Sampling (signal processing)
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In signal processing, sampling is the reduction of a signal (information theory) from continuous time to discrete time.
A digital signal is often a discrete representation of a continuous analog signal (for example, a real-world signal that might represent a pressure or a velocity). The continuous signal is usually sampled at regular intervals by an analog to digital converter (ADC) and the value of the continuous signal in that interval is represented by a discrete value. The representation often introduces some error into the data. The error depends mostly on the sampling frequency, and the number of bits used for the representation. The sampling frequency or sampling rate is then the rate at which new samples are taken from the continuous signal. It represents the temporal or spatial accuracy of the discrete signal. The number of bits used for one value of the discrete signal indicates how accurately the signal magnitude is represented.
In a theoretical sampler, a continuous signal is multiplied by a Dirac comb, yielding another continuous signal. Only when this signal is quantized does it become a digital audio signal where all three indices are discretized.
The Nyquist-Shannon sampling theorem, a fundamental theorem of signal processing, states that a sampled signal cannot unambiguously represent signal components with frequencies above half the sampling frequency. This frequency (half the sampling frequency) is called the Nyquist frequency. Frequencies above the Nyquist frequency N can be observed in the digital signal, but their frequency is ambiguous. That is, a frequency component with frequency f cannot be distinguished from another component with frequency 2N-f, 2N+f, 4N-f, etc. This is called aliasing. To handle this problem as gracefully as possible, most analog signals are filtered with an anti-aliasing filter (usually a low-pass filter) at the Nyquist frequency before conversion to the digital representation.
Within the limitations of the sampling theorem, the original signal can be completely reconstructed (to within the resolution of the sample values) from the set of ideal samples by expanding each sample into a signal component constructed from the sinc function, using the Nyquist-Shannon interpolation formula.