Zak transform
Inmathematics,theZak transform[1][2](also known as theGelfandmapping) is a certain operation which takes as input a function of one variable and produces as output a function of two variables. The output function is called the Zak transform of the input function. The transform is defined as aninfinite seriesin which each term is a product of adilationof atranslationby anintegerof the function and anexponential function.In applications of Zak transform tosignal processingthe input function represents asignaland the transform will be a mixedtime–frequencyrepresentation of the signal. The signal may bereal valuedorcomplex-valued,defined on a continuous set (for example, the real numbers) or adiscrete set(for example, the integers or a finite subset of integers). The Zak transform is a generalization of thediscrete Fourier transform.[1][2]
The Zak transform had been discovered by several people in different fields and was called by different names. It was called the "Gelfand mapping" becauseIsrael Gelfandintroduced it in his work oneigenfunctionexpansions. The transform was rediscovered independently byJoshua Zakin 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.[1][2]
Continuous-time Zak transform: Definition
[edit]In defining the continuous-time Zak transform, the input function is a function of a real variable. So, letf(t) be a function of a real variablet.The continuous-time Zak transform off(t) is a function of two real variables one of which ist.The other variable may be denoted byw.The continuous-time Zak transform has been defined variously.
Definition 1
[edit]Letabe a positive constant. The Zak transform off(t), denoted byZa[f], is a function oftandwdefined by[1]
- .
={\sqrt {a}}\sum _{k=-\infty }^{\infty }f(at+ak)e^{-2\pi kwi}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2716c01afca29abf8e2713e17ad6ebf387863084)
Definition 2
[edit]The special case of Definition 1 obtained by takinga= 1 is sometimes taken as the definition of the Zak transform.[2]In this special case, the Zak transform off(t) is denoted byZ[f].
- .
=\sum _{k=-\infty }^{\infty }f(t+k)e^{-2\pi kwi}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14c43d62184c3a075028d9c18b74292496c651fe)
Definition 3
[edit]The notationZ[f] is used to denote another form of the Zak transform. In this form, the Zak transform off(t) is defined as follows:
- .
=\sum _{k=-\infty }^{\infty }f(t+k)e^{-k\nu i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03e958a39d25befa4a6fd445276fc4bb970f7c44)
Definition 4
[edit]LetTbe a positive constant. The Zak transform off(t), denoted byZT[f], is a function oftandwdefined by[2]
- .
={\sqrt {T}}\sum _{k=-\infty }^{\infty }f(t+kT)e^{-2\pi kwTi}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a79bff80c299271e777c4b8722027a04f18faade)
Heretandware assumed to satisfy the conditions 0 ≤t≤Tand 0 ≤w≤ 1/T.
Example
[edit]The Zak transform of the function
is given by
=e^{-2\pi \lceil -t\rceil wi}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ab11bd42c3ec35805b4eec37bc3b4dcef991701)
wheredenotes the smallest integer not less than(theceil function).
Properties of the Zak transform
[edit]In the following it will be assumed that the Zak transform is as given in Definition 2.
1. Linearity
Letaandbbe any real or complex numbers. Then
=aZ[f](t,w)+bZ[g](t,w)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ac7575e6eeca9669b6834f6797e916830a55e4)
2. Periodicity
=Z[f](t,w)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/364571182e34ac8149fdc4f9694713001cb373c0)
3. Quasi-periodicity
=e^{2\pi wi}Z[f](t,w)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa701c5ded113f55a854751c5404319e66ded1d7)
4. Conjugation
={\overline {Z[f]}}(t,-w)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c119a4a1d4013401b61f7081987514013f66d7c)
5. Symmetry
- Iff(t) is even then
=Z[f](-t,-w)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9340d5d01aceebbb56b1b6a8371bc9c295eb9a7)
- Iff(t) is odd then
=-Z[f](-t,-w)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89e675761f9de84fa1d3d6eb74a3e21c7b1517d0)
6. Convolution
Letdenoteconvolutionwith respect to the variablet.
=Z[f](t,w)\star Z[g](t,w)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ea544b7c8197316779bb619506a5cb1f409b92c)
Inversion formula
[edit]Given the Zak transform of a function, the function can be reconstructed using the following formula:
\,dw.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4052cca8190c1fb5fcc884030a68b685dd68f008)
Discrete Zak transform: Definition
[edit]Letbe a function of an integer variable(asequence). The discrete Zak transform ofis a function of two real variables, one of which is the integer variable.The other variable is a real variable which may be denoted by.The discrete Zak transform has also been defined variously. However, only one of the definitions is given below.
Definition
[edit]The discrete Zak transform of the functionwhereis an integer variable, denoted by,is defined by
![{\displaystyle Z[f]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e901e66f65cb937c058578ef158093ef853b3b5)
=\sum _{k=-\infty }^{\infty }f(n+k)e^{-2\pi kwi}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76b8e17b2749a8f92d67b64212b5f07d98c3e7f1)
Inversion formula
[edit]Given the discrete transform of a function,the function can be reconstructed using the following formula:
\,dw.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b88f2f755b0afd39e0a95f0c142fddff20caa30d)
Applications
[edit]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. The Zak transform has also applications in mathematics. For example, it has been used in the Gabor representation problem.