Re: one-dimensional and two-dimensional convolution

*To*: mathgroup at smc.vnet.net*Subject*: [mg69090] Re: one-dimensional and two-dimensional convolution*From*: "Jens-Peer Kuska" <kuska at informatik.uni-leipzig.de>*Date*: Wed, 30 Aug 2006 06:32:21 -0400 (EDT)*Organization*: Uni Leipzig*References*: <ed0v1q$qe$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Hi, and when you read future in your signal processing book you will find out, that discrete convolutions/correlations with real signals can be expressed in the discrete Fourier domain in a simple way ... Continue reading ! Regards Jens "bd satish" <bdsatish at gmail.com> schrieb im Newsbeitrag news:ed0v1q$qe$1 at smc.vnet.net... | | Hi buddies , | | Here is the definition of a 1-dimensional discrete-time convolution | (encountered in Signals & Systems ,etc) of two functions x[n] and | h[n] : | | y[n] = Sum[ x[k] * h[n-k] , { k , -Infiniy , Infinity }] | | The above command works when x[n] and y[n] are functions in the strict | sense (say , x[n] = Exp[-2*pi*n/6] UnitStep[n] and h[n] = Exp[-4*Pi*n/7] | UnitStep[n] ) | But in Signal processing , we often have lists : | | X= { 1,2,3,1,-1 } , H = { 1,-4,5,6} etc. where the entries are the | function's values at different values of n . i.e. | | X = { x[0] , x[1] , x[2] , x[3] , x[4] } . Similarly H = { h[0] , h[1] | , h[2] , ... } (, say ) | | Without using Sum ( and DiscreteDelta ) , is there any way to directly (& | quickly) find the convolution sum. | I'm finding convolution of lists whose lengths are typically 700 to 1000. So | the code needs be really fast. | | The Mathematica command " ListConvolve " did not help me. Anybody plz help | me out !! | | Also , the 2-dimensional convolution of two-variable functions , | f[x,y] and h[x,y] is defined as : | | z[x,y] = Sum[ Sum[ f[m,n] * h[x-m , y-n] , { m , -Infinity, Infinity } ] , | { n , -Infinity , Infinity } ] | | But in image processing , we often have nested lists. In this case , | f[x,y] and h[x,y] are matrices : | | f = { { 1,2,3 } , { 4,5,6 } ,{7,8,9} } etc. Similarly h . There are no | restrictions on the dimensions of matrices f and h . | | Mathgroup , plz help me in solving these 1-D and 2-D convolutions !! | |