State and prove Nyquist sampling theorem

Statement

A band-limited signal $m(t)$ with no frequency components above $W$ Hz ($M(f) = 0$ for $|f| > W$) is uniquely determined by its samples taken at uniform intervals $T_s \le \frac{1}{2W}$, i.e. at a rate $f_s \ge 2W$. It can be recovered exactly from the samples by an ideal low-pass filter of bandwidth $W$.

The minimum rate $2W$ is the Nyquist rate; $1/(2W)$ is the Nyquist interval.

Proof

1. Ideal sampling. Multiply $m(t)$ by a periodic impulse train:

$$m_\delta(t) = m(t)\sum_{n=-\infty}^{\infty}\delta(t - nT_s) = \sum_{n=-\infty}^{\infty} m(nT_s)\,\delta(t - nT_s)$$

2. Spectrum of the impulse train. It is periodic, with Fourier series coefficients $1/T_s$:

$$\sum_n \delta(t - nT_s) = \frac{1}{T_s}\sum_{k=-\infty}^{\infty} e^{j2\pi k f_s t} \;\longleftrightarrow\; f_s\sum_{k}\delta(f - kf_s)$$

3. Spectrum of the sampled signal. Multiplication in time is convolution in frequency:

$$M_\delta(f) = M(f) * f_s\sum_k\delta(f - kf_s) = f_s\sum_{k=-\infty}^{\infty} M(f - kf_s)$$

So $M_\delta(f)$ is $M(f)$ repeated every $f_s$ Hz, scaled by $f_s$.

f_s >= 2W (no overlap):
     /\        /\        /\
    /  \      /  \      /  \
 --/----\----/----\----/----\---> f
  -W  0  W fs-W fs   2fs
f_s < 2W: copies overlap -> aliasing

4. Condition for no overlap. The copy at $f_s$ occupies $f_s - W$ to $f_s + W$. It does not overlap the baseband copy ($-W$ to $W$) if

$$f_s - W \ge W \;\Rightarrow\; f_s \ge 2W$$

5. Recovery. If $f_s \ge 2W$, pass $m_\delta(t)$ through an ideal LPF with $H(f) = T_s$ for $|f| < W$ (any cut-off between $W$ and $f_s - W$):

$$M_\delta(f)\,H(f) = M(f)$$

so $m(t)$ is recovered exactly.

6. Interpolation formula. With $f_s = 2W$, the LPF impulse response is $h(t) = \text{sinc}(2Wt)$, so

$$m(t) = \sum_{n=-\infty}^{\infty} m\!\left(\frac{n}{2W}\right)\text{sinc}(2Wt - n)$$

Each sample is replaced by a sinc pulse and the sum rebuilds $m(t)$ at all instants. This proves the theorem. If $f_s < 2W$, the copies overlap (aliasing) and $m(t)$ cannot be recovered; an anti-aliasing filter is therefore used before sampling.