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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 (a continuous signal) to a sequence of samples (a discrete-time signal).

A sample is a value or set of values at a point in time and/or space. 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 is retrievable from a sequence of samples, up to the Nyquist limit, by passing the sequence of samples through a type of low pass filter called a reconstruction filter.

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.

## Complex sampling

Complex sampling (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.<

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.

## Complex sampling

Complex sampling (I/Q sampling) is the simultaneous sampling of two different, but related, waveforms, resulting in p

Complex sampling (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.[A]  When one waveform$,{\hat {s}}(t),$ is the Hilbert transform of the other waveform$,s(t),\,$ the complex-valued function,  $s_{a}(t)\triangleq s(t)+i\cdot {\hat {s}}(t),$ 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).[B] More apparently, the equivalent baseband waveform,  $s_{a}(t)\cdot e^{-i2\pi {\frac {B}{2}}t},$ 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 ${\hat {s}}(t),$ by processing the product sequence$,\left[s(nT)\cdot e^{-i2\pi {\frac {B}{2}}Tn}\right],$ [C]  through a digital lowpass filter whose cutoff frequency is B/2.[D] 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.