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Topic: The Case Of Scaling Factor In Convolution
Replies: 0

 sunny Posts: 5 Registered: 1/10/08
The Case Of Scaling Factor In Convolution
Posted: Jan 10, 2008 3:54 PM

Hello,

I know this may sound a common or even a basic question to many,
however I will appreciate if someone can give me insight on this. I
read some of the older posts on related topics but a fraction of doubt
still remains.

I am trying to find the correct scaling factor for the output of
convolution (in MATLAB). I have a signal x[n] of length N. I have a
unit-amplitide discrete-time rectangular pulse function h[n] of length
L. I used conv(x,h) and ended up with the output y[n] of length N+L-1.
After plotting it I found the amplitude of the output to be much
higher compared to what I will get if I would use continuous-time
convolution (however the shape was same).

I figured out that the approximation of continuous-time convolution by
a discrete-time convolution requires scaling of output by the number
of samples. So I divided y[n] by N+L-1. However the output turned out
to be much smaller (but same in shape) compared to x[n] plot. After
trying various factors which include N, L, N+L-1 and their square-
roots, I discovered that 1/sqrt(L) was the factor which gave me the
correctly scaled amplitude.

However, it remains unclear to me (a) why the output should depend on
the *sqrt of the no of samples* when the discrete-time approximation
of continuous-time convolution clearly states that the sacling factor
is just the *no of samples*? (b) why the scaling of output should
depend on only L and not on N?

Sincerely,

Sunny.