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?