Sampling (signal processing)

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.