Sample a sound at twice its highest pitch and nothing is lost
The Nyquist–Shannon sampling theorem makes a startling promise: a signal can be rebuilt exactly from snapshots, provided they come faster than twice its highest frequency. That rule underpins digital signal processing, the number crunching inside mobile phone calls, MRI scanners, hearing aids and radar.
Before a computer can touch a real-world signal, an analog-to-digital converter must turn it into numbers in two steps. Discretisation chops the signal into equal time slices and records one amplitude for each. Quantisation then rounds every measurement to a value from a limited set, much as real numbers get rounded to integers. Engineers usually sample well above the theoretical minimum and add an anti-aliasing filter to cap the bandwidth, though an imperfect filter lets some distortion leak into the rebuilt signal. After processing, a digital-to-analog converter often turns the result back into a continuous signal.
Much of the craft lies in choosing where to look at a signal. In the time domain, or the space domain for images where position means pixel location, the workhorse is filtering: each new output is a weighted mix of neighbouring samples, calculated by convolving the input with an impulse response. Finite impulse response filters are always stable, while infinite impulse response designs contain feedback that can run away into oscillation, a risk analysed with the Z-transform. Alternatively the Fourier transform converts a signal into the strength and phase of each frequency, revealing which tones are present and which are missing. Wavelet methods capture timing and frequency together, within limits set by an uncertainty principle.
Hardware follows the job. Offline work on stored files can run on an ordinary computer using techniques like the FFT, whereas real-time tasks call for dedicated signal processors, programmable FPGA chips or, for the most demanding or high-volume products, custom ASICs. Music software splits along similar lines: Logic Pro and Cubase rely on the computer's own processor, while Pro Tools HD leans on extra DSP hardware.
Digital processing beats analog in areas like correcting transmission errors and compressing data. Hearing aids have used it since 1996 for automatic directional microphones, and it also shapes room correction in hi-fi systems, digital synthesizers and CAT scans.
Source: Digital signal processing
