View source for Discrete-time Fourier transform

{{distinguish|text=the [[discrete Fourier transform]]}}
{{short description|Fourier analysis technique applied to sequences}}
{{Fourier transforms}}

In [[mathematics]], the '''discrete-time Fourier transform''' ('''DTFT''') is a form of [[Fourier analysis]] that is applicable to a sequence of values.

The DTFT is often used to analyze samples of a continuous function.  The term ''discrete-time'' refers to the fact that the transform operates on discrete data, often samples whose interval has units of time.  From uniformly spaced samples it produces a function of frequency that is a [[periodic summation]] of the [[continuous Fourier transform]] of the original continuous function.  Under certain theoretical conditions, described by the [[sampling theorem]], the original continuous function can be recovered perfectly from the DTFT and thus from the original discrete samples.  The DTFT itself is a continuous function of frequency, but discrete samples of it can be readily calculated via the [[discrete Fourier transform]] (DFT) (see {{slink|#Sampling the DTFT}}), which is by far the most common method of modern Fourier analysis.

Both transforms are invertible.  The inverse DTFT is the original sampled data sequence.  The inverse DFT is a periodic summation of the original sequence.  The [[fast Fourier transform]] (FFT) is an algorithm for computing one cycle of the DFT, and its inverse produces one cycle of the inverse DFT.

== Definition ==
The discrete-time Fourier transform of a discrete set of real or complex numbers {{math|''x''[''n'']}}, for all [[Number#Integers|integers]] {{mvar|n}}, is a [[Fourier series]], which produces a periodic function of a frequency variable.  When the frequency variable, ω, has [[Normalized frequency (digital signal processing)|normalized units]] of ''radians/sample'', the periodicity is {{math|2π}}, and the Fourier series is:

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk|:|<math>X_{2\pi}(\omega) = \sum_{n=-\infty}^{\infty} x[n] \,e^{-i \omega n}.</math>|{{EquationRef|Eq.1}}}}
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

The utility of this frequency domain function is rooted in the [[Poisson summation formula]].  Let {{math|''X''(''f'')}} be the Fourier transform of any function, {{math|''x''(''t'')}}, whose samples at some interval {{mvar|T}} (''seconds'') are equal (or proportional) to the {{math|''x''[''n'']}} sequence, i.e. {{math|''T''⋅''x''(''nT'') {{=}} ''x''[''n'']}}.  Then the periodic function represented by the Fourier series is a periodic summation of {{math|''X''(''f'')}} in terms of frequency {{mvar|f}} in [[hertz]] (''cycles/sec''):

{{NumBlk|:|<math>
X_{1/T}(f) = X_{2\pi}(2\pi f T)\ \triangleq
\sum_{n=-\infty}^{\infty} \underbrace{T\cdot x(nT)}_{x[n]}\ e^{-i 2\pi f T n}\;
\stackrel{\mathrm{Poisson\;f.}}{=} \;
 \sum_{k=-\infty}^{\infty} X\left(f - k/T\right).
</math>|{{EquationRef|Eq.2}}}}

[[File:Fourier transform, Fourier series, DTFT, DFT.svg|thumb|400px|Fig 1. Depiction of a Fourier transform (upper left) and its periodic summation (DTFT) in the lower left corner. The lower right corner depicts samples of the DTFT that are computed by a discrete Fourier transform (DFT).]]

The integer {{mvar|k}} has units of ''cycles/sample'', and {{math|1/''T''}} is the sample-rate, {{mvar|f<sub>s</sub>}} (''samples/sec'').  So {{math|''X''<sub>1/''T''</sub>(''f'')}} comprises exact copies of {{math|''X''(''f'')}} that are shifted by multiples of {{mvar|f<sub>s</sub>}} hertz and combined by addition. For sufficiently large {{mvar|f<sub>s</sub>}} the {{math|''k'' {{=}} 0}} term can be observed in the region {{math|[−''f<sub>s</sub>/2, f<sub>s</sub>/2'']}} with little or no distortion ([[aliasing]]) from the other terms.  In Fig.1, the extremities of the distribution in the upper left corner are masked by aliasing in the periodic summation (lower left).

We also note that {{math|''e''<sup>−''i2πfTn''</sup>}} is the Fourier transform of {{math|''δ''(''t'' − ''nT'')}}.  Therefore, an alternative definition of DTFT is:{{
efn-ua|In fact {{EquationNote|Eq.2}} is often justified as follows:

:<math>\begin{align}
\mathcal{F}\left \{\sum_{n=-\infty}^{\infty} T\cdot x(nT) \cdot \delta(t-nT)\right \} &=\mathcal{F}\left \{x(t)\cdot T \sum_{n=-\infty}^{\infty} \delta(t-nT)\right \}\\
&= X(f) * \mathcal{F}\left \{T \sum_{n=-\infty}^{\infty} \delta(t-nT)\right \} \\
&= X(f) * \sum_{k=-\infty}^{\infty} \delta \left(f - \frac{k}{T}\right) \\
&= \sum_{k=-\infty}^{\infty} X\left(f - \frac{k}{T}\right).
\end{align}</math>
}}

{{NumBlk|:|<math>X_{1/T}(f) = \mathcal{F}\left \{\sum_{n=-\infty}^{\infty} x[n] \cdot \delta(t-nT)\right \}
</math>|{{EquationRef|Eq.3}}}}

The modulated [[Dirac comb]] function is a mathematical abstraction sometimes referred to as ''impulse sampling''.<ref name="Rao"/>

== Inverse transform ==
An operation that recovers the discrete data sequence from the DTFT function is called an ''inverse DTFT''.  For instance, the inverse continuous Fourier transform of both sides of {{EquationNote|Eq.3}} produces the sequence in the form of a modulated Dirac comb function:

:<math>\sum_{n=-\infty}^{\infty} x[n]\cdot \delta(t-n T) = \mathcal{F}^{-1}\left \{X_{1/T}(f)\right\} \ \triangleq \int_{-\infty}^\infty X_{1/T}(f)\cdot e^{i 2 \pi f t} df.</math>

However, noting that {{math|''X''<sub>1/''T''</sub>(''f'')}} is periodic, all the necessary information is contained within any interval of length {{math|1/''T''}}.  In both {{EquationNote|Eq.1}} and {{EquationNote|Eq.2}}, the summations over n are a [[Fourier series#Complex Fourier coefficients|Fourier series]], with coefficients {{math|''x''[''n'']}}.  The standard formulas for the Fourier coefficients are also the inverse transforms:

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk|:|<math>\begin{align}
x[n] &= T \int_{\frac{1}{T}} X_{1/T}(f)\cdot e^{i 2 \pi f nT} df \quad \scriptstyle{\text{(integral over any interval of length }1/T\textrm{)}} \\
\displaystyle &= \frac{1}{2 \pi}\int_{2\pi} X_{2\pi}(\omega)\cdot e^{i \omega n} d\omega \quad \scriptstyle{\text{(integral over any interval of length }2\pi\textrm{)}}
\end{align}</math>|{{EquationRef|Eq.4}}}}
|cellpadding= 6
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|border colour = #0073CF
|background colour=#F5FFFA}}

== Periodic data ==
When the input data sequence {{math|''x''[''n'']}} is {{mvar|N}}-periodic, {{EquationNote|Eq.2}} can be computationally reduced to a discrete Fourier transform (DFT), because:
* All the available information is contained within {{mvar|N}} samples.
* {{math|''X''<sub>1/''T''</sub>(''f'')}} converges to zero everywhere except at integer multiples of {{math|1/(''NT'')}}, known as [[harmonic]] frequencies.
* The DTFT is periodic, so the maximum number of unique harmonic amplitudes is {{math|(1/''T'') / (1/(''NT'')) {{=}} ''N''}}

The kernel {{math|''x''[''n''] ''e''<sup>−''i2πfTn''</sup>}} is {{mvar|N}}-periodic at the harmonic frequencies, {{math|''f'' {{=}} ''k''/(''NT'')}}. Introducing the notation <math>\sum_{N}</math> to represent a sum over any {{mvar|n}}-sequence of length {{mvar|N}}, we can write''':'''

:<math>
\begin{align}
X_{1/T}\left(\frac{k}{NT}\right) &= \sum_{m=-\infty}^{\infty} \left(\sum_{N} x[n-mN]\cdot e^{-i 2 \pi \frac{k}{N}(n-mN)}\right) \\
&= \sum_{m=-\infty}^{\infty} \left(\sum_{N} x[n]\cdot e^{-i 2 \pi \frac{k}{N}n}\right)
= T \underbrace{\left(\sum_{N} x(nT)\cdot e^{-i 2 \pi \frac{k}{N}n}\right)}_{X[k] \quad \text{(DFT)}} \cdot \left(\sum_{m=-\infty}^{\infty}1\right).
\end{align}
</math>

Therefore, the DTFT diverges at the harmonic frequencies, but at different frequency-dependent rates.  And those rates are given by the DFT of one cycle of the {{math|''x''[''n'']}} sequence.  In terms of a Dirac comb function, this is represented by''':'''

:<math>X_{1/T}(f) =  \sum_{k=-\infty}^{\infty} (T\cdot X[k])\ \cdot\  \frac{1}{NT}\delta\left(f-\frac{k}{NT}\right)
= \underbrace{\frac{1}{N} \sum_{k=-\infty}^{\infty} X[k]\ \cdot\ \delta\left(f-\frac{k}{NT}\right)}_{\text{DTFT of a periodic sequence}}. </math>&nbsp; &nbsp; &nbsp; {{
efn-ua|Substituting this expression into formula &nbsp;<math>x(nT) = \int_{\frac{1}{T}} X_{1/T}(f)\cdot e^{i 2 \pi f nT} df</math>&nbsp; produces the correctly scaled inverse DFT for the {{math|''x''(''nT'')}} sequence.
}}{{
efn-ua|The [[generalized function]] <math>\delta\left(f-\tfrac{k}{NT}\right)</math> is not unitless.  It has the same units as {{mvar|''T''}}.
}}

== Sampling the DTFT ==
When the DTFT is continuous, a common practice is to compute an arbitrary number of samples ({{mvar|N}}) of one cycle of the periodic function {{math|''X''<sub>1/''T''</sub>}}:

:<math>\begin{align}
\underbrace{X_{1/T}\left(\frac{k}{NT}\right)}_{X_k} &= \sum_{n=-\infty}^\infty x[n]\cdot e^{-i 2\pi \frac{kn}{N}} \quad \quad k = 0, \dots, N-1 \\
&= \underbrace{\sum_{N} x_{_N}[n]\cdot e^{-i 2\pi \frac{kn}{N}},}_{DFT}\quad \scriptstyle{\text{(sum over any }n\text{-sequence of length }N)}
\end{align} </math>

where <math>x_{_N}</math> is a periodic summation''':'''

:<math>x_{_N}[n]\ \triangleq\ \sum_{m=-\infty}^{\infty} x[n-mN].</math>

The <math>x_{_N}</math> sequence is the inverse DFT.  Thus, our sampling of the DTFT causes the inverse transform to become periodic.  The array of {{math|{{abs|''X''<sub>k</sub>}}<sup>2</sup>}} values is known as a '''[[periodogram]]''', and the parameter {{mvar|N}} is called NFFT in the Matlab function of the same name.<ref name="Matlab"/>

In order to evaluate one cycle of <math>x_{_N}</math> numerically, we require a finite-length {{math|''x''[''n'']}} sequence.  For instance, a long sequence might be truncated by a [[window function]] of length {{mvar|L}} resulting in three cases worthy of special mention.  For notational simplicity, consider the {{math|''x''[''n'']}} values below to represent the values modified by the window function.

'''Case: Frequency decimation.'''  {{math|''L'' {{=}} ''N'' ⋅ ''I''}}, for some integer {{mvar|I}} (typically 6 or 8)

A cycle of <math>x_{_N}</math> reduces to a summation of {{mvar|I}} ''blocks'' of length {{mvar|N}}, or ''circular addition''.{{
efn-ua|Here we borrow a concept from [[Circular shift]] and [[Circular convolution]].
}}&nbsp; The DFT then goes by various names, such as''':'''
*''window-presum FFT''<ref name="Gumas"/>
*''Weight, overlap, add (WOLA)''<ref name="Wang"/><ref name="Lyons"/>{{
efn-ua|WOLA is not to be confused with the [[Overlap-add method]] of piecewise convolution.
}}
*''polyphase FFT''<ref name="Lillington"/>
*''polyphase filter bank''<ref name="Chennamangalam"/>
*''multiple block windowing'' and ''time-aliasing''.<ref name="Dahl"/>

Recall that decimation of sampled data in one domain (time or frequency) produces overlap (sometimes known as [[aliasing]]) in the other, and vice versa.  Compared to an {{mvar|L}}-length DFT, the <math>x_{_N}</math> summation/overlap causes decimation in frequency, leaving only DTFT samples least affected by [[spectral leakage]].  That is usually a priority when implementing an FFT [[filter bank|filter-bank]] (channelizer).  With a conventional window function of length {{mvar|L}}, [[Spectral leakage#Window tradeoffs|scalloping loss]] would be unacceptable.  So multi-block windows are created using [[FIR filter]] design tools.<ref>{{cite journal | last1 =Lin | first1 =Yuan-Pei | last2 =Vaidyanathan | first2 =P.P. | title =A Kaiser Window Approach for the Design of Prototype Filters of Cosine Modulated Filterbanks | journal =IEEE Signal Processing Letters | volume =5 | issue =6 | pages =132–134 | date =June 1998 | url =http://authors.library.caltech.edu/6891/1/LINieeespl98.pdf | access-date =2017-03-16| doi =10.1109/97.681427 | bibcode =1998ISPL....5..132L }}</ref><ref>{{Citation | title =cmfb.m | publisher =Caltech | url =http://www.systems.caltech.edu/dsp/software/conventional/cmfb.m | access-date = 2017-03-16}}</ref>&nbsp;  Their frequency profile is flat at the highest point and falls off quickly at the midpoint between the remaining DTFT samples.  The larger the value of parameter {{mvar|I}}, the better the potential performance.

'''Case: {{math|''L'' {{=}} ''N''+1}},''' where {{mvar|N}} is even-valued

This case arises in the context of [[Window function#Symmetry|Window function]] design, out of a desire for real-valued DFT coefficients.<ref>{{cite journal |ref=refHarris |doi=10.1109/PROC.1978.10837 |last=Harris |first=Fredric J. |title=On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform |journal=Proceedings of the IEEE |volume=66 |issue=1 |pages=51–83 |date=Jan 1978 |url=http://web.mit.edu/xiphmont/Public/windows.pdf |citeseerx=10.1.1.649.9880|bibcode=1978IEEEP..66...51H }}</ref>&nbsp; When a symmetric sequence is associated with the indices {{nowrap|[-M ≤ n ≤ M],}} known as a ''[[finite Fourier transform]] data window'', its DTFT, a continuous function of frequency <math>(f),</math> is real-valued.  When the sequence is shifted into a ''DFT data window'', {{nowrap|[0 ≤ n ≤ 2M],}} the DTFT is multiplied by a complex-valued phase function:  <math>e^{-i2\pi f M}</math>.  But when sampled at frequencies <math>f = k/2M,</math> for integer values of <math>k,</math> the samples are all real-valued.  To achieve that goal, we can perform a <math>2M</math>-length DFT on a periodic summation with 1-sample of overlap. Specifically, the last sample of a data sequence is deleted and its value added to the first sample.  Then a window function, shortened by 1 sample, is applied, and the DFT is performed.  The shortened, even-length window function is sometimes called ''DFT-even''.  In actual practice, people commonly use DFT-even windows without overlapping the data, because the detrimental effects on spectral leakage are negligible for long sequences (typically hundreds of samples).{{
efn-ua|An example of the effects for short sequences can be seen at [[:File:Comparison_of_symmetric_and_periodic_Gaussian_windows.svg]], where the 9-sample symmetric sequence (green DTFT) has lower spectral leakage metrics than the 8-sample truncated sequence (blue).
}}

[[Image:zeropad.png|thumb|350px|Fig 2. DFT of {{math|''e''<sup>i2πn/8</sup>}} for {{math|''L'' {{=}} 64}} and {{math|''N'' {{=}} 256}}]]
[[Image:no-zeropad.png|thumb|350px|Fig 3. DFT of {{math|''e''<sup>i2πn/8</sup>}} for {{math|''L'' {{=}} 64}} and {{math|''N'' {{=}} 64}}]]

'''Case: Frequency interpolation.'''  {{math|''L'' ≤ ''N''}}

In this case,the DFT simplifies to a more familiar form:

:<math>X_k = \sum_{n=0}^{N-1} x[n]\cdot e^{-i 2\pi \frac{kn}{N}}.</math>

In order to take advantage of a fast Fourier transform algorithm for computing the DFT, the summation is usually performed over all {{mvar|N}} terms, even though {{math|''N'' − ''L''}} of them are zeros.  Therefore, the case {{math|''L'' < ''N''}} is often referred to as "zero-padding".

Spectral leakage, which increases as {{mvar|L}} decreases, is detrimental to certain important performance metrics, such as resolution of multiple frequency components and the amount of noise measured by each DTFT sample.  But those things don't always matter, for instance when the {{math|''x''[''n'']}} sequence is a noiseless sinusoid (or a constant), shaped by a window function.  Then it is a common practice to use ''zero-padding'' to graphically display and compare the detailed leakage patterns of window functions.  To illustrate that for a rectangular window, consider the sequence:

:<math>x[n] = e^{i 2\pi \frac{1}{8} n},\quad </math> and <math>L=64.</math>

'''Figures 2 and 3''' are plots of the magnitude of two different sized DFTs, as indicated in their labels.  In both cases, the dominant component is at the signal frequency: {{math|''f'' {{=}} 1/8 {{=}} 0.125}}.  Also visible in '''Fig 2''' is the spectral leakage pattern of the {{math|''L'' {{=}} 64}} rectangular window.  The illusion in '''Fig 3''' is a result of sampling the DTFT at just its zero-crossings.  Rather than the DTFT of a finite-length sequence, it gives the impression of an infinitely long sinusoidal sequence.  Contributing factors to the illusion are the use of a rectangular window, and the choice of a frequency (1/8 = 8/64) with exactly 8 (an integer) cycles per 64 samples.  A [[Window function#Hann window|Hann window]] would produce a similar result, except the peak would be widened to 3 samples (see [https://commons.wikimedia.org/wiki/File:DFT-even_Hann_window_&_spectral_leakage.png DFT-even Hann window]).

== Convolution ==
The [[convolution theorem]] for sequences is:

:<math>x * y\ =\ \scriptstyle \text{DTFT}^{-1} \displaystyle \left[\scriptstyle \text{DTFT} \displaystyle \{x\}\cdot \scriptstyle \text{DTFT} \displaystyle \{y\}\right].</math>

An important special case is the [[circular convolution]] of sequences {{mvar|x}} and {{mvar|y}} defined by <math>x_{_N}*y,</math> where <math>x_{_N}</math> is a periodic summation.  The discrete-frequency nature of {{math|DTFT{''x''<sub><sub>N</sub></sub>}}} "selects" only discrete values from the continuous function {{math|DTFT{''y''}}}, which results in considerable simplification of the inverse transform.  As shown at [[Convolution theorem#Functions of discrete variable sequences]]:

:<math>x_{_N} * y\ =\ \scriptstyle \text{DTFT}^{-1} \displaystyle \left[\scriptstyle \text{DTFT} \displaystyle \{x_{_N}\}\cdot \scriptstyle \text{DTFT} \displaystyle \{y\}\right]\ =\ \scriptstyle \text{DFT}^{-1} \displaystyle \left[\scriptstyle \text{DFT} \displaystyle \{x_{_N}\}\cdot \scriptstyle \text{DFT} \displaystyle \{y_{_N}\}\right].</math>

For {{mvar|x}} and {{mvar|y}} sequences whose non-zero duration is less than or equal to {{mvar|N}}, a final simplification is:

:<math>x_{_N} * y\ =\ \scriptstyle \text{DFT}^{-1} \displaystyle  \left[\scriptstyle \text{DFT} \displaystyle \{x\}\cdot \scriptstyle \text{DFT} \displaystyle \{y\}\right].</math>

The significance of this result is expounded at [[Circular convolution#Example|Circular convolution]] and [[Convolution#Fast convolution algorithms|Fast convolution algorithms]].

== Symmetry properties ==
When the real and imaginary parts of a complex function are decomposed into their [[Even and odd functions#Even–odd decomposition|even and odd parts]], there are four components, denoted below by the subscripts RE, RO, IE, and IO.  And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform''':'''<ref name=ProakisManolakis/>{{rp|p. 291}}

<math>
\begin{align}
\mathsf{Time\ domain} \quad &\ x \quad &= \quad & x_{_{RE}} \quad &+ \quad & x_{_{RO}} \quad &+ \quad i\ & x_{_{IE}} \quad &+ \quad &\underbrace{i\ x_{_{IO}}} \\
&\Bigg\Updownarrow\mathcal{F} & &\Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F}\\
\mathsf{Frequency\ domain} \quad &X \quad &= \quad & X_{RE} \quad &+ \quad &\overbrace{i\ X_{IO}} \quad &+ \quad i\ & X_{IE} \quad &+ \quad & X_{RO}
\end{align}
</math>

From this, various relationships are apparent, for example''':'''
*The transform of a real-valued function ({{math|''x''{{sub|{{sub|RE}}}}+ ''x''{{sub|{{sub|RO}}}}}}) is the [[Even and odd functions#Complex-valued functions|even symmetric]] function {{math|X{{sub|{{sub|RE}}}}+ i X{{sub|{{sub|IO}}}}}}.  Conversely, an even-symmetric transform implies a real-valued time-domain.
*The transform of an imaginary-valued function ({{math|i ''x''{{sub|{{sub|IE}}}}+ i ''x''{{sub|{{sub|IO}}}}}}) is the [[Even and odd functions#Complex-valued functions|odd symmetric]] function {{math|X{{sub|{{sub|RO}}}}+ i X{{sub|{{sub|IE}}}}}}, and the converse is true.
*The transform of an even-symmetric function ({{math|''x''{{sub|{{sub|RE}}}}+ i ''x''{{sub|{{sub|IO}}}}}}) is the real-valued function {{math|X{{sub|{{sub|RE}}}}+ X{{sub|{{sub|RO}}}}}}, and the converse is true.
*The transform of an odd-symmetric function ({{math|''x''{{sub|{{sub|RO}}}}+ i ''x''{{sub|{{sub|IE}}}}}}) is the imaginary-valued function {{math|i X{{sub|{{sub|IE}}}}+ i X{{sub|{{sub|IO}}}}}}, and the converse is true.

== Relationship to the Z-transform ==
<math>X_{2\pi}(\omega)</math> is a [[Fourier series]] that can also be expressed in terms of the bilateral [[Z-transform]].&nbsp; I.e.:

:<math>X_{2\pi}(\omega) = \left. \widehat X(z) \, \right|_{z = e^{i \omega}} = \widehat X(e^{i \omega}),</math>

where the <math>\widehat X</math> notation distinguishes the Z-transform from the Fourier transform.  Therefore, we can also express a portion of the Z-transform in terms of the Fourier transform:

:<math>
\begin{align}
\widehat X(e^{i \omega}) &= \ X_{1/T}\left(\tfrac{\omega}{2\pi T}\right)
 \ = \ \sum_{k=-\infty}^{\infty} X\left(\tfrac{\omega}{2\pi T} - k/T\right)\\
&= \sum_{k=-\infty}^{\infty} X\left(\tfrac{\omega - 2\pi k}{2\pi T} \right).
\end{align}
</math>

Note that when parameter {{mvar|T}} changes, the terms of <math>X_{2\pi}(\omega)</math> remain a constant separation <math>2 \pi</math> apart, and their width scales up or down. The terms of {{math|''X''<sub>1/''T''</sub>(''f'')}} remain a constant width and their separation {{math|1/''T''}} scales up or down.

== Table of discrete-time Fourier transforms ==
Some common transform pairs are shown in the table below.  The following notation applies:

*<math>\omega=2 \pi f T</math> is a real number representing continuous angular frequency (in radians per sample). (<math>f</math> is in cycles/sec, and <math>T</math> is in sec/sample.)  In all cases in the table, the DTFT is 2π-periodic (in <math>\omega</math>).
*<math>X_{2\pi}(\omega)</math> designates a function defined on <math>-\infty < \omega < \infty </math>.
*<math>X_o(\omega)</math> designates a function defined on <math>-\pi < \omega \le \pi</math>, and zero elsewhere.  Then:
::<math>X_{2\pi}(\omega)\ \triangleq \sum_{k=-\infty}^{\infty} X_o(\omega - 2\pi k).</math>

* <math>\delta ( \omega )</math> is the [[Dirac delta function]]
* <math>\operatorname{sinc} (t)</math>  is the normalized [[sinc function]]
* <math>\operatorname{rect}(t)</math> is the [[rectangle function]]
* <math>\operatorname{tri} (t)</math> is the [[triangle function]]
* {{mvar|n}} is an integer representing the discrete-time domain (in samples)
* <math>u[n]</math> is the discrete-time [[Heaviside step function#Discrete form|unit step function]]
* <math>\delta[n]</math> is the [[Kronecker delta]] <math>\delta_{n,0}</math>

{| class="wikitable"
|-
! Time domain <br /> ''x''[''n'']
! Frequency domain <br />X<sub>2π</sub>(ω)
! Remarks
! Reference
|-
| <math>\delta[n]</math>
| <math>X_{2\pi}(\omega) = 1</math>
| 
| <ref name=ProakisManolakis/>{{rp|p. 305}}
|-
| <math>\delta[n-M]</math>
| <math>X_{2\pi}(\omega) = e^{-i\omega M}</math>
| integer <math>M</math>
| 
|-
|<math>\sum_{m = -\infty}^{\infty} \delta[n - M m] \!</math>
|<math>X_{2\pi}(\omega) = \sum_{m = -\infty}^{\infty} e^{-i \omega M m} = \frac{2\pi}{M}\sum_{k = -\infty}^{\infty} \delta \left( \omega - \frac{2\pi k}{M} \right) \,</math><br>
<math>X_o(\omega) = \frac{2\pi}{M}\sum_{k = -(M-1)/2}^{(M-1)/2} \delta \left(\omega - \frac{2\pi k}{M} \right) \,</math> &nbsp; &nbsp; odd ''M''<br>
<math>X_o(\omega) = \frac{2\pi}{M}\sum_{k = -M/2+1}^{M/2} \delta \left(\omega - \frac{2\pi k}{M} \right) \,</math> &nbsp; &nbsp; even ''M''<br>
| integer <math>M > 0 </math>
| 
|-
| <math>u[n]</math>
| <math>X_{2\pi}(\omega) = \frac{1}{1-e^{-i \omega}} + \pi \sum_{k=-\infty}^{\infty} \delta (\omega - 2\pi k)\!</math><br>
<math>X_o(\omega) = \frac{1}{1-e^{-i \omega}} + \pi \cdot \delta (\omega)\!</math>
|The <math>1/(1-e^{-i \omega})</math> term must be interpreted as a [[distribution (mathematics)|distribution]] in the sense of a [[Cauchy principal value]] around its [[pole (complex analysis)|poles]] at <math>\omega=2 \pi k</math>.
| 
|-
| <math>a^n u[n]</math>
| <math>X_{2\pi}(\omega) = \frac{1}{1-a e^{-i \omega}}\!</math>
| <math>0 < |a| < 1 </math>
| <ref name=ProakisManolakis/>{{rp|p. 305}}
|-
| <math>e^{-i a n}</math>
| <math>X_o(\omega) = 2\pi\cdot \delta (\omega +a),</math> &nbsp; &nbsp; -π < a < π<br>
<math>X_{2\pi}(\omega) = 2\pi \sum_{k=-\infty}^{\infty} \delta (\omega +a -2\pi k)</math>
| real number <math>a</math>
| 
|-
| <math>\cos(a\cdot n)</math>
| <math>X_o(\omega) = \pi \left[\delta \left(\omega -a\right)+\delta \left(\omega +a\right)\right],</math><br>
<math>X_{2\pi}(\omega)\ \triangleq \sum_{k=-\infty}^{\infty} X_o(\omega - 2\pi k)</math>
| real number <math>a</math> with <math>-\pi < a < \pi</math>
| 
|-
| <math>\sin(a\cdot n)</math>
| <math>X_o(\omega) = \frac{\pi}{i} \left[\delta \left(\omega -a\right)-\delta \left(\omega +a\right)\right]</math>
| real number <math>a</math> with <math>-\pi < a < \pi</math>
| 
|-
| <math>\operatorname{rect} \left[ { n - M \over N  } \right] </math>
| <math>X_o(\omega) = { \sin[ N \omega / 2 ] \over \sin( \omega / 2 ) } \,  e^{ -i \omega M } \!</math>
| integers <math>M</math> and  <math>N</math>
| 
|-
| <math>\operatorname{sinc} ( W (n+a))</math>
| <math>X_o(\omega) =  \frac{1}{W} \operatorname{rect} \left( { \omega \over 2\pi W } \right) e^{ia\omega}</math>
| real numbers <math>W,a</math> with <math>0 < W < 1</math>
| 
|-
| <math>\operatorname{sinc}^2(W n)\,</math>
| <math>X_o(\omega) = \frac{1}{W} \operatorname{tri} \left( { \omega \over 2\pi W } \right)</math>
| real number <math>W</math>, <math>0 < W < 0.5</math>
| 
|-
| <math> \begin{cases}
0 & n=0 \\
\frac{(-1)^n}{n} & \mbox{elsewhere}
\end{cases}</math>
| <math>X_o(\omega) = j \omega</math>
|it works as a [[differentiator]] filter
|-
| <math>\frac{1}{(n + a)} \left\{ \cos [ \pi W (n+a)] - \operatorname{sinc} [ W (n+a)] \right\}</math>
| <math>X_o(\omega) = \frac{j \omega}{W} \cdot \operatorname{rect} \left( { \omega \over \pi W } \right) e^{j a \omega}</math>
| real numbers <math>W,a</math> with <math>0 < W < 1 </math>
| 
|-
| <math>\begin{cases}
\frac{\pi}{2}  & n = 0 \\
\frac{(-1)^n - 1}{\pi n^2} & \mbox{ otherwise}
\end{cases}</math>
| <math>X_o(\omega) = |\omega|</math>
| 
|
|-
| <math>\begin{cases}
0; & n \text{ even} \\
\frac{2}{\pi n} ; & n \text{ odd}
\end{cases}</math>
| <math>X_o(\omega) = \begin{cases}
j & \omega < 0 \\
0 & \omega = 0 \\
-j & \omega > 0
\end{cases}</math>
|[[Hilbert transform]]
| 
|-
| <math>\frac{C (A + B)}{2 \pi} \cdot \operatorname{sinc} \left[ \frac{A - B}{2\pi} n \right] \cdot \operatorname{sinc} \left[ \frac{A + B}{2\pi} n \right]</math>
| <math>X_o(\omega) = </math>[[Image:Trapezoid signal.svg|250px]]
| real numbers <math>A,B</math> <br /> complex <math>C</math>
| 
|}

==Properties==
This table shows some mathematical operations in the time domain and the corresponding effects in the frequency domain.
* <math>*\!</math> is the [[Convolution#Discrete convolution|discrete convolution]] of two sequences
* <math>x[n]^{*}</math> is the [[complex conjugate]] of {{math|''x''[''n'']}}.

{| class="wikitable"
|-
! Property
! Time domain<br/>{{math|''x''[''n'']}}
! Frequency domain<br/><math>X_{2\pi}(\omega)</math>
! Remarks
! Reference
|-
| Linearity
| <math>a\cdot x[n] + b\cdot y[n]</math>
| <math>a\cdot X_{2\pi}(\omega) + b\cdot Y_{2\pi}(\omega)</math>
| complex numbers <math>a,b</math>
| <ref name=ProakisManolakis>{{Citation | last =Proakis | first =John G. | last2 =Manolakis | first2 =Dimitri G. | title =Digital Signal Processing: Principles, Algorithms and Applications | place =New Jersey | publisher =Prentice-Hall International | year =1996 | edition =3 | language =English | id =sAcfAQAAIAAJ | isbn =9780133942897| bibcode =1996dspp.book.....P }}</ref>{{rp|p. 294}}
|-
| Time reversal / Frequency reversal
| <math>x[-n]</math>
| <math>X_{2\pi}(-\omega) \!</math>
|
| <ref name=ProakisManolakis/>{{rp|p. 297}}
|-
| Time conjugation
| <math>x[n]^*</math>
| <math>X_{2\pi}(-\omega)^* \!</math>
|
| <ref name=ProakisManolakis/>{{rp|p. 291}}
|-
| Time reversal & conjugation
| <math>x[-n]^*</math>
| <math>X_{2\pi}(\omega)^* \!</math>
|
| <ref name=ProakisManolakis/>{{rp|p. 291}}
|-
| Real part in time
| <math>\Re{(x[n])}</math>
| <math>\frac{1}{2}(X_{2\pi}(\omega) + X_{2\pi}^*(-\omega))</math>
|
| <ref name=ProakisManolakis/>{{rp|p. 291}}
|-
| Imaginary part in time
| <math>\Im{(x[n])}</math>
| <math>\frac{1}{2i}(X_{2\pi}(\omega) - X_{2\pi}^*(-\omega))</math>
|
| <ref name=ProakisManolakis/>{{rp|p. 291}}
|-
| Real part in frequency
| <math>\frac{1}{2}(x[n]+x^*[-n])</math>
| <math>\Re{(X_{2\pi}(\omega))}</math>
|
| <ref name=ProakisManolakis/>{{rp|p. 291}}
|-
| Imaginary part in frequency
| <math>\frac{1}{2i}(x[n]-x^*[-n])</math>
| <math>\Im{(X_{2\pi}(\omega))}</math>
|
| <ref name=ProakisManolakis/>{{rp|p. 291}}
|-
| Shift in time / Modulation in frequency
| <math>x[n-k]</math>
| <math>X_{2\pi}(\omega)\cdot e^{-i\omega k}</math>
| integer {{mvar|k}}
| <ref name=ProakisManolakis/>{{rp|p. 296}}
|-
| Shift in frequency / Modulation in time
| <math>x[n]\cdot e^{ian} \!</math>
| <math>X_{2\pi}(\omega-a) \!</math>
| real number <math>a</math>
| <ref name=ProakisManolakis/>{{rp|p. 300}}
|-
| Decimation
| <math>x[nM]</math>
| <math>\frac{1}{M}\sum_{m=0}^{M-1} X_{2\pi}\left(\tfrac{\omega - 2\pi m}{M}\right) \!</math>&nbsp; {{
efn-ua|This expression is derived as follows:
:<math>
\begin{align}
\sum_{n=-\infty}^{\infty} x(nMT)\ e^{-i\omega n}
&= \frac{1}{MT}\sum_{k=-\infty}^{\infty} X\left(\tfrac{\omega}{2\pi MT} - \tfrac{k}{MT}\right)\\
&= \frac{1}{MT}\sum_{m=0}^{M-1} \quad \sum_{n=-\infty}^{\infty} X\left(\tfrac{\omega}{2\pi MT} - \tfrac{m}{MT} - \tfrac{n}{T}\right),
\quad \text{where} \quad k \rightarrow m + nM\\
&=\frac{1}{M}\sum_{m=0}^{M-1} \quad \frac{1}{T}\sum_{n=-\infty}^{\infty}X\left(\tfrac{(\omega - 2\pi m)/M}{2\pi T} - \tfrac{n}{T}\right)\\
&= \frac{1}{M}\sum_{m=0}^{M-1} \quad X_{2\pi}\left(\tfrac{\omega - 2\pi m}{M}\right)
\end{align}
</math>
}}
| integer <math>M</math>
|
|-
| Time Expansion
| <math> \scriptstyle \begin{cases}
x[n/M] & n=\text{multiple of M} \\
0 & \text{otherwise}
\end{cases}</math>
| <math>X_{2\pi}(M \omega) \!</math>
| integer <math>M</math>
|
|-
| Derivative in frequency
| <math>\frac{n}{i} x[n] \!</math>
| <math>\frac{d X_{2\pi}(\omega)}{d \omega} \!</math>
|
| <ref name=ProakisManolakis/>{{rp|p. 303}}
|-
| Integration in frequency
| <math> \!</math>
| <math> \!</math> 
|
|
|-
| Differencing in time
| <math> x[n]-x[n-1] \!</math> 
| <math> \left( 1-e^{-i \omega} \right) X_{2\pi}(\omega)  \!</math>
|
|
|-
| Summation in time
| <math> \sum_{m=-\infty}^{n} x[m] \!</math> 
| <math> \frac{1}{\left( 1-e^{-i \omega} \right)} X_{2\pi}(\omega) + \pi X(0) \sum_{k=-\infty}^{\infty} \delta(\omega-2\pi k) \!</math>
|
|
|-
| Convolution in time / Multiplication in frequency
| <math>x[n] * y[n] \!</math>
| <math>X_{2\pi}(\omega) \cdot Y_{2\pi}(\omega) \!</math>
|
| <ref name=ProakisManolakis/>{{rp|p. 297}}
|-
| Multiplication in time / Convolution in frequency
| <math>x[n] \cdot y[n] \!</math>
| <math>\frac{1}{2\pi}\int_{-\pi}^{\pi}X_{2\pi}(\nu) \cdot Y_{2\pi}(\omega-\nu) d\nu \!</math>
| [[Periodic convolution]]
| <ref name=ProakisManolakis/>{{rp|p. 302}}
|-
| [[Cross correlation]]
| <math>\rho_{xy} [n] = x[-n]^* * y[n] \!</math>
| <math>R_{xy} (\omega) = X_{2\pi}(\omega)^* \cdot Y_{2\pi}(\omega) \!</math> 
|
|
|-
| [[Parseval's theorem]]  
| <math>E_{xy} = \sum_{n=-\infty}^{\infty} {x[n] \cdot y[n]^*} \!</math>
| <math>E_{xy} = \frac{1}{2\pi}\int_{-\pi}^{\pi}{X_{2\pi}(\omega) \cdot Y_{2\pi}(\omega)^* d\omega} \!</math>
|
| <ref name=ProakisManolakis/>{{rp|p. 302}}
|}

==See also==
* [[Multidimensional transform]]
* [[Zak transform]]

== Notes ==
{{notelist-ua}}

== References ==
{{reflist|refs=

<ref name="Rao">{{cite book |title=Signals and Systems |author=Rao, R. |isbn=9788120338593 |url=https://books.google.com/books?id=4z3BrI717sMC |publisher=Prentice-Hall Of India Pvt. Limited|year=2008 }}</ref>

<ref name="Matlab">{{cite web|url=https://www.mathworks.com/help/signal/ref/periodogram.html|title=Periodogram power spectral density estimate - MATLAB periodogram}}</ref>

<ref name="Gumas">{{cite journal|last1=Gumas |first1=Charles Constantine |date=July 1997 |title=Window-presum FFT achieves high-dynamic range, resolution |journal=Personal Engineering & Instrumentation News |pages=58–64 |url=http://www.chipcenter.com/dsp/DSP000315F1.html |url-status=bot: unknown |archiveurl=https://web.archive.org/web/20010210052902/http://www.chipcenter.com/dsp/DSP000315F1.html |archivedate=2001-02-10 }}</ref>

<ref name="Lillington">{{Cite web |last=Lillington|first=John |title=Comparison of Wideband Channelisation Architectures |publisher=RF Engines Ltd |date= |url=http://www.edn.com/Pdf/ViewPdf?contentItemId=4133641 | access-date =2016-10-30
}}</ref>

<ref name="Chennamangalam">{{Cite web |last=Chennamangalam|first=Jayanth|title=The Polyphase Filter Bank Technique |publisher=CASPER Group |date=2016-10-18 |url=https://casper.berkeley.edu/wiki/The_Polyphase_Filter_Bank_Technique | access-date =2016-10-30
}}</ref>

<ref name="Lyons">{{Cite web |last=Lyons |first=Richard G. |title=DSP Tricks: Building a practical spectrum analyzer |publisher=EE Times |date=June 2008 |url=http://www.embedded.com/design/real-time-and-performance/4007611/DSP-Tricks-Building-a-practical-spectrum-analyzer}} &nbsp; Note however, that it contains a link labeled ''weighted overlap-add structure'' which incorrectly goes to [[Overlap-add method]].</ref>

<ref name="Dahl">{{cite thesis |last=Dahl |first=Jason F. |date=2003-02-06 |title=Time Aliasing Methods of Spectrum Estimation |type=Ph.D. |publisher=Brigham Young University |url=http://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=1049&context=etd |access-date=2016-10-31}}</ref>

<ref name="Wang">{{cite journal | url =https://ieeexplore.ieee.org/document/4148464 | title =Channelized Receiver with WOLA Filterbank | first1 =Hong | last1 =Wang | first2 =Youxin | last2 =Lu | first3 =Xuegang | last3 =Wang | journal = 2006 CIE International Conference on Radar | pages =1–3 | date = 16 October 2006 | doi = 10.1109/ICR.2006.343463 | publisher = IEEE | location = Shanghai, China | isbn =0-7803-9582-4}}</ref>
}}
{{refbegin|}}

== Further reading==
* {{cite book|title =Multirate Digital Signal Processing|last1 =Crochiere|first1 =R.E.|last2 =Rabiner|first2 =L.R.|date =1983|publisher =Prentice Hall|isbn =0-13-605162-6|location = Englewood Cliffs, NJ|pages =313–326}}
* {{cite book|title=Discrete-Time Signal Processing|last=Oppenheim|first=Alan V.|last2=Schafer|first2=Ronald W.|publisher=Prentice Hall Signal Processing Series|year=1999|isbn=0-13-754920-2|edition=2nd|location=|pages=|url-access=registration|url=https://archive.org/details/discretetimesign00alan}}
* {{cite book|title=A Course in Digital Signal Processing|last=Porat|first=Boaz|date=1996|publisher=John Wiley and Sons|isbn=0-471-14961-6|location=|pages=27–29 and 104–105}}
* {{cite book|title=Circuits, Signals, and Systems|last=Siebert|first=William M.|publisher=MIT Press|year=1986|isbn=0262690950|location=MIT Electrical Engineering and Computer Science Series. Cambridge, MA|pages=}}
* {{cite book|title=Understanding Digital Signal Processing|last=Lyons|first=Richard G.|publisher=Prentice Hall|year=2010|isbn=978-0137027415|edition=3rd|location=|pages=}}

{{refend}}

{{DSP}}

[[Category:Transforms]]
[[Category:Fourier analysis]]
[[Category:Digital signal processing]]