Sampling

Sampling is the process of converting a continuous-time signal into a discrete-time signal by measuring its value at regular intervals.

Mathematical Model

Ideal sampling multiplies the continuous signal $x(t)$ by a Dirac comb:

$$ x_s(t) = x(t) \cdot \sum_{n=-\infty}^{\infty} \delta(t - nT_s) $$

where $T_s$ is the sampling period and $f_s = 1/T_s$ is the sampling frequency.

Effect in Frequency Domain

Sampling causes the spectrum to become periodic (replicated at multiples of $f_s$):

$$ X_s(f) = \frac{1}{T_s} \sum_{k=-\infty}^{\infty} X(f - k f_s) $$

Nyquist-Shannon Sampling Theorem

A bandlimited signal with maximum frequency $f_{max}$ can be perfectly reconstructed from its samples if:

$$ f_s > 2 f_{max} $$

The minimum sampling rate $2 f_{max}$ is called the Nyquist rate.

See: Whittaker-Shannon sampling theorem

Aliasing

When $f_s < 2 f_{max}$, the replicated spectra overlap. High frequencies “fold back” and appear as lower frequencies — this is aliasing.

Aliasing cannot be corrected after sampling. Prevention requires:

1. Anti-aliasing filter: Low-pass filter before sampling
2. Oversampling: Sample faster than strictly necessary

Reconstruction

Ideal reconstruction uses a low-pass filter (sinc interpolation):

$$ x(t) = \sum_{n=-\infty}^{\infty} x[n] \, \text{sinc}\left(\frac{t - nT_s}{T_s}\right) $$

See: Sinc function

Practical Considerations

• Real signals are never perfectly bandlimited
• Anti-aliasing filters have finite rolloff
• Reconstruction filters are causal approximations
• Quantization adds another source of error

See Also

• Whittaker-Shannon sampling theorem
• Sinc function
• Nyquist criterion