T decompositions P = P P, Ω = ΩΩ are used, and up to a constant J(A) can be written as T J ( a) = Γa y, Γ = P Ω, y= Γ s where a and s are the vectors generated by concatenating the columns of A and S, and is the Kronecker product. Design Phase To identify an index set of the most appropriate locations in a to be set to zero in order to satisfy a target sparsity. jω This is done subject to maintaining closeness of H ( e, p) to H ( ω, p). d 7

8 The target sparsity is achieved by introducing sparsity promoting l -norm of a into an objective function: minimize μ a + Γ a y () a where a n = i= a i. Problem () is a convex, for which many fast algorithms are available. E.g. using FISTA, () can be solved with a small number of iterations. Input: Data Γ and y, parameter μ and iteration number M. Step. Compute A = Γ y, a = A (:). Set b = a, t =, and m =. Step. Compute a m = S μ/l {b m ( /L)Γ T (Γb m y)}, where Sα ( u) = sgn( u) max{ u α,} Step 3. Update t m+ = (+ + 4 t m ) / Step 4. Update b m+ = a m + (( tm ) / t m + ) (a m a m ). Step 5. If m < M, set m = m + and repeat from Step ; otherwise stop and output a m as solution â. 8

9 Hard thresholding is applied to vector â with an appropriate value of threshold ε so that the length of the index set I = {, i aˆ () i < ε } equals to target sparsity N z. The index set I is a key ingredient of design phase. Design Phase To find a coefficient matrix A that minimizes the WLS error J(A) subject to the sparsity constraint. This part of the design is carried out by solving the convex problem 9 Γ minimize J ( a) = a y a subject to: ai ( ) = for i I By simply substituting the constraints into the objective function, the above problem becomes an unconstrained least square problem whose solution is given by

10 ( I ) Γ s and ( I ) a = y a = where I denotes the set of index not in I and Γ s is composed of those columns of Γ with indices in set I. 4. A Design Example The Design Problem Design a VFD FIR filter of order (N = 65 and K = 7) with cutoff ωc =.9π. Design performance is evaluated in terms of maximum error emax = max { e( ω, p), ω.9 π, p } with e( ω, p) = log H( ω, p) H ( ω, p) and L-error d

11 .9π e = H( ω, p) Hd ( ω, p) dpdω / The weighting function was set to W( ω, p) = W( ω) W( p) with W (p) = for p in [, ] and for ω [,.88 π ) W ( ω) = 3 for ω [.88 π,.8994 π) for ω [.88 π, π] The target sparsity was set to N z = 98 which means a 37.5% of coefficients were set to zero. To achieve this sparsity, the two key parameters in phase- were set to μ 5 3 =, ε =. It took 6 FISTA iterations for the algorithm in phase to converge.

12 Phase of the design then produced an optimal A* with sparsity(a*) = 98. Below are the numerical evaluation results: maximum error e max =.. L-error e = 75.5 db. ======== Comparison ======= Compare with an equivalent nonsparse VFD filter of order (N = 65, K = 4) with the same specifications: maximum error e max =.69. L-error e = 45.8 db.

13 Profile of frequency response error e( ω, p) over ω.9 π, p : the sparse VFD filter: 5 e(,p) (db) Fractional delay p Normalized frequency 3

14 Profile of frequency response error e( ω, p) over ω.9 π, p : the nonsparse VFD filter: 5 e(,p) (db) Fractional delay p Normalized frequency 4

15 ======== Comparison ======= To justify phase of the design, we compare the above result with the following: We design a conversional (nonsparse) VFD filter of order (N = 65, K = 7), then applied hard thresholding to generate exactly 98 locations that may be considered appropriate to set to zero. We then went on the carried out phase to yield a sparse VFD filter. It was found that maximum error e max =.5 (vs. with phase ) L-error e = 73.4 db (vs 75.5 db with phase ). 5

16 Profile of frequency response error e( ω, p) over ω.9 π, p : the sparse VFD filter without phase : 5 e(,p) (db) Fractional delay p Normalized frequency 6

17 ======== Comparison 3 ======= Fractional delay over ω.9 π, p. The sparse filter:.8 Fractional delay Fractional delay p Normalized frequency.8 7

18 Fractional delay over ω.9 π, p. The equivalent nonsparse filter:.8 Fractional delay Fractional delay p Normalized frequency.8 8

19 5. Concluding Remarks A two-phase technique for the WLS design of VFD FIR filters subject to a target coefficient sparsity constraint has been proposed. The design algorithm is easy to implement abd computationally efficient because it is based on l l convex optimization. The performance of the filter appears to be satisfactory compared with its nonsparse counterpart. 9