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A Band-limited signal is defined as a signal whose Fourier Transform is zero above a specified frequency. Shannon Sampling Theorem is also stated for a strictly band-limited functions,i.e,

"For a band-limited case, 2W numbers per second are sufficient."

Where, "W" is the max. frequency content.

My confusion is that why band-limit a signal? What would be the situation if a signal is not perfectly band-limited?

My understanding says that unless we have a band-limited signal, we can't get a usable "W" and aliasing would also occur. Does it make sense as an answer?

I think I know answers intuitively, but struggling to frame it in words. Please guide me.

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A discrete-time signal represents a continuous-time signal that is band-limited to at most half the sampling frequency. Sampling a continuous-time signal that contains higher frequencies than half the sampling frequency will result in aliasing: Samples of those high frequencies are identical to and cannot be discerned from samples of other frequencies below half the sampling frequency. To avoid contamination of the signal by the aliased frequencies, you would first band-limit (lowpass filter) the continuous-time signal and only then sample it.

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All the frequency component above W will be aliased, but if they are small, it would be nothing more than noise, everything below will be properly sampled.

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"why band-limit a signal?"

There will be components of a system that have bandwidth limitations. For example, ADCs have a limit as to how fast they can sample. Therefore the frequency content would need to be limited otherwise you would get aliased frequencies. This is why it is common to find low pass filters before ADCs which prevents erroneous frequency content being mixed into the discrete time (sampled) representation of the signal.

Its interesting to note that in communications, intentionally aliasing signals are sometimes used to 'foldback' a signal of interest into range of an ADC.

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