In mathematics, the Zak transform[1][2] (also known as the Gelfand mapping) is a certain operation which takes as input a function of one variable and produces as output a function of two variables.
The transform is defined as an infinite series in which each term is a product of a dilation of a translation by an integer of the function and an exponential function.
The signal may be real valued or complex-valued, defined on a continuous set (for example, the real numbers) or a discrete set (for example, the integers or a finite subset of integers).
[1][2] The Zak transform had been discovered by several people in different fields and was called by different names.
The transform was rediscovered independently by Joshua Zak in 1967 who called it the "k-q representation".
There seems to be a general consensus among experts in the field to call it the Zak transform, since Zak was the first to systematically study that transform in a more general setting and recognize its usefulness.
So, let f(t) be a function of a real variable t. The continuous-time Zak transform of f(t) is a function of two real variables one of which is t. The other variable may be denoted by w. The continuous-time Zak transform has been defined variously.
The Zak transform of f(t), denoted by Za[f], is a function of t and w defined by[1] The special case of Definition 1 obtained by taking a = 1 is sometimes taken as the definition of the Zak transform.
[2] In this special case, the Zak transform of f(t) is denoted by Z[f].
The notation Z[f] is used to denote another form of the Zak transform.
In this form, the Zak transform of f(t) is defined as follows: Let T be a positive constant.
The Zak transform of f(t), denoted by ZT[f], is a function of t and w defined by[2] Here t and w are assumed to satisfy the conditions 0 ≤ t ≤ T and 0 ≤ w ≤ 1/T.
The discrete Zak transform has also been defined variously.
, the function can be reconstructed using the following formula: The Zak transform has been successfully used in physics in quantum field theory,[3] in electrical engineering in time-frequency representation of signals, and in digital data transmission.