Sampling Theorem and Nyquist Rate

The Fundamental Problem

Natural signals—speech, EEG, radar returns, seismic waves—are continuous-time. Computers can only process discrete-time sequences. The bridge between these worlds is sampling.

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Why Sampling Is Necessary

A continuous-time signal $x_a(t)$ contains an infinite continuum of values—impossible for digital systems to store or process. Sampling captures values at uniform intervals $T$ (the sampling period), producing a discrete sequence:

\[x[n] = x_a(nT) \quad \text{for } n = 0, \pm 1, \pm 2, \dots\]

The sampling frequency $F_s = 1/T$ (Hz) determines how many samples per second we take.

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The Sampling Theorem (Nyquist–Shannon)

If a continuous-time signal $x_a(t)$ contains no frequency components above $F_{\text{max}}$ Hz, it can be perfectly reconstructed from its samples provided the sampling frequency satisfies:

\[F_s \ge 2 F_{\text{max}}\]

The minimum rate $F_s = 2 F_{\text{max}}$ is the Nyquist rate. The corresponding frequency $F_s/2$ is the Nyquist frequency.

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