For a low-pass FIR filter h[n], the common normalization rule is with regards to H(j0), i.e., sum(h[n]). Then, how about the normalization rule for finite difference filters? For example, given h1[n] = {1, -1} and h2[n] = {1/2, 0, -1/2}, and they have been properly normalized according to the meaning of finite difference. Notice that both have a positive number and a negative number, but the normalization factors are quite different. Then, what is the normalization rule for a general finite difference filter?
Convolution is normalized by the lengths of the lists being convolved. Look at Bracewell's Fourier Transforms and Applications.
Dear Luisiana Cundin , thank you for your information. In Bracewell's book, I only found normalization for correlation. My question is about normalization rule for finite difference filters, e.g., h1[n] = {1, -1} and h2[n] = {1/2, 0, -1/2}.
See page 24, Chapter 3 on Convolutions. Another path for understanding your filter is to see them as wavelets. They are a version of a Haar wavelet. Normalization is the same.
So, normalized Haar wavelets are {1,-1}/Sqrt[2],
https://en.wikipedia.org/wiki/Haar_wavelet
Dear Luisiana Cundin , thank you for your further information.
In Bracewell's book, I guess what you point to is about the sequence in Page 33, "{1/n, 1/n, ..., 1/n}" and "1/7{1, 1, 1, 1, 1, 1, 1}". If treated as filters, these two are special low-pass filters, and the common normalization rule is sum(h[n]), i.e., the lengths (1/n and 1/7) are also special cases of sum(h[n]).
My question is about normalization rule for finite difference filters. For example, both h1[n] = {1, -1} and h2[n] = {1/2, 0, -1/2} are 1-st order finite difference filters, for which h1[n] is the forward/backward finite difference filter and h2[n] is the central finite difference filter, but it seems that h2[n] is nornalized with 2.
Therefore, if we obtain a finite difference filter by any precedure, how should we normalize the coefficients with a general rule?
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