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As we all know that the inverse Fourier transform of a frequency domain unit pulse/rectangular function (which looks like low pass filter) is Sinc function. So, I was under the impression that any low pass filter impulse response would look like Sinc function.

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However, the RC low pass filters impulse response look like an exponentially decaying function.

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Since both are low pass filters with the similar looking frequency domain transfer functions, why RC low pass filter response does not look like Sinc function?

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    \$\begingroup\$ No, sinc function in the time domain only applies to rectangular frequency response (and vice versa). A first order (RC) LPF does have a decaying exponential. And there are other characteristics : uniquely, a Gaussian LPF has a Gaussian time domain response... There used to be a time when math courses taught the commonest Laplace transforms like these. \$\endgroup\$ – user_1818839 Jan 11 at 23:02
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    \$\begingroup\$ Is a rectangular pulse the same as an exponentially decaying waveform? Not only they are not, but one is symmetric, the other is not. According to your logic -- "Since both are low pass filters with the similar looking frequency domain transfer functions" -- a Bessel lowpass is the same as an elliptic/Cauer lowpass. The transfer functions are really not the same. \$\endgroup\$ – a concerned citizen Jan 11 at 23:02
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Here's the short answer: a first-order lowpass and a rectangular-in-frequency filter don't look the same in the frequency domain. So why should they be the same when you take the inverse Fourier transform?

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  • \$\begingroup\$ SINC pulses do not need to be square. e.g. Blackman, Welsh etc \$\endgroup\$ – Tony Stewart EE75 Jan 13 at 16:16
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    \$\begingroup\$ There is exactly one sinc function, and its Fourier transform is square. There may be functions like the sinc function -- but their Fourier transforms are going to be way more square than \$1/(\tau s + 1)\$. So I think my point stands. \$\endgroup\$ – TimWescott Jan 13 at 19:19
  • \$\begingroup\$ You think only square windows make SINC functions.? fiiir.com \$\endgroup\$ – Tony Stewart EE75 Jan 13 at 22:02
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    \$\begingroup\$ By definition \$\mathrm{sinc} x = \lim_{\chi \to x} \frac{\sin \pi \chi}{\pi \chi}\$. That page you reference is generating what it calls "windowed sinc". Not plain ol' sinc. \$\endgroup\$ – TimWescott Jan 14 at 1:04
  • \$\begingroup\$ So in theoretical terms, it must be a "boxcar" but in practical terms it doesn't. \$\endgroup\$ – Tony Stewart EE75 Jan 14 at 1:22

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