Discrete-time Fourier transform

For a sequence of complex numbers: $x[n],\;n\in \mathbb {Z}$ (integers), the discrete-time Fourier transform (or DTFT) is the function:

$X(e^{i\omega })=\sum _{n=-\infty }^{\infty }x[n]\,e^{-i\omega n}\,$

Relation to continuous Fourier transform

Consider a continuous-time function, $x(t)\,$, and the Fourier transform:

$X(f)=\int _{-\infty }^{\infty }x(t)\cdot e^{-i2\pi ft}\,dt$

A common operation is to digitize $x(t)\,$ [aka sampling], which alters the Fourier transform. The altered transform is the DTFT. A mathematical model of the sampling process is to multiply $x(t)\,$ by the Dirac comb function, which is an infinite sequence of impulses occurring at integer multiples of the sampling interval $T\,$ (seconds):

$x_{\mathrm {T} }(t)=x(t)\cdot \sum _{n=-\infty }^{\infty }\delta (t-nT)=\sum _{n=-\infty }^{\infty }x(nT)\,\delta (t-nT)$

The Fourier transform of this product is:

$X_{\mathrm {T} }(f)\,$	$=\int _{-\infty }^{\infty }\left[\sum _{n=-\infty }^{\infty }x(nT)\,\delta (t-nT)\right]e^{-i2\pi ft}\,dt$
	$=\sum _{n=-\infty }^{\infty }x(nT)\int _{-\infty }^{\infty }\left[\delta (t-nT)\cdot e^{-i2\pi ft}\right]\,dt$
	$=\sum _{n=-\infty }^{\infty }x(nT)\cdot e^{-i2\pi fnT}\,$

With the associations:

$x[n]=x(nT)\,$

$\omega =2\pi fT\,$

this becomes identical to the expression for $X(e^{i\omega })\,$.
Since $f\,$ represents ordinary frequency (cycles per second) and $T\,$ has units of seconds per sample, the units of $fT\,$ are cycles per sample. It is common notational practice to replace this product with a single variable, called normalized frequency. Since $1/T\,$ represents the sample rate (samples per second), $fT\,$ represents actual frequencies as multiples (usually fractional) of the sample rate.   $\omega \,$, as defined above, is also a normalized frequency, but its units are radians per sample.

$X_{\mathrm {T} }(f)\,$ can also be expressed in terms of $X(f)\,$, by means of the Fourier transform pair:

$\sum _{n=-\infty }^{\infty }\delta (t-nT)\quad \Longleftrightarrow \quad {1 \over T}\sum _{k=-\infty }^{\infty }\delta \left(f-{k \over T}\right)$

So the transform of the product is this convolution:

$X_{\mathrm {T} }(f)\quad =X(f)*{1 \over T}\sum _{k=-\infty }^{\infty }\delta (f-{k \over T})\quad ={1 \over T}\sum _{k=-\infty }^{\infty }X(f-{k \over T})$

which uses the shifting property of the Dirac delta under convolution. This result reveals that the DTFT can be constructed by summing "copies" of $X(f)\,$ superimposed at intervals of $1/T\,$ (the sample-rate). Therefore the higher the rate, the farther apart are the copies. (Also see aliasing and Nyquist-Shannon sampling theorem.)

Periodicity

Sampling $x(t)\,$ causes its spectrum (DTFT) to become periodic. In terms of ordinary frequency, $f\,$ (cycles per second), the period is the sample rate,  $1/T$.   In terms of normalized frequency, $fT\,$ (cycles per sample), the period is $1$.  And in terms of $\omega \,$ (radians per sample), the period is $2\pi$, which also follows directly from the periodicity of $e^{-i\omega n}\,$. I.e.:

$e^{-i(\omega +2\pi k)n}=e^{-i\omega n}\,$

where both n and k are arbitrary integers. Therefore:

$X(e^{i(\omega +2\pi k)})=X(e^{i\omega })\,$

The notation $X(e^{i\omega })\,$:

1. highlights the periodicity property, and
2. helps distinguish between the DTFT and underlying Fourier transform of $x(t)\,$, i.e. $X(f)\,$ (or $X(\omega )\,$), and
3. emphasizes the relationship of the DTFT to the Z-transform.

However, its relevance is obscured when the DTFT is formed by the frequency domain method (superposition), as discussed above. So the notation $X(\omega )\,$ is also commonly used. And the periodicity property is:

$X(\omega +2\pi k)=X(\omega )\,$

Inverse transform

It is easily shown that the following inverse transform recovers the discrete-time sequence:

$x[n]={\frac {1}{2\pi }}\int _{-\pi }^{\pi }X(e^{i\omega })\,e^{i\omega n}\,d\omega$

Notes:

• With minor differences, this is a Fourier series decomposition.
• With infinite limits of integration (i.e., continuous-time inverse Fourier transform), this would recover the Dirac impulses as well.

Finite length sequences

For practical evaluation of the DTFT numerically, a finite-length sequence is obviously needed. For instance, a long sequence might be modified by a rectangular window function, resulting in:

$X(e^{i\omega })=\sum _{n=0}^{L-1}x[n]\,e^{-i\omega n}\,$,   where $L\,$ is the modified sequence length.

This is often a useful approximation of the spectrum of the unmodified sequence. The difference is a loss of clarity (resolution), which improves as L increases.
It is common to evaluate $X(\omega )$ at an arbitrary number $(N)$ of uniformly-spaced frequencies across one period (2π):

$\omega _{k}={\frac {2\pi }{N}}k\,$,     for $k=0,1,\dots ,N-1\,$

which gives:

$X[k]=X(e^{i\omega _{k}})=\sum _{n=0}^{L-1}x[n]\,e^{-i2\pi {\frac {k}{N}}n}$

When $N\geq L\,$, this can also be written:

$X[k]=\sum _{n=0}^{N-1}x[n]\,e^{-i2\pi {\frac {k}{N}}n}$,   because we define $x[n]=0\,$ for $n\geq L\,$.

With that cosmetic adjustment, the $X[k]\,$ sequence is now recognizable as a discrete Fourier transform (DFT). While $N$ defines the resolution at which we sample the DTFT, $L$ limits the inherent resolution of the DTFT itself. So they are usually similar (or equal) values. And while it is common to choose $N>L$, the only reason to include the zero-valued terms in the summation is to take advantage of a fast Fourier transform algorithm for computing the DFT. However, when that is done it is often given undue significance, such as zero-padded DFT and/or interpolated DFT. But obviously the exact same DFT can be calculated straightforwardly without the zero-valued terms. One can also compute the DTFT for the case of $N<L$ (or for other frequency samplings) where it is not equivalent to a DFT.

To illustrate why $N>L$ is common, consider the sequence:

$x[n]=e^{i2\pi {\frac {1}{8}}n}$, and $L=64$.

The two figures below 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: $f={\begin{matrix}{\frac {1}{8}}\end{matrix}}=0.125\,$. Also visible on the right is the spectral leakage pattern of the $L=64$ rectangular window. The illusion on the left is a result of sampling the DTFT at all of 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 (${\begin{matrix}{\frac {1}{8}}\end{matrix}}={\begin{matrix}{\frac {8}{64}}\end{matrix}}$) with exactly 8 (an integer) cycles per 64 samples.

DFT for L = 64 and N = 64

DFT for L = 64 and N = 256

Difference between the DTFT and other Fourier transforms

Essentially, the DTFT is the reverse of the Fourier series, in that the latter has a continuous, periodic input and a discrete spectrum. The applications of the two transforms, however, are quite different.

The DFT and the DTFT can be viewed as the logical result of applying the standard continuous Fourier transform to discrete data. From that perspective, we have the satisfying result that it's not the transform that varies, it's just the form of the input:

• If it is discrete, the Fourier transform becomes a DTFT.
• If it is periodic, the Fourier transform becomes a Fourier series.
• If it is both, the Fourier transform becomes a DFT.

Relationship to the Z-transform

The DTFT is a special case of the Z-transform. The bilateral Z-transform is defined as:

$X(z)=\sum _{n=-\infty }^{\infty }x[n]\,z^{-n}$

So the special case is: $z=e^{i\omega }\,$.   Since $|e^{i\omega }|=1\,$, it is the evaluation of the Z-transform around the unit circle in the complex plane.

Table of discrete-time Fourier transforms

Some common transform pairs are shown below. The following notation applies:

• $n\!$ is an integer representing the discrete-time domain (in samples)
• $\omega \!$ is a real number in $(-\pi ,\ \pi )$, representing continuous angular frequency (in radians per sample).
  • The remainder of the transform $(|\omega |>\pi \,)$ is defined by: $X(\omega +2\pi k)=X(\omega )\,$
• $u[n]\!$ is the discrete-time unit step function
• $\operatorname {sinc} (t)\!$ is the sinc function for real-valued argument t
• $\delta (\omega )\!$ is the Dirac delta function
• $\delta [n]\!$ is the Kronecker delta $\delta _{n,0}\!$
• $\operatorname {rect} (t)$ is the rectangle function for arbitrary real-valued t:

$\mathrm {rect} (t)=\sqcap (t)={\begin{cases}0&{\mbox{if }}|t|>{\frac {1}{2}}\\[3pt]{\frac {1}{2}}&{\mbox{if }}|t|={\frac {1}{2}}\\[3pt]1&{\mbox{if }}|t|<{\frac {1}{2}}\end{cases}}$

• $\operatorname {tri} (t)$ is the triangle function for arbitrary real-valued t:

$\operatorname {tri} (t)=\land (t)={\begin{cases}1+t;&-1\leq t\leq 0\\1-t;&0<t\leq 1\\0&{\mbox{otherwise}}\end{cases}}$

Time domain $x[n]\,$	Frequency domain $X(\omega )\,$	Remarks
$\delta [n]\!$	$1\!$
$\delta [n-k]\!$	$e^{-ik\omega }\!$	integer k
$u[n]\!$	${\frac {1}{1-e^{-i\omega }}}\!$
$e^{-an}\!$	$\delta (\omega +a)\,$	real number a
$\cos(an)\!$	${\frac {1}{2}}\left[\delta (\omega -a)+\delta (\omega +a)\right]$	real number a
$\sin(an)\!$	${\frac {1}{2i}}\left[\delta (\omega -a)-\delta (\omega +a)\right]$	real number a
$\mathrm {rect} \left[{(n-M/2) \over M}\right]$	${\sin[\omega (M+1)/2] \over \sin(\omega /2)}\,e^{-i\omega M/2}$	integer M
$\operatorname {sinc} [\pi (a+n)]$	$e^{ia\omega }\!$	real number a
${\frac {W^{2}}{2\pi }}\operatorname {sinc} ^{2}\left[{\frac {W}{2}}n\right]$	$\operatorname {tri} \left({\omega \over W}\right)$	real number W
${\frac {W}{\pi }}\operatorname {sinc} [W(n+a)]$	$\operatorname {rect} \left({\omega \over 2W}\right)\cdot e^{ja\omega }$	real numbers W, a $0<W\leq \pi$
${\frac {W}{\pi (n+a)}}\left\{\cos[W(n+a)]-\operatorname {sinc} [W(n+a)]\right\}$	$j\omega \cdot \operatorname {rect} \left({\omega \over W}\right)e^{ja\omega }$	real numbers W, a
${\frac {1}{\pi n^{2}}}[(-1)^{n}-1]$	$|\omega |\!$
${\frac {C(A+B)}{2\pi }}\cdot \operatorname {sinc} \left[{\frac {A-B}{2}}n\right]\cdot \operatorname {sinc} \left[{\frac {A+B}{2}}n\right]$		real numbers A, B complex C

Properties

This table shows the relationships between generic discrete-time Fourier transforms. We use the following notation:

• $*\!$ is the convolution between two signals
• $x[n]^{*}\!$ is the complex conjugate of the function x[n]
• $\rho _{xy}[n]\!$ represents the correlation between x[n] and y[n].

The first column provides a description of the property, the second column shows the function in the time domain, the third column shows the spectrum in the frequency domain:

Property	Time domain $x[n]\!$	Frequency domain $X(\omega )\!$	Remarks
Linearity	$ax[n]+by[n]\!$	$aX(\omega )+bY(\omega )\!$
Shift in time	$x[n-k]\!$	$X(e^{i\omega })e^{-i\omega k}\!$	integer k
Shift in frequency	$x[n]e^{ian}\!$	$X(e^{i(\omega -a)})\!$	real number a
Time reversal	$x[-n]\!$	$X(e^{-i\omega })\!$
Time conjugation	$x[n]^{*}\!$	$X(e^{-i\omega })^{*}\!$
Time reversal & conjugation	$x[-n]^{*}\!$	$X(e^{i\omega })^{*}\!$
Derivative in frequency	${\frac {n}{i}}x[n]\!$	${\frac {dX(e^{i\omega })}{d\omega }}\!$
Integral in frequency	${\frac {i}{n}}x[n]\!$	$\int _{-\pi }^{\omega }X(e^{i\vartheta })d\vartheta \!$
Convolve in time	$x[n]*y[n]\!$	$X(e^{i\omega })\cdot Y(e^{i\omega })\!$
Multiply in time	$x[n]\cdot y[n]\!$	${\frac {1}{2\pi }}X(e^{i\omega })*Y(e^{i\omega })\!$
Correlation	$\rho _{xy}[n]=x[-n]^{*}*y[n]\!$	$R_{xy}(\omega )=X(e^{i\omega })^{*}\cdot Y(e^{i\omega })\!$

References

• Alan V. Oppenheim and Ronald W. Schafer (1999). Discrete-Time Signal Processing (2nd Edition ed.). Prentice Hall Signal Processing Series. ISBN 0-13-754920-2. {{cite book}}: |edition= has extra text (help)

• William McC. Siebert (1986). Circuits, Signals, and Systems. MIT Electrical Engineering and Computer Science Series. Cambridge, MA: MIT Press.

• Boaz Porat. A Course in Digital Signal Processing. John Wiley and Sons. pp. pp. 27-29 and 104-105. ISBN 0-471-14961-6. {{cite book}}: |pages= has extra text (help)