Zak transform

<h2 id="continuous-time-zak-transform-definition">Continuous-time Zak transform: Definition</h2>
In defining the continuous-time Zak transform, the input function is a function of a real variable. 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.

<h3>Definition 1</h3>
Let a be a positive constant. The Zak transform of f(t), denoted by Za[f], is a function of t and w defined by<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

Z
 
 a
 
 
 [
 f
 ]
 (
 t
 ,
 w
 )
 =
 
 
 a
 
 
 
 ∑
 
 k
 =
 −
 ∞
 
 
 ∞
 
 
 f
 (
 a
 t
 +
 a
 k
 )
 
 e
 
 −
 2
 π
 k
 w
 i
 
 
 
 
 {\displaystyle Z_{a}[f](t,w)={\sqrt {a}}\sum _{k=-\infty }^{\infty }f(at+ak)e^{-2\pi kwi}}
 
.
<h3>Definition 2</h3>
The special case of Definition 1 obtained by taking a = 1 is sometimes taken as the definition of the Zak transform.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> In this special case, the Zak transform of f(t) is denoted by Z[f].

Z
 [
 f
 ]
 (
 t
 ,
 w
 )
 =
 
 ∑
 
 k
 =
 −
 ∞
 
 
 ∞
 
 
 f
 (
 t
 +
 k
 )
 
 e
 
 −
 2
 π
 k
 w
 i
 
 
 
 
 {\displaystyle Z[f](t,w)=\sum _{k=-\infty }^{\infty }f(t+k)e^{-2\pi kwi}}
 
.
<h3>Definition 3</h3>
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:

Z
 [
 f
 ]
 (
 t
 ,
 ν
 )
 =
 
 ∑
 
 k
 =
 −
 ∞
 
 
 ∞
 
 
 f
 (
 t
 +
 k
 )
 
 e
 
 −
 k
 ν
 i
 
 
 
 
 {\displaystyle Z[f](t,\nu )=\sum _{k=-\infty }^{\infty }f(t+k)e^{-k\nu i}}
 
.
<h3>Definition 4</h3>
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<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

Z
 
 T
 
 
 [
 f
 ]
 (
 t
 ,
 w
 )
 =
 
 
 T
 
 
 
 ∑
 
 k
 =
 −
 ∞
 
 
 ∞
 
 
 f
 (
 t
 +
 k
 T
 )
 
 e
 
 −
 2
 π
 k
 w
 T
 i
 
 
 
 
 {\displaystyle Z_{T}[f](t,w)={\sqrt {T}}\sum _{k=-\infty }^{\infty }f(t+kT)e^{-2\pi kwTi}}
 
.
Here t and w are assumed to satisfy the conditions 0 ≤ t ≤ T and 0 ≤ w ≤ 1/T.

<h2 id="example">Example</h2>
The Zak transform of the function

ϕ
 (
 t
 )
 =
 
 
 {
 
 
 
 1
 ,
 
 
 0
 ≤
 t
 <
 1
 
 
 
 
 0
 ,
 
 
 
 otherwise
 
 
 
 
 
 
 
 
 
 {\displaystyle \phi (t)={\begin{cases}1,&0\leq t<1\\0,&{\text{otherwise}}\end{cases}}}

is given by

Z
        [
        ϕ
        ]
        (
        t
        ,
        w
        )
        =
        
          e
          
            −
            2
            π
            ⌈
            −
            t
            ⌉
            w
            i
          
        
      
    
    {\displaystyle Z[\phi ](t,w)=e^{-2\pi \lceil -t\rceil wi}}

where 
 
 
 
 ⌈
 −
 t
 ⌉
 
 
 {\displaystyle \lceil -t\rceil }
 
 denotes the smallest integer not less than 
 
 
 
 −
 t
 
 
 {\displaystyle -t}
 
 (the <a href="/facts/Ceiling_function/ssehyMMf">ceiling function</a>).

<h2 id="properties-of-the-zak-transform">Properties of the Zak transform</h2>
In the following it will be assumed that the Zak transform is as given in Definition 2.
1. Linearity
Let a and b be any real or complex numbers. Then

Z
        [
        a
        f
        +
        b
        g
        ]
        (
        t
        ,
        w
        )
        =
        a
        Z
        [
        f
        ]
        (
        t
        ,
        w
        )
        +
        b
        Z
        [
        g
        ]
        (
        t
        ,
        w
        )
      
    
    {\displaystyle Z[af+bg](t,w)=aZ[f](t,w)+bZ[g](t,w)}

2. Periodicity

Z
        [
        f
        ]
        (
        t
        ,
        w
        +
        1
        )
        =
        Z
        [
        f
        ]
        (
        t
        ,
        w
        )
      
    
    {\displaystyle Z[f](t,w+1)=Z[f](t,w)}

3. Quasi-periodicity

Z
        [
        f
        ]
        (
        t
        +
        1
        ,
        w
        )
        =
        
          e
          
            2
            π
            w
            i
          
        
        Z
        [
        f
        ]
        (
        t
        ,
        w
        )
      
    
    {\displaystyle Z[f](t+1,w)=e^{2\pi wi}Z[f](t,w)}

4. Conjugation

Z
        [
        
          
            
              f
              ¯
            
          
        
        ]
        (
        t
        ,
        w
        )
        =
        
          
            
              Z
              [
              f
              ]
            
            ¯
          
        
        (
        t
        ,
        −
        w
        )
      
    
    {\displaystyle Z[{\bar {f}}](t,w)={\overline {Z[f]}}(t,-w)}

5. Symmetry

If f(t) is even then 
 
 
 
 Z
 [
 f
 ]
 (
 t
 ,
 w
 )
 =
 Z
 [
 f
 ]
 (
 −
 t
 ,
 −
 w
 )
 
 
 {\displaystyle Z[f](t,w)=Z[f](-t,-w)}

If f(t) is odd then 
 
 
 
 Z
 [
 f
 ]
 (
 t
 ,
 w
 )
 =
 −
 Z
 [
 f
 ]
 (
 −
 t
 ,
 −
 w
 )
 
 
 {\displaystyle Z[f](t,w)=-Z[f](-t,-w)}

6. Convolution
Let 
 
 
 
 ⋆
 
 
 {\displaystyle \star }
 
 denote <a href="/facts/Convolution/PVPrdz9J">convolution</a> with respect to the variable t.

Z
        [
        f
        ⋆
        g
        ]
        (
        t
        ,
        w
        )
        =
        Z
        [
        f
        ]
        (
        t
        ,
        w
        )
        ⋆
        Z
        [
        g
        ]
        (
        t
        ,
        w
        )
      
    
    {\displaystyle Z[f\star g](t,w)=Z[f](t,w)\star Z[g](t,w)}

<h2 id="inversion-formula">Inversion formula</h2>
Given the Zak transform of a function, the function can be reconstructed using the following formula:

f
        (
        t
        )
        =
        
          ∫
          
            0
          
          
            1
          
        
        Z
        [
        f
        ]
        (
        t
        ,
        w
        )
        
        d
        w
        .
      
    
    {\displaystyle f(t)=\int _{0}^{1}Z[f](t,w)\,dw.}

<h2 id="discrete-zak-transform-definition">Discrete Zak transform: Definition</h2>
Let 
 
 
 
 f
 (
 n
 )
 
 
 {\displaystyle f(n)}
 
 be a function of an integer variable 
 
 
 
 n
 ∈
 
 Z
 
 
 
 {\displaystyle n\in \mathbb {Z} }
 
 (a <a href="/facts/Sequence/gV2S4uqp">sequence</a>). The discrete Zak transform of 
 
 
 
 f
 (
 n
 )
 
 
 {\displaystyle f(n)}
 
 is a function of two real variables, one of which is the integer variable 
 
 
 
 n
 
 
 {\displaystyle n}
 
. The other variable is a real variable which may be denoted by 
 
 
 
 w
 
 
 {\displaystyle w}
 
. The discrete Zak transform has also been defined variously. However, only one of the definitions is given below.

<h3>Definition</h3>
The discrete Zak transform of the function 
 
 
 
 f
 (
 n
 )
 
 
 {\displaystyle f(n)}
 
 where 
 
 
 
 n
 
 
 {\displaystyle n}
 
 is an integer variable, denoted by 
 
 
 
 Z
 [
 f
 ]
 
 
 {\displaystyle Z[f]}
 
, is defined by

Z
        [
        f
        ]
        (
        n
        ,
        w
        )
        =
        
          ∑
          
            k
            =
            −
            ∞
          
          
            ∞
          
        
        f
        (
        n
        +
        k
        )
        
          e
          
            −
            2
            π
            k
            w
            i
          
        
        .
      
    
    {\displaystyle Z[f](n,w)=\sum _{k=-\infty }^{\infty }f(n+k)e^{-2\pi kwi}.}

<h3>Inversion formula</h3>
Given the discrete transform of a function 
 
 
 
 f
 (
 n
 )
 
 
 {\displaystyle f(n)}
 
, the function can be reconstructed using the following formula:

f
        (
        n
        )
        =
        
          ∫
          
            0
          
          
            1
          
        
        Z
        [
        f
        ]
        (
        n
        ,
        w
        )
        
        d
        w
        .
      
    
    {\displaystyle f(n)=\int _{0}^{1}Z[f](n,w)\,dw.}

<h2 id="applications">Applications</h2>
The Zak transform has been successfully used in physics in quantum field theory,<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> 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.

<h2 id="references">References</h2>

<ol>
<li id="fn:1">"Zak transform". Encyclopedia of Mathematics. Retrieved 15 December 2014. <a href="http://www.encyclopediaofmath.org/index.php/Zak_transform" target="_blank">http://www.encyclopediaofmath.org/index.php/Zak_transform</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Alexander D. Poularikas, ed. (2010). Transforms and Applications Handbook (3rd ed.). CRC Press. pp. 16.1 – 16.21. ISBN 978-1-4200-6652-4. <a href="978-1-4200-6652-4" target="_blank">978-1-4200-6652-4</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">"Zak transform". Encyclopedia of Mathematics. Retrieved 15 December 2014. <a href="http://www.encyclopediaofmath.org/index.php/Zak_transform" target="_blank">http://www.encyclopediaofmath.org/index.php/Zak_transform</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Alexander D. Poularikas, ed. (2010). Transforms and Applications Handbook (3rd ed.). CRC Press. pp. 16.1 – 16.21. ISBN 978-1-4200-6652-4. <a href="978-1-4200-6652-4" target="_blank">978-1-4200-6652-4</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">"Zak transform". Encyclopedia of Mathematics. Retrieved 15 December 2014. <a href="http://www.encyclopediaofmath.org/index.php/Zak_transform" target="_blank">http://www.encyclopediaofmath.org/index.php/Zak_transform</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Alexander D. Poularikas, ed. (2010). Transforms and Applications Handbook (3rd ed.). CRC Press. pp. 16.1 – 16.21. ISBN 978-1-4200-6652-4. <a href="978-1-4200-6652-4" target="_blank">978-1-4200-6652-4</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">"Zak transform". Encyclopedia of Mathematics. Retrieved 15 December 2014. <a href="http://www.encyclopediaofmath.org/index.php/Zak_transform" target="_blank">http://www.encyclopediaofmath.org/index.php/Zak_transform</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Alexander D. Poularikas, ed. (2010). Transforms and Applications Handbook (3rd ed.). CRC Press. pp. 16.1 – 16.21. ISBN 978-1-4200-6652-4. <a href="978-1-4200-6652-4" target="_blank">978-1-4200-6652-4</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Alexander D. Poularikas, ed. (2010). Transforms and Applications Handbook (3rd ed.). CRC Press. pp. 16.1 – 16.21. ISBN 978-1-4200-6652-4. <a href="978-1-4200-6652-4" target="_blank">978-1-4200-6652-4</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">J. Klauder, B.S. Skagerstam (1985). Coherent States. World Scientific. <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
</ol>

Zak transform open-in-new

Zak transform