Digital signal processing

Digital signal processing (DSP) is the manipulation of signals after they have been converted into sequences of numbers. A continuous, analogue signal such as sound or a measured voltage is first sampled at regular intervals and each sample is rounded to a digital value, a step called quantisation. Once in numerical form the signal can be filtered, analysed and transformed by arithmetic operations, which can be made far more precise and repeatable than the equivalent analogue circuits.

Sampling rate is governed by a key result. The Nyquist sampling theorem, associated with the work of Nyquist and Shannon, states that a signal must be sampled at more than twice its highest frequency to be reconstructed faithfully. If it is sampled too slowly, high frequencies fold down and masquerade as lower ones, a corruption called aliasing; an anti-aliasing filter before the converter prevents this. Quantisation introduces a small rounding error that limits the resolution and sets the noise floor, with more bits per sample giving finer resolution.

Much DSP is understood in two complementary views. In the time domain, operations such as convolution describe how a filter responds to an input, and correlation measures how similar two signals are. The Fourier transform moves a signal into the frequency domain, revealing the strength of each frequency it contains and making filtering and spectral analysis straightforward. A common building block is the finite impulse response (FIR) filter, which computes each output as a weighted sum of recent input samples; it is always stable and can be designed to have a linear phase response, which preserves the shape of waveforms.

Frequently asked questions

What does the Nyquist theorem require?

A signal must be sampled at more than twice its highest frequency, otherwise it cannot be reconstructed correctly and aliasing occurs.

What is aliasing?

When sampling is too slow, frequencies above half the sampling rate fold back and appear as false lower frequencies, distorting the signal; an anti-aliasing filter prevents it.

Why are FIR filters popular?

A finite impulse response filter is inherently stable and can be designed with linear phase, so it does not distort the relative timing of frequency components.