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For example, we need the frequency range of $0\to3000\text{hz}$ to represent a speech signal. The book, Digital Signal Processing (Proakis), stated that we need a low-pass filter to admit only the said frequency range and to avoid aliasing distortion effects.

My question is, if we only need up to $3000$hz, shouldn't we simply sample with a sampling frequency that is sufficient not to have aliasing with the said range? Instead of adding complexity using filters, can we simply sample with a rate that is only sufficient for $0\to3000$hz, ignoring the aliasing for frequency above this value? In other words, we use the aliasing itself as a filter?

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Aliasing is a phenomenon that occurs when the Nyquist frequency $\frac{f_s}{2}$ is lower than the highest frequency in a signal.

If you set $f_s=6\text{Hz}$ and sample these three sine waves one at a time $x_1(t)=\sin(t) \: \:$, $x_2(t) = \sin(5t) \: \:$, $x_3(t) = \sin(7t)$ the sample points will look identical for all of them.

If we were given the set of samples and were asked to draw the continuous time sinusoids we would be faced with an ambiguity. Should we draw a 1Hz, 5Hz or 7Hz sinusoid? The three signals are said to be each others aliases, hence the name aliasing. So you have a problem: -

enter image description here

If you sample with a sampling frequency, where $\frac{f_s}{2}<f_{max}$ you have a problem, because the aliased signals will "corrupt" your measurements, making it seem like some frequency components have higher intensity than they actually have.

There are two solutions to this aliasing problem.

  1. Increase your sampling frequency such that $\frac{f_s}{2}>f_{max}$
  2. Apply an anti-aliasing filter to dampen the frequencies above $\frac{f_s}{2}$ to an insignificant intensity (magnitude).

enter image description here

A thing to note is that when you use an anti-aliasing filter, the filter also dampens some of the unaliased frequencies. So you should consider which filter type to use, Butterworth, Chebyshev, Elliptic, etc.

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It's OK if you are sure that the signal is bandlimited within 0~3kHz. Apparently speech signal is not. It contains higher frequency components leading to aliasing if you only sample at 6kHz, which makes you get the wrong results under 3kHz.

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