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.^{[upper-alpha 1]}

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.

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) be a continuous function (or "signal") to be sampled, and let sampling be performed by measuring the value of the continuous function every T seconds, which is called the sampling interval or sampling period.^[1]  Then the sampled function is given by the sequence:

S(nT),   for integer values of n.

The sampling frequency or sampling rate, f_s, is the average number of samples obtained in one second, thus f_s = 1/T, with the unit samples per second, sometimes referred to as hertz, for example e.g. 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), 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). That mathematical abstraction is sometimes referred to as impulse sampling.^[2]

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) 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 cycle/sample × f_s samples/second = f_s/2, known as the Nyquist frequency of the sampler. Therefore, 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.^[3]

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:

• Aliasing. Some amount of aliasing is inevitable because only theoretical, infinitely long, functions can have no frequency content above the Nyquist frequency. Aliasing can be made arbitrarily small by using a sufficiently large order of the anti-aliasing filter.
• Aperture error results from the fact that the sample is obtained as a time average within a sampling region, rather than just being equal to the signal value at the sampling instant.^[4] In a capacitor-based sample and hold circuit, aperture errors are introduced by multiple mechanisms. For example, the capacitor cannot instantly track the input signal and the capacitor can not instantly be isolated from the input signal.
• Jitter or deviation from the precise sample timing intervals.
• Noise, including thermal sensor noise, analog circuit noise, etc.
• Slew rate limit error, caused by the inability of the ADC input value to change sufficiently rapidly.
• Quantization as a consequence of the finite precision of words that represent the converted values.
• Error due to other non-linear effects of the mapping of input voltage to converted output value (in addition to the effects of quantization).

Although the use of oversampling can completely eliminate aperture error and aliasing by shifting them out of the passband, this technique cannot be practically used above a few GHz, and may be prohibitively expensive at much lower frequencies. Furthermore, while oversampling can reduce quantization error and non-linearity, it cannot eliminate these entirely. Consequently, practical ADCs at audio frequencies typically do not exhibit aliasing, aperture error, and are not limited by quantization error. Instead, analog noise dominates. At RF and microwave frequencies where oversampling is impractical and filters are expensive, aperture error, quantization error and aliasing can be significant limitations.

Jitter, noise, and quantization are often analyzed by modeling them as random errors added to the sample values. Integration and zero-order hold effects can be analyzed as a form of low-pass filtering. The non-linearities of either ADC or DAC are analyzed by replacing the ideal linear function mapping with a proposed nonlinear function.

When a bandpass signal is sampled slower than its Nyquist rate, the samples are indistinguishable from samples of a low-frequency alias of the high-frequency signal. That is often done purposefully in such a way that the lowest-frequency alias satisfies the Nyquist criterion, because the bandpass signal is still uniquely represented and recoverable. Such undersampling is also known as bandpass sampling, harmonic sampling, IF sampling, and direct IF to digital conversion.^[22]

Oversampling is used in most modern analog-to-digital converters to reduce the distortion introduced by practical digital-to-analog converters, such as a zero-order hold instead of idealizations like the Whittaker–Shannon interpolation formula.^[23]

Complex sampling (or I/Q sampling) is the simultaneous sampling of two different, but related, waveforms, resulting in pairs of samples that are subsequently treated as complex numbers.^{[upper-alpha 3]}  When one waveform${\displaystyle ,{\hat {s}}(t),}$  is the Hilbert transform of the other waveform${\displaystyle ,s(t),\,}$  the complex-valued function,  ${\displaystyle s_{a}(t)\triangleq s(t)+i\cdot {\hat {s}}(t),}$  is called an analytic signal,  whose Fourier transform is zero for all negative values of frequency. In that case, the Nyquist rate for a waveform with no frequencies ≥ B can be reduced to just B (complex samples/sec), instead of 2B (real samples/sec).^{[upper-alpha 4]} More apparently, the equivalent baseband waveform,  ${\displaystyle s_{a}(t)\cdot e^{-i2\pi {\frac {B}{2}}t},}$  also has a Nyquist rate of B, because all of its non-zero frequency content is shifted into the interval [-B/2, B/2).

Although complex-valued samples can be obtained as described above, they are also created by manipulating samples of a real-valued waveform. For instance, the equivalent baseband waveform can be created without explicitly computing ${\displaystyle {\hat {s}}(t),}$  by processing the product sequence${\displaystyle ,\left[s(nT)\cdot e^{-i2\pi {\frac {B}{2}}Tn}\right],}$^{[upper-alpha 5]}  through a digital low-pass filter whose cutoff frequency is B/2.^{[upper-alpha 6]} Computing only every other sample of the output sequence reduces the sample-rate commensurate with the reduced Nyquist rate. The result is half as many complex-valued samples as the original number of real samples. No information is lost, and the original s(t) waveform can be recovered, if necessary.

• Crystal oscillator frequencies
• Downsampling
• Upsampling
• Multidimensional sampling
• In-phase and quadrature components and I/Q data
• Sample rate conversion
• Digitizing
• Sample and hold
• Beta encoder
• Kell factor
• Bit rate
• Normalized frequency