Aliasing; Nyquist of 6cos50πt + …

Aliasing effect

Aliasing is the distortion that occurs when a signal is sampled at a rate lower than its Nyquist rate ($f_s < 2f_M$). The shifted copies of the spectrum in the sampled signal overlap, so a high-frequency component "folds back" and appears as a lower frequency (its alias). After overlap the original spectrum cannot be separated by any filter, so the signal cannot be recovered exactly.

Example: a 700 Hz tone sampled at $f_s = 1000$ Hz gives the same samples as a $1000-700 = 300$ Hz tone.

How to overcome aliasing:

1. Sample fast enough: choose $f_s > 2f_M$ (in practice $f_s$ is kept 2.5 to 5 times $f_M$ to leave a guard band).
2. Anti-aliasing filter: pass the signal through an analog low-pass filter with cutoff $f_s/2$ before sampling, so that no component above $f_s/2$ reaches the sampler. Real signals and noise are never strictly band-limited, so this filter is always used in practice.
3. Oversampling: sample at a much higher rate, then low-pass filter and decimate digitally. This relaxes the sharpness needed in the analog anti-aliasing filter.

Nyquist rate of $x(t) = 6\cos 50\pi t + 20\sin 300\pi t - 10\cos 100\pi t$

$f_M = 150$ Hz.

$$f_N = 2 f_M = 2\times150 = 300\ \text{Hz}$$

Answer: Nyquist rate = 300 Hz (300 samples/s, $\omega_N = 600\pi$ rad/s). (Sampling at exactly 300 Hz would make every sample of $\sin 300\pi t$ zero, so in practice $f_s$ must be strictly greater than 300 Hz.)