I would like to compute the minimum sampling frequency for the signal $x(t)$ given below:

$x(t)= sinc ^3(300t) * sinc ^5(600t)$ and $ *$ stands for convolution.

where the sinc function is defined as $sinc (w t) = sin (\pi wt)/(\pi wt)$.

I have tried like this : $sinc^3(300t)$ will correspond convolution of 3 rectangular functions in frequency domain which results in a frequency limited function to 3 times 150 Hz ie 450 Hz. And the other function corresponding to maximum of frequency 5 times 300 Hz which is 1500 Hz. Now finally convolution operator get transformed to multiplication in frequency domain , is the maximum frequency content limited to 450 Hz and minimum sampling frequency 900 Hz?

It would be helpful if someone shows me the derivation of fourier transform of sinc function for this purpose.

  • $\begingroup$ Have you consulted your textbook or class notes? $\endgroup$ – John Dec 7 '13 at 17:07

There's several different ways you can go about solving this problem just in the time domain. Use trig identities or Eulers formula to reduce the equation. Both are simple enough and just require some time to work through. Feel free to use WolframAlpha to check your work. I find it sometimes helpful for tedious algebra.

Yes, the Fourier transform of a sinc function is a rectangular function, so you will have what you've described as convolved rectangular functions. This is more work than is necessary as after you've simplified the time-domain expression you will be able to find the maximum frequency, and subsequently the required sampling rate, for that signal.

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  • $\begingroup$ wolframalpha is really great. I really appreciate your help. $\endgroup$ – dexterdev Dec 9 '13 at 13:15

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