Sampling

Sampling is the act of recording a continuous-time signal $x(t)$ at evenly-spaced moments to produce a discrete sequence. Pick a sampling period $T_s$ in seconds, and define the samples by

$x[n]=x(n{T}_s),n=\dots ,-1,0,1,2,\dots$

The list of samples $x[n]$ is a discrete-time signal. The sampling rate (or sampling frequency) is $f_s=1/T_s$, in hertz — samples per second.

Impulse sampling

The cleanest mathematical model — and the one that connects to the Fourier transform — is impulse sampling. Instead of producing a list of numbers, we multiply $x(t)$ by a periodic impulse train of period $T_s$:

$x_{\delta }(t)=x(t)\cdot {\delta }_{T_s}(t)=x(t){\sum }_{n=-\infty }^{\infty }\delta (t-n{T}_s)={\sum }_{n=-\infty }^{\infty }x(n{T}_s)\delta (t-n{T}_s).$

(The last equality uses the equivalence property of the impulse.)

The result $x_{\delta }(t)$ is a continuous-time signal — still on the real line — but zero except at the sample instants, where it carries impulses with strengths equal to the sample values.

This is a mathematical fiction (no real sampler produces zero-width impulses), but it’s useful because it preserves all the sample information and has a clean Fourier transform.

The spectrum of a sampled signal

Take the Fourier transform of $x_{\delta }(t)$ using the convolution theorem:

$X_{\delta }(f)=X(f)\ast \mathcal{F}\{{\delta }_{T_s}(t)\}=X(f)\ast f_s{\delta }_{f_s}(f)=f_s{\sum }_{n=-\infty }^{\infty }X(f-n{f}_s).$

(Convolution with a train of impulses just shifts and sums copies.)

The spectrum of the sampled signal is the original spectrum replicated at multiples of $f_s$, with each copy scaled by $f_s$. The $n=0$ copy is the original; copies at $\pm f_s,\pm 2f_s,\dots$ are images.

This single formula is the key to everything in sampling theory:

• If the copies don’t overlap → original signal can be recovered. Sampling theorem guarantees this when $f_s$ is more than twice the highest frequency in the signal.
• If they overlap → aliasing mixes high-frequency content into low frequencies, and information is irreversibly lost.

The Nyquist condition

For a bandlimited signal with maximum frequency $f_m$ (so $X(f)=0$ for $|f|>f_m$), the copies in $X_{\delta }(f)$ don’t overlap when

$f_s>2f_m.$

The critical rate $2f_m$ is the Nyquist rate. See Sampling theorem for the full statement.

From samples back to continuous signal

If $f_s>2f_m$, the original $X(f)$ can be recovered by lowpass-filtering $X_{\delta }(f)$ with cutoff between $f_m$ and $f_s-f_m$. The corresponding time-domain operation is sinc interpolation:

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

In practice, real systems use approximations: zero-order hold, first-order hold, FIR filters that approximate the sinc with finitely many taps.

In the analog-to-digital pipeline

Sampling is the first step in converting an analog signal to digital:

$\text{continuous }x(t)\stackrel{\text{sample}}{\to }x[n]\stackrel{\text{[[Quantization|quantize]]}}{\to }q[n]\stackrel{\text{[[Encoding (digital)|encode]]}}{\to }\text{bits}$

Anti-aliasing filtering usually precedes sampling, to enforce bandlimiting (no real signal is perfectly bandlimited).