Sampling theorem

The Sampling Theorem (also called the Nyquist or Nyquist–Shannon theorem) states:

If a continuous-time signal $x(t)$ is bandlimited with maximum frequency $f_m$ (that is, $X(f)=0$ for $|f|>f_m$), and is sampled at a rate $f_s>2f_m$, then the samples $x(n{T}_s)$ contain enough information to reconstruct $x(t)$ exactly.

The critical sampling rate $2f_m$ — twice the highest frequency — is the Nyquist rate. A signal sampled at $f_s>2f_m$ is oversampled; at $f_s<2f_m$ it is undersampled.

Why it works

Sampling produces a spectrum $X_{\delta }(f)=f_s{\sum }_{n}X(f-n{f}_s)$ — copies of $X(f)$ spaced $f_s$ apart in frequency. For the copies not to overlap, the right edge of the $n=0$ copy (at $f_m$) must be below the left edge of the $n=1$ copy (at $f_s-f_m$):

$f_m<f_s-f_m⟺f_s>2f_m.$

When this holds, the copies are cleanly separated, and the $n=0$ copy is just $(1/f_s)X(f)$ — a faithful replica of the original spectrum. Lowpass-filter to isolate it (multiply by $(1/f_s)\mathrm{rect}(f/f_s)$ in frequency), and the result is $X(f)$.

In time, multiplying by a frequency rect becomes convolving with a sinc — that’s the sinc interpolation formula for recovering $x(t)$ from $x[n]$:

$x(t)={\sum }_{n=-\infty }^{\infty }x(n{T}_s)\mathrm{sinc}\left(\frac{t-n{T}_s}{T_s}\right).$

If you sample below the Nyquist rate, the copies overlap and aliasing destroys information.

Things to internalize

It’s a strict inequality. $f_s>2f_m$, not $\ge$. Sampling exactly at the Nyquist rate is a boundary case where reconstruction is not always guaranteed. For example, sampling $\cos (2\pi f_{m}t)$ at $f_s=2f_m$ gives samples that line up with the zero crossings — every sample is the same value, and you lose the cosine. Always sample strictly above Nyquist.

It’s about bandlimited signals. Real signals are never perfectly bandlimited — every real signal has some high-frequency components (noise, harmonics, sharp edges). The practical workaround is to apply an anti-aliasing filter before sampling, enforcing bandlimiting.

It guarantees exact reconstruction in principle. “Reconstructible exactly” doesn’t mean “easy to reconstruct in practice.” The sinc-interpolation formula involves an infinite sum of sincs, which is impossible to compute exactly with finitely many samples. Real systems use practical approximations (zero-order hold, FIR filters).

The Nyquist rate depends on the highest frequency, not the bandwidth. A signal in the band $(100,110)\text{kHz}$ has bandwidth $10\text{kHz}$ but highest frequency $110\text{kHz}$, so the Nyquist rate is $220\text{kHz}$, not $20\text{kHz}$. (There are tricks — bandpass sampling — for narrowband signals far from DC, but they use a more careful argument; the basic theorem uses $2f_m$.)

The practical signal chain

A complete digital signal chain looks like:

analog signal → anti-aliasing filter → sampler → digital processing → reconstruction filter → analog out

The anti-aliasing filter prevents aliasing during sampling; the reconstruction filter smooths the discrete output back into a continuous-time signal (approximating sinc reconstruction). Both filters are essential — without them, the digital chain would mangle the signal.

Why it’s one of the most consequential results in engineering

The sampling theorem says, fundamentally, that a continuous signal can be completely represented by a discrete sequence — no information is lost — as long as you sample fast enough. This is the mathematical foundation of every digital audio system, every digital camera, every digital communication link. Without it, the entire digital revolution would have lacked a theoretical basis.

Named after Harry Nyquist (1928) and Claude Shannon (1949), though variants were known earlier (Whittaker, Kotelnikov).