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I've encountered a test that uses the cumulative value of a finite time series to deterime the data set's stationarity.

I would like to know the characteristics of this test in frequency space, so would therefore need to know the fourier transform of this cummalative function.

I was assuming that the function could be writen as the product of the initial time series function, f, and a heavyside step function, H. With the defining function written as:

$g(t) = \sum^{t}_{n=1} f(n) H(t-n) $

I would like to know fourier transform of the function g(t). Is there a straightforward solution to this function or does the use of the discontinuous step function make a solution impossible.

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Discrete-Time Signal Processing (2nd Edition) (Prentice-Hall Signal Processing Series) [Hardcover] Alan V. Oppenheim (Author), Ronald W. Schafer (Author), John R. Buck (Author)

and all will be revealed.

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