A Unified Approach to the Design of Interpolated and Frequency Response Masking FIR Filters

Transcription

1 A Unified Approach to the Design of Interpolated and Frequency Response Masking FIR Filters Wu Sheng Lu akao Hinamoto University of Victoria Hiroshima University Victoria, Canada Higashi Hiroshima, Japan May

2 Outline Early Work Filter Structures Convex Concave Procedure (CCP) Design of Interpolated FIR (IFIR) Filters Design of Frequency Response Masking (FRM) FIR Filters Design Examples 2

3 1. Early Work Interpolated FIR (IFIR) Filters Y. Neuvo, C. Y. Dong, and S. K. Mitra, Saramӓki, Y. Neuvo, and S. K. Mitra, Frequency Response Masking (FRM) FIR Filters Y. C. Lim, Y. C. Lim and Y. Lian, Many variants since 1990s. 3

4 Interpolated FIR (IFIR) Filters 2. Filter Structures F(z L ) M(z) L H ( z) = F( z ) M( z) 4

5 Frequency Response Masking (FRM) FIR Filters F(z L ) M a (z) z L(N 1)/2 + M c (z) + H z F z M z z F z M z L L( N 1)/2 L ( ) = ( ) a( ) + ( ) c( ) 5

6 3. Convex Concave Procedure (CCP) CCP refers to a heuristic method to solve a general class of nonconvex problems of the form minimize f ( x) g( x) subject to: f ( x) g ( x) for i = 1,2,..., m i i where f ( x ), g( x ), fi ( x ), and gi ( x ) for i = 1, 2,..., m are convex. he basic CCP algorithm is an iterative procedure including two steps: (i) Convexify the objective function and constraints by replacing g( x ) and g ( x ), respectively, with their affine approximations i gˆ( xx, k) = g( xk) + g( xk) ( x x k) gˆ ( xx, ) = g ( x) + g ( x) ( x x ) for i= 1,2,..., m i k i k i k k 6

7 (ii) Solve the convex problem minimize f( x) gˆ ( x, xk ) subject to: f ( x) gˆ ( x, x ) for i = 1,2,..., m i i k Property 1 If x 0 is feasible for the original problem, x 0 is also a feasible point for the convexified problem. If x k+ 1 is produced by solving the convexified problem, then x k+ 1 is also feasible for the original problem. Property 2 CCP is a descent algorithm, namely, { f( x k ), k = 0,1,...} decreases monotonically. Property 3 Iterates { x k, k = 0,1,...} converge to a critical point of the original problem. 7

8 4. Design of Interpolated FIR (IFIR) Filters Frequency response of an IFIR filter: jω jlω jω H ( e ) = F( e ) M( e ) Its zero phase frequency response: (, ) ( ) H ( ) 0 x ω = aftf Lω amt m ω where a f and a m are coefficient vectors determined by the impulse responses of F( z ) and M ( z ) respectively, and t f ( ω) and t m( ω) are vectors with trigonometric components determined by the filter lengths and types Let Hd ( ω ) be the desired zero phase response of the IFIR filter, the frequency weighted minimax design of an IFIR filter amounts to finding a f and a m that solve the nonconvex minimax problem minimize max w( ω) at f f( Lω) at m m( ω) Hd( ω) a f, a m ω Ω where w( ω ) > 0 is a frequency selective weight over ω Ω. 8

9 Converting the problem to: minimize δ subject to: at f f( Lω) at m m( ω) δw + Hd( ω) at f f( Lω) at m m( ω) δw Hd( ω) Convexifying the problem by adding 1 2 p( x, ω) with hence the constraints become 2 2 f f L m m p( x, ω) = a t ( ω) + a t ( ω) 2 at f f( Lω) + at m m( ω) p( x, ω) + 2δw + 2 Hd( ω) 2 at f f( Lω) at m m( ω) p( x, ω) + 2δw 2 Hd( ω) which fit nicely into a CCP, hence the convexification is done by linearizing p( x, ω) on the right hand sides of the constraints. 9

10 Summarizing, the k th iteration in the CCP solves the convex problem where minimize δ ηi( xx, k, ω) ti ( ω) x subject to: for ω d, i 0,1 0 Ω = ti ( ω) x 1 i η = p( x, ω) + p( x, ω) ( x x ) + 2 δ + ( 1) 2 H ( ω), i k k k w d t i t f ( Lω) ( ω) = i ( 1) m( ω) t Ω d = { ω j, j = 1,2,..., K} Ω In words, the k th iteration of the design algorithm solves an SDP problem involving a total of 2K 2 by 2 matrices that are required to be positive semidefinite. 10

11 5. Design of Frequency Response Masking (FRM) FIR Filters Frequency response of an FRM filter: jω jlω jω jl( N 1) ω/2 jlω jω H ( e ) = F( e ) Ma( e ) + e F( e ) Mc( e ) Its zero phase frequency response: H( x, ω) = a ( ) ( ) ( ) ftf Lω aata ω actc ω + act c( ω) where a f, a a and ac are coefficient vectors determined by the impulse responses of F( z ), M a ( z ) and M c( z ), respectively. Let H d ( ω ) be the desired zero phase response of the IFIR filter, the frequency weighted minimax design of an IFIR filter amounts to finding a f a a, and a c that solve the nonconvex minimax problem minimize max w( ω) H( x, ω) H d ( ω) a, a, a f a c ω Ω where w( ω ) > 0 is a frequency selective weight over ω Ω. 11

12 Converting the problem to: minimize δ subject to: H( x, ω) δ H ( ω) 0, ω Ω w H( x, ω) δ + H ( ω) 0, ω Ω Convexifying the problem by adding the term w 12 d d f f L 2 a a 2 c c p( x, ω) = a t ( ω) + a t ( ω) + a t ( ω) hence the constraints become u( x, ω) p( x, ω) v( x, ω) p( x, ω) where are convex because u( x, ω) = p( x, ω) + H( x, ω) δ H ( ω) v( x, ω) = p( x, ω) H( x, ω) δ + H ( ω) w w d d

13 p( x, ω) + H( x, ω) = [ a a ] M a + [ a a ] N a + a t p( x, ω) H( x, ω) = [ a a a ] M + [ a a ] N a a t with positive semidefinite m i 1 f 1 f 2 f a 0 2 f c 1 c c a a ac 1 f 1 f 2 f a 1 2 f c 0 c c a a ac M = mm and N = nn where i i i i i i tf( Lω) tf( Lω) = and i for i 0,1 i i ( 1) a( ω) n = = ( 1) c( ω) t t he above form of constraints fits nicely into a CCP, hence convexification can be done by linearizing p( x, ω) on the right hand sides of the constraints, i.e., u( x, ω) p ( x, x, ω) for ω Ω v( x, ω) p ( x, x, ω) for ω Ω where p ( xx,, ω) = p( x, ω) + p( x, ω) ( x x ) with k k k k k k 13

14 2[ at f f( Lω)] tf( Lω) p( xk, ω) = [ aata( ω)] ta( ω) [ at c c( ω)] tc( ω) Summarizing, the k th iteration in the CCP solves the convex problem minimize δ subject to: u( x, ω ) p ( x, x, ω ) for j = 1,..., K where { ω j, j = 1,2,..., K} Ω. j k j v( x, ω ) p ( x, x, ω ) for j = 1,..., K j k j In words, the k th iteration of the design algorithm solves an SOCP problem minimizing a linear objective subject to a total of 2K quadratic constraints. 14

15 6. Design Examples Example 1 he first algorithm was applied to design a lowpass IFIR filter with normalized passband edge ω p = 0.15π, stopband edge ω a = 0.2π. he sparsity factor was set to L = 4, and orders of F(z) and M(z) are 31 and 17, respectively. he frequency weight w(ω) was set to w(ω) 1 for ω in the passband and w(ω) 2 for ω in the stopband. An initial was generated by the standard technique proposed in [3]. A total of K = 1400 frequency grids were uniformly placed in [0, ω p ] [ω a, π] to form the discrete set Ω d for problem. It took the algorithm 91 iterations to converge to an IFIR filter with A p = db and A a = db. he same design problem was addressed as Example in [4] using the method described in [Saramӓki, 1993]. he method was implemented as function ifir in Signal Processing oolbox of MALAB. With [F,M] = ifir(4, low, [0.15, 0.2], [0.002, 0.001], advanced ) the function returns with optimized impulses of filter F(z) of order 31 and M(z) of order 17 (in Example of [4], the order of M(z) was said to be 16, however the order of M(z) produced by the above MALAB code was actually 17), with A p = db and A a = db. 15

16 Amplitude Response in db (a) 16

17 Passband Ripple ( db vs db) Proposed Method of [27], [4] (a) Normalized frequency 17

18 Stopband Attenuation (60.84 db vs db) 58 Proposed Method of [27], [4] (b) Normalized frequency 18

19 Example 2 he second algorithm was applied to design a lowpass FRM filter with the same design specifications as in the first example in [3] and [10]. he normalized passband and stopband edges were ω p = 0.6π and ω a = 0.61π. he sparsity factor was set to L = 9, and orders of F(z), M a (z), and M c (z) were 44, 40, and 32, respectively. A trivial weight w(ω) 1 was utilized. With K = 1100, it took the algorithm 60 iterations to converge to an FRM filter with A p = db and A a = db, which are favorably compared with those achieved in [3] (A p = db and A a = db), which has been a benchmark for FRM filters, and those reported in [10] (A p = db and A a = db). 19

20 Amplitude Response in db

21 Passband Ripple in db

22 hank you. Q & A 22