help! Using discrete convolution to approximate continuous convolution...?
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From: vashkevich on 19 Nov 2009 01:21 >Hi all, > >I am helping a friend on this. We would like to evaluate the >convolution of > >f(x)=exp(3 * x) / (1 + exp(x) ^ 5). > >However, in Maple and Mathematica, using symbolic calculations, the >convolution results in numerical oscillation (lots of spikeness). We >don't know why. We just couldn't get rid of the spikeness. And we >thought at the end of day, our end-goal is to numerically evaluate >some functions involving the convolution of f(x). So it might be >better to directly handle the convolution in discrete domain and using >Matlab's conv function. > >So we wrote the following paragraph; but we just couldn't get the >scaling and positioning right. For example, let's call the convolution >g(x)=f(x) ** f(x), g(0) should correspond to the 20000th element in >the resultant z, but z(20000) doesn't give a correct number, compared >to the theoretical result of the convolution. > >Could anybody please help us? We need a way to figure out the scaling >factor on the convolution result and how does it map into the >continuous convolution result. And hopefully with lots of samples, we >would be able to approximate the continuous convolution using discrete >convolution. We have to take this approach because the closed-form >expression for g(x) gives a lot spikeness and we just couldn't get a >stable convolution result via evaluating the closed-form expression at >all points. > >Thanks a lot! > >------------------------------------------------------- > >deltat=0.001; > >x= [-10:deltat:10-deltat]; > >y= exp(3 * x) ./ (1 + exp(x) .^ 5); > >z=conv(y, y)*deltat; > >figure; > >plot(z); > If understand convolution as g(x) = Integral( f(y)*f(x-y) dy), then convolution in your case may be calculated g(x) = f**f = x*exp(3*x)/(exp(5*x) - 1) Now it is clear, why you obtained oscillation. Analytical result has uncertainty at zero, i.e. with x=0, but it can be deleted in first order of Taylor series g~x*(1+3x)/(1+5x -1) = x(1+3x)/5x = (1+3x)/5 If I not to mistaken in calculation, it's true. Good luck |