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While studying the convergence of Fourier transform, I got to know two conditions.

  • $$\sum_{n=-\infty}^{\infty}|x(n)|<\infty$$
  • $$\sum|x(n)|^{2} \leq [\sum|x(n)|]^{2}$$

While I was reading the text, I found this paragraph quite confusing. I didn't understood this.

If a sequence is not absolutely summable but has finite energy, one may employ a type of convergence in which the series converges so the mean square error is 0. The "attendant Gibbs oscillations" at a discontinuity are of practical significance in filter design.

I don't know what Gibbs oscillations mean here. So if someone could please explain the meaning of this paragraph (taken for DSP by Alan V. Oppenheim).

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The first condition mentioned in your question is absolute summability, which is sufficient for the discrete-time Fourier transform (DTFT) to exist. In this case, the sum given by the DTFT of a sequence converges uniformly. The other condition you probably mean is square summability:

$$\sum_{n=-\infty}^{\infty}|x[n]|^2\lt\infty\tag{1}$$

in which case the DTFT exists if we relax the condition of uniform convergence. That other type of convergence is called mean-square convergence. In this case, the result of the sum oscillates around points of discontinuities, and those oscillations do not decrease with increasing number of elements in the sum. You can find a more in-depth explanation of this so-called Gibbs phenomenon in this article.

The importance of the Gibbs phenomenon in digital filter design is that you might want to approximate an ideal frequency response (with discontinuities) by a filter of finite length (finite impulse response, FIR), and this results in large errors close to the discontinuities. This problem can be alleviated by introducing transition bands and/or by (non-rectangular) windowing instead of simple truncation of the sum.

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  • $\begingroup$ I am not able to understand the last paragraph. Maybe because we haven't been taught this thing till now in our course. We haven't reached to filter design. By the way thank you very much. I am now able to relate the absolute summability with convergence of Fourier Transform. $\endgroup$ – Himanshu Sharma Sep 21 '18 at 1:10

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