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I am aware that due to gibbs phenomenon, ripples are induced at the vicinity of transition in the frequency response of a filter and ripples occur in frequency domain when there are discontinuities in the time domain signal.

Now my doubt is how and why discontinuities are present in time domain signal?

Is it due to periodic extension of aperiodic signal in time domain to compute DFT?

please correct me if I am wrong in my understanding and provide an explanation.

Thanks!!

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It's mostly related to what's called the uncertainty principle of the Fourier Transform: "the more compact a signal is in one domain the more spread out it is in the either". A sharp transients in the time domain has a wide spectrum. A sharp transition in the frequency domain has a long time domain ringing. See for example http://www.ams.org/samplings/feature-column/fcarc-uncertainty

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If I understand correctly your confusion is about filter length, frequency response or DFT of the filter and its relation to Gibbs phenomenon. Gibbs phenomenon is due to sharp transition / discontinuity. This is because smooth functions have rapidly decreasing Fourier coefficients but discontinuous function have very slowly decreasing Fourier coefficient hence the Fourier series does not converge easily for discontinuous functions.

However this is not due to periodic extension of aperiodic signal to compute DFT. There is discontinuity because all real life signals are finite. When we design a filter in time domain, we need to limit ourselves to finite number of coefficient to approximate the desired frequency response. However even increasing the number of coefficients is not enough to get good convergence of Fourier series and avoid oscillations. More on the theory of this can be found here : http://www.ee.ic.ac.uk/hp/staff/dmb/courses/E1Fourier/00500_GibbsPhenomenon_p.pdf

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  • $\begingroup$ As u said gibbs phenomenon is due to discontinuity and finite length which means the inpulse response must be finite. But we also have filters which has infinite inpulse response (IIR filters) such as chebyshev. There are ripples in passband of type one chebyshev filter. Why is that? $\endgroup$ – pavan sunder Jun 27 '17 at 11:43
  • $\begingroup$ Impulse response is infinite in IIR Filters but the filter coefficients are still finite. To understand why exactly there are ripples you need to go into a bit of mathematics and Dirichlet theorem. For this you can refer to the reference i have given. $\endgroup$ – Perscitius Jun 27 '17 at 12:01

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