In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of "samples". A sample is a value of the signal at a point in time and/or space; this definition differs from the term's usage in statistics, which refers to a set of such values. A sampler is a subsystem or operation that extracts samples from a continuous signal. A theoretical ideal sampler produces samples equivalent to the instantaneous value of the continuous signal at the desired points. The original signal can be reconstructed from a sequence of samples, up to the Nyquist limit, by passing the sequence of samples through a reconstruction filter.
Theory
Functions of space, time, or any other dimension can be sampled, and similarly in two or more dimensions. For functions that vary with time, let s ( t ) {\displaystyle s(t)} be a continuous function (or "signal") to be sampled, and let sampling be performed by measuring the value of the continuous function every T {\displaystyle T} seconds, which is called the sampling interval or sampling period. Then the sampled function is given by the sequence:
s ( n T ) {\displaystyle s(nT)} , for integer values of n {\displaystyle n} . The sampling frequency or sampling rate, f s {\displaystyle f_{s}} , is the average number of samples obtained in one second, thus f s = 1 / T {\displaystyle f_{s}=1/T} , with the unit samples per second, sometimes referred to as hertz, for example 48 kHz is 48,000 samples per second. Reconstructing a continuous function from samples is done by interpolation algorithms. The Whittaker–Shannon interpolation formula is mathematically equivalent to an ideal low-pass filter whose input is a sequence of Dirac delta functions that are modulated (multiplied) by the sample values. When the time interval between adjacent samples is a constant ( T ) {\displaystyle (T)} , the sequence of delta functions is called a Dirac comb. Mathematically, the modulated Dirac comb is equivalent to the product of the comb function with s ( t ) {\displaystyle s(t)} . That mathematical abstraction is sometimes referred to as impulse sampling. Most sampled signals are not simply stored and reconstructed. The fidelity of a theoretical reconstruction is a common measure of the effectiveness of sampling. That fidelity is reduced when s ( t ) {\displaystyle s(t)} contains frequency components whose cycle length (period) is less than 2 sample intervals (see Aliasing). The corresponding frequency limit, in cycles per second (hertz), is 0.5 {\displaystyle 0.5} cycle/sample × f s {\displaystyle f_{s}} samples/second = f s / 2 {\displaystyle f_{s}/2} , known as the Nyquist frequency of the sampler. Therefore, s ( t ) {\displaystyle s(t)} is usually the output of a low-pass filter, functionally known as an anti-aliasing filter. Without an anti-aliasing filter, frequencies higher than the Nyquist frequency will influence the samples in a way that is misinterpreted by the interpolation process.
Practical considerations In practice, the continuous signal is sampled using an analog-to-digital converter (ADC), a device with various physical limitations. This results in deviations from the theoretically perfect reconstruction, collectively referred to as distortion. Various types of distortion can occur, including:
… excerpt ends here. Continue reading the full article.
