Frequency domain optimization of multistage separable FIR filters. I. Optimization algorithms

Algorithms for frequency domain optimization of thc multistage scparable (MSS) FIR filter are presented. .The major attribute o i this two-dimensional digital filter s t ~ c t u r c is the parallelism of its design which makes it an ideal candidate for parallel processing. This fact gives the filter a much higher throughput ralc than o t h e r two-dimens iona l digi ta l f i l l e r structures. Due to the non-linear nature of thc o p t i m i r a t i o n problem. i te ra t ive a lgor i thms arc prescnled. The filter's parallelism. combined with the opt imizat ion algorithms. give an alternative f i l ter s1ructurc having a high throughput rate while still maintaining a given measure of quality in the frequency domain. 1. Filler Strurture l ' hc dptimrration algorithm, prrsentcd apply to thc filter stmcture shown in Figure 1 Figure 1: Structure of a I-stagc multistagc scpnrnblc f i l t e r . This structure has the following frcquency response: Taking the inverse Fourier transform leads to the following impulse response: 2 . I m o a t h n s e of the S k u U u The strengths of this structure are its flexibility. increased t h r o u g h p u t ra te . and c o m p u t a t i o n a l cfficiency. These attributes make the stcuclure very attractiveand useful. Flexibility: Any finite-extent impulse response can be implemented by a multistage separablc FIR filter. Thcrcfore. this structure can be used to implement a widc class of filters. Given an arbitrary MxN matrix. i t has bccn shown [ I ] that the matrix can bc broken down into a finite sum of MxN matrices. Each matrix in this sum can be factored into the product of a column vector of length M and a row Y C C ~ O ~ o l length N. Thcrcfore. given any MxN impulse response. that impulse response can be broken down into .a finite sum of MxN impulse responses. Each impulse rcsponrc can be factored into the product of a one-dimensional impulse rcsponse of length M (c(n2)) and a one-dimensional impulse response of length N (r (n1)) . Increased throughput ra te through thc usc of parallel processing: This stmcture lends itsclf easily to parallel processing. Each stage of the filter has a structure identical 10 evcry other stage and proccsser an identical input vector. Each stagc can thercfore bc implemented with identical algorithms. The standard FIR f i l ter does not enjoy this advantagc. This parallelism is useful if one wants to build hardware for thc proposcd structure. I t is also useful i f one implements the proposed stmcture on a computer that is capable of parallel processing. since many of today's computers do have this capability. By using a computer ( o r o t h e r hardware) with para l le l p rocess ing capability. the proposed s t ructure would result in a lower computa t iona l t ime. t h c r c f o r e . a l o w e r computational cost. One can rcalizt t h i s advantage cvcn if the number of filter stagcs is not small and the number of coefficients ncedcd is greater than that of a standard FIR filter with the same region of suppon in the spatial domain 111. Note: a "stage" is one of the parallel branches of the s t ~ c t u r e given in Figure I . Computational efficiency: I f J ( the number o f stages) is small. thcn the number o i coelficients of the multislage separable FIR filler will be less than that of a standard FIR filter with the same region of support in the spatial domain. This means fewer multiplies and