Convolution Theorem: Hamming Window on a Time Series and Fourier Domain If we have a set of time series data, y, consisting of 100 data points. One can apply a N (odd) Hamming window as a weighted moving average to decrease the noise. Say, if we choose 7 point Hamming window, H,  [0.0800    0.3100    0.7700    1.0000    0.7700    0.3100    0.0800], and perform a weighted moving average on y. In Matlab, I am doing conv(y, H./sum(H), 'same').

Since this is a convolution, I am wondering that by convolution theorem, one should be able to obtain identical results in Fourier domain, because convolution should become multiplication in the Fourier domain.

However, the FFT of y will have 100 complex numbers, and Hamming window will also have 7 complex numbers. What would be an equivalent operation of a weighted moving average in the Fourier domain?

Thanks.
Fistly, you need to pad the data (y) and the filter (Hamming window). The minimum required length to avoid circular convolution = data length + filter length - 1. In this case, it is 100 + 7 - 1 = 106. For convenient, you may choose power of 2 length, in this case 128.

So, pad the data to 128, do fft on it.

Pad the filter to 128, do fft on it.

Multiply both ffts.

Do ifft.