Time-recursive Architectures and Wavelet Transform

The time-recursive computation has been proved as a particularly useful tool in real-time data compression and in transform domain adaptive filtering, with applications in the areas of audio, radar, sonar and video. Unlike the FFT based ones, the time-recursive architectures require only local communication. Also, they are modular and regular, thus they are very appropriate for VLSI implementation and they allow high degree of parallelism. In t.his paper, we propose an architectural framework for parallel time-recursive computation. We consider a class of linear operators that consists of the discrete time, time invariant, compactly supported, but otherwise arbitrary kernel functions. We define a shift p rop erty of the linear operators and reveal its relation with the time-recursive implementation. We demonstrate the potential of the proposed framework by designing a time-recursive architecture for the Discrete Wavelet Transform. 1 I N T R O D U C T I O N In many signal processing applications the key computation consists of a mapping operator [ho h1 ... hnr-11 : z(.) -+ X ( ) , which operates on the semi-infinite sequence of scalar data z(.) and produces the sequence X ( . ) as follows: N-I X ( t ) = h,z(t + n N + I) , t = 0 , 1 , . . . . (1) n=O A time-recursive implementation of a mapping operator [hn hl . . . hN-11 is the one that. is based on an update computation of the type X ( t + 1) = U ( X ( t ) , z ( f + 1) ) . For example, if we have [h, = 1, n = 0 , 1 , . . . , N 11, then X ( t ) will be the sum of the last N values in the input stream. The recursive algorithmic implementation of this operator will be simply the computation X ( t + 1) = X ( t ) + ~ ( t + 1 ) ~ ( t N + 1). The time-recursive computation has been proved as a particularly useful tool in real-time data compression [l, 2,3] and in transform domain adaptive filtering [4, 5 , 61, with applications in the areas of audio, radar, sonar and video. There is a common infrastructure among t,he mapping o p erators that are involved in these diverse applications. The unifying feature is a shift property we discuss in the following Section. We also show how this pr0pert.y dictates t,he time-recursive architectural design. In Sect,ion 3, we design a time-recursive archit.ecture for the Discret.e Wa.velet. Transform (DWT). We conclude wit,h Sect,ion 4. 2 A R C H I T E C T U R A L FR.AMEWORK We can specify a mapping opemtor [ / i o h l . . . I t x i ] with a function f(.), for which t.he values at. t.he poiiit,s 0, I , . . . , N 1 are the prescribed coefficient,s: h,, = f (n ) , n = 0, 1 , . . . , N 1 . In t,lie sequel. we will use the term kernel function or simply kernel for t.his funct.ioii f ( .). Furthermore, we will call kerncl CJYOU~J a vect,or of kernel functions f(.) = [fo(.) f l ( . ) . . ' f . z r l ( . ) l r . Shift Property: A kernel grotrp f ( .) sati.$fies the .diift property (SP), i j it satisfie.5 the (ninti.i.i.) difierence eqiicitioii f(n 1) = Rf ( , I . ) , ) i = I . 2 . . . . . N , ( 2 ) with specified final condition f(LV), iohcrr-c. R i.5 a conafont matrix of site M x A l . Lemma 1 A recursive implementation of (I, kernel grotip f(.) is feusible zj th is kernel grotcp satisfie-s the shift I J ~ O ~ erty. Proof: (2) gives: AI-I f p ( ' i I , 1) = c r , , s , c I t ) .