EE123 Digital Signal Processing

Transcription

1 Optimality H d (e j! ) EE13 Digital Signal Processing Lecture 3 Least Squares: Z! p! s Z Don t Care!care Variation: weighted least-squares H(e j! ) H d (e j! ) d! W (!) H(e j! ) H d (e j! ) d! Design Through Optimization Example: Least Squares Idea: Sample/discretize the frequency response H(e j! ) ) H(e j! k ) Target: Design M+1= N+1 filter First design non-causal Then, shift to make causal H(e j! ) and hence h[n] Sample points are fixed M+1 is the filter order! k = k P apple! 1 < <! p apple P >> M + 1 ( rule of thumb P=1M) Yields a (good) approximation of the original problem h[n] = h[n M/] H(e j! )=e j M H(e j! )

2 Example: Least Squares Matrix formulation: i T h = h h[ N], h[ N + 1],, h[n] b = H d (e j! 1 ),,H d (e j! P ) T A = e j! 1( N) e j! 1(+N) e j! P ( N) e j! P (+N) 3 Least Squares Solution: argmin h A h b h =(A A) 1 A b Result will generally be non-symmetric and complex valued However, if H(e j! ) is real, h[n] should have symmetry! argmin h A h b Design of Linear-Phase LP Filter Suppose: H(e j! ) is real-symmetric M is even (M+1 taps) Then: h[n] is real-symmetric around midpoint So: H(e j! ) = h[0] + h[1]e j! + h[ 1]e +j! + h[]e j! + h[ ]e +j! = h[0] + cos(!) h[1] + cos(!) h[] + Least-Squares Linear-Phase Filter Don t Care H d (e j! )! p! s Given M, ωp, ωs find the best LS filter: 1 cos( M! 3 b =[1, 1,, 1, 0, 0,, 0] T A = 1 cos( M! p) 1 cos( M! s) 1 cos( M! P )

3 Least-Squares Linear-Phase Filter Given M, ωp, ωs find the best LS filter: 1 cos( M! 3 b =[1, 1,, 1, 0, 0,, 0] T A = 1 cos( M! p) 1 cos( M! s) 1 cos( M! P ) h + =[ h[0],, h[ M ]]T =(A A) 1 A b h = h+ [n] n apple 0 h + [ n] n<0 h[n] = h[n M/] Extension: LS has no preference for pass band or stop band Use weighting of LS to change ratio want to solve the discrete version of: Z W (!) H(e j! ) H d (e j! ) d! where W(ω) is δp in the pass band and δs in stop band Similarly: W(ω) is 1 in the pass band and δp/δs in stop band Weighted Least-Squares argmin h+ (A h + b) W (A h b) Solution: h + =(A W A) 1 W A b W = p s p s 3 + Min-Max optimal Filters Chebychev Design (min-max)!care max H(e j! ) H d (e j! ) Parks-McClellan algorithm - equi-ripple Also known as Remez exchange algorithms (signalremez) Also with convex optimization

4 Specifications 1+ p 1 1 p s Don t Care H d (e j! )! p! s Min-Max Filter Design Minimize: max pass-band ripple 1 p apple H(e j! ) apple 1+ p, 0 apple w apple! p min-max stop-band ripple Filter specifications are given in terms of boundaries H(e j! ) apple s,! s apple w apple Min-max Ripple Design Recall, is symmetric and real ~ Given ωp ωs M, find δ,h+: Subject to : H(e j! 1+ ) 1 = 1 apple H(e j! k ) apple 1+ 0 apple! k apple! p apple H(e j! k ) apple > 0 Solution is a linear program in δ,h+ A well studied class of problems ~! s apple! k apple Min-Max Ripple via Linear Programming subject to : 1 cos(! 1 ) cos( M! 3 A p = 1 cos(! p ) cos( M! p) 1 cos(! s ) cos( M! 3 A s = 1 cos(! P ) cos( M! P ) - 1 A p h+ 1+ > 0 capital P A s h+

5 Convex Optimization Many tools and Solvers Tools: CVX (Matlab) CVXOPT, CVXMOD (Python) Engines: Sedumi (Free) MOSEK (commercial) Take EE1! Using CVX (in Matlab) M = 1; wp = 0*pi; ws = 0*pi; MM = 1*M; w = linspace(0,pi,mm); idxp = find(w <=wp); idxs = find(w >=ws); Ap = [ones(length(idxp), *cos(kron(w(idxp)', [1:M/]))]; As = [ones(length(idxs), *cos(kron(w(idxs)', [1:M/]))]; % optimization cvx_begin variable hh(m/+1,; variable d(1,; (d) subject to Ap*hh <=1+d; Ap*hh >=1-d; As*hh < d; As*hh > -d; ds>0; cvx_end h = [hh(end:-1: ; hh(:end)]; Variations: Convex Problems: Fix δs optimize for δp Fix δp optimize for δs Linear constraints on h[n] Quasi-Convex (feasible through bisection) Fix δp, δs, M, Δω=ωs-ωp Fix δp, δs, Δω=ωs-ωp, M Bisection Example: Minimize M given δp, δs, Δω=ωs-ωp Initialize problem with: Set Mmin to be small and hence infeasible Set Mmax to be large and hence feasible Set M = floor(mmax/ + Mmin/) Given M, δp, Δω=ωs-ωp solve for minimum δs If δs violates constrains, set Mmin = M if δs within constraints, set Mmax = M Set M = floor(mmax/ + Mmin/) Repeat till M is tight

6 IIR Design Historically Continuous IIR design was advanced Use results from CT to DT CT IIT designs have closed form, easy to use Easy to control Magnitude, not easy to control phase Common Types: Butterworth - monotonic, no ripple Chebyshev - Type I, pass band ripple, Type II stop band ripple Elliptic - Ripples in both bands Design of DT IIR Filters from Analog Discretize by one of many techniques Hc(s) H(z) Must satisfy: Imaginary axis is mapped to unit circle stability of Hc(s) should result in stable H(z) Two methods: Impulse invariance - match impulse response Bilinear transformation