Robust Digital Filters Part 1: Minimax FIR Filters Wu-Sheng Lu - PowerPoint PPT Presentation

Robust Digital Filters Part 1: Minimax FIR Filters Wu-Sheng Lu Takao Hinamoto University of Victoria Hiroshima University Victoria, Canada Higashi-Hiroshima, Japan May 2019 1

Outline • Measures for Performance Robustness • Design Formulations • Properties of Error Functions • Design of Robust Minimax FIR Filters • An Example 2

1. Measures for Performance Robustness • Motivation Typical error measures 1/2         L : 2 W ( ) | H ( , ) x H ( ) | d   2 d  L  : max     W ( ) | H ( , ) x H ( ) | d    Let x ∗ be a minimizer under a certain measure. The optimal performance of the filter, i.e. H   , would be achieved only if its implementation is perfectly ( , x ) accurate. In practice, however, neither hardware nor software utilized in a practical implementation are of infinite precision, thus only an approximate version of H  x is actually realized. This approximation may be modeled as a frequency  ( , )   response for some variation  due to various reasons ranging from H   ( , x ) power-of-two constraints on filter coefficients to rounding errors in multiplications using fixed-point arithmetic.   Since the perturbed design represented by  is no longer a minimizer, the x   performance degradation at  is inevitable even for small  . x 3

• We seek to develop design methods for digital filters that achieve performance optimality subject to variations of filter coefficients. To this end we introduce new L and L  error measures for filters with robust 2 performance against coefficient variations: 1/2            2 e 2 ( ) x max W ( ) | H ( , x ) H ( ) | d   d    B r and        e ( ) x max max W ( ) | H ( , x ) H ( ) |  d      B r where B r is a region for parameter variation  . Choices of B r include · Bounding box      B { :| | r i , 0,1,..., K } r i i · Ball     B { :|| || r } r 2 · Ellipsoid B      with { :|| D || 1} D diag{ d d , ,..., d } r 2 0 1 K 4

2. Design Formulations • Given filter’s order and type (FIR or IIR) and a region of permitted parameter variations B r , robust lease-square and minimax designs are obtained by solving x minimize e ( ) 2 and e  x minimize ( ) • Constraints on stability of H ( z ) need to be imposed for IIR designs. • This paper addresses linear-phase FIR filters only, and we consider FIR filters of odd length N with transfer function and frequency response  N 1     x with K = ( N − 1)/2.     i jK T H z ( ) h z , H ( , ) x e c ( ) i  i 0 • Let      be the desired frequency response. The two error jK H ( ) e A ( ) d d functions become 1/2       T     2  e 2 ( ) x max W ( ) | ( ) ( c x ) A ( ) | d   d    B r    T     e ( ) x max max W ( ) | ( ) ( c x ) A ( ) |  d     B r 5

3. Properties of Error Functions • Property 1: Functions x and e  x are convex. Hence the two design e 2 ( ) ( ) problems can be addressed as convex optimization problems. • Property 2: The sub-differential of with respect to x is given by e  x ( )        (1a) g x ( )  e ( ) x W ( ) ( c )( | y |)     T       y [ ( c ) ( x ) A ( )] d where   , and          T ( , ) arg (max max W ( ) | ( ) ( c x ) A ( ) |) d     B   , r   1 if y 0  (1b)     | y | 1 if y 0     [ 1,1] if y 0  6

4. Design of Robust Minimax FIR Filters This paper is focused on the design of robust minimax linear-phase FIR filters:   T     minimize max max W ( ) | ( ) ( c x ) A ( ) | d     x B r • Available options include a variation of the gradient descent (GD) method, known as the heavy ball method (Polyak, 1987), and GD with momentum . The heavy ball algorithm updates iterate x to k (2a)       x x g ( x x )   k 1 k k k k k k 1 which starts with k = 0 with point set to x . We see the last term of the above x  1 0 update uses past iterates to provide momentum that pushes the current iterate like a heavy ball to move down hill faster. The step size  is calculated using k    ( ) k e ( x ) e   (2b)  k best k k 2 || g || k 2 where x keeps track of the best performance achieved so far,  > 0 ( ) k  e min e  ( ) best i k  i 1,..., k       is a sequence satisfying and , and  satisfies and k    k  2      k k k k    0 0 k 0   . Here  were simply set to be proportional to 1/( k + 1) .  2    and k k k  k 0 7

• Computing Sub-gradient g k A major step of the algorithm is to calculate sub-gradient g which is given by k         g x ( ) e ( ) x W ( ) ( c )( | y |)     T       y [ ( c ) ( x ) A ( )] d where        T     ( , ) arg(max max W ( ) | ( ) ( c x ) A ( ) |) d     B   , r is not trivial to compute. • Formulas for calculating (    were derived in Sec. 3B of the paper for the  , ) case of B r being a bounding box:    K          (3a) T max W ( ) r | cos( i ) | A ( ) c ( ) x | i d  i 0    d     (3b)    A    T   sgn( ( ) c ( ) x ){sgn( ( c ))} r d   T where collects permitted upper bounds for individual design  r r r  r 0 1 K variables. ◊ Note that (3a) is a 1 -D maximization problem and hence straightforward to perform. 8

Algorithm for Robust Minimax FIR Filters inputs: desired amplitude response A  , filter length N , weight W  , frequency grids ( ) ( ) d  , initial design x 0 , and number of iterations N t . d for k = 0, 1, . . . , N t , Step 1 : use (3a) and (3b) to compute   and  ; use (1) to compute  g . k Step 2 : use (2a) and (2b) to compute . x k  1 end 9

5. An Example We consider designing a robust minimax low-pass FIR filter of length N = 21 with normalized pass-band edge  = 0.4  and stop-band edge  = 0.5  . Assume p a W   and a bounding box ( ) 1      B { :| | 0.005, i 0,1,...,10} r i The set  consists of 100 frequency grids that are evenly placed over the union of d the pass-band and stop-band [0, 0.4 ]    . We use a least-squares low-pass  [0.6 , ] filter with the same passband and stopband edges as the initial point of the proposed algorithm. The parameters  and  were set to k k 0.16 0.08   and    k k  k 1 k 1 The algorithm was able to reduce the object function from 0.2837 to 0.0906 in 100 iterations, see below. Fig. 2 depicts the amplitude response of the robust filter obtained after 104 iterations, at which the objective function was reduced to   . e ( x ) 0.0898  10

0.3 0.25 0.2 0.15 0.1 0.05 0 10 20 30 40 50 60 70 80 90 100 iteration 11

5 0 -5 -10 -15 -20 -25 -30 -35 -40 0 0.5 1 1.5 2 2.5 3 normalized frequency 12

For comparison, a conventional linear-phase minimax FIR filter of length 21 was designed using the Parks-McClellan (PM) algorithm. Let be its parameter (PM) x vector, the robustness measure of the PM filter was found to be  . (PM) e  ( x ) 0.1099 This is to say, the approximation error of an FIR filter whose coefficients vary from the PM filter within the bounding box B r is bounded by 0.1099 , while the (PM) x approximation error of an FIR filter whose coefficients vary from the robust minimax filter x within the bounding box B r is guaranteed not exceeding 0.0898 ,  representing a 18.23% reduction. The first 11 coefficients of the robust minimax and PM filters are given in the Table below. 13

TABLE I First 11 Coefficients Robust Minimax PM filter 0.037739257611475 0.038776212428159 0.003955891303322 0.002476747389280 −0.030947825075270 −0.030327895328723 −0.017617078016662 −0.018181020733260 0.034656023078736 0.035632966126753 0.040944804608671 0.039290923363811 −0.045087860127533 −0.045102726246706 −0.091116616564451 −0.092430917385436 0.046215874348786 0.047087467600674 0.313562923929549 0.311837559676779 0.449135853464320 0.448729827357856 14

Thank you. Q & A 15