Discrete-time Fourier transform

In mathematics, the discrete-time Fourier transform (DTFT) is one of the specific forms of Fourier analysis. As such, it transforms one function into another, which is called the frequency domain representation, or simply the "DTFT", of the original function (which is often a function in the time-domain). The DTFT requires an input function that is discrete. Such inputs are often created by digitally sampling a continuous function, like a person's voice.

The DTFT frequency-domain representation is always a periodic function. Since one period of the function contains all of the unique information, it is sometimes convenient to say that the DTFT is a transform to a "finite" frequency-domain (the length of one period), rather than to the entire real line.

Definition

Given a discrete set of real or complex numbers: $x[n],\;n\in \mathbb {Z}$ (integers), the discrete-time Fourier transform (or DTFT) of $x[n]\,$ is usually written:

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

(Eq.1)

Relationship to sampling

Often the x[n] sequence represents the values (aka samples) of a continuous-time function, x(t), at discrete moments in time: t = nT, where T is the sampling interval (in seconds), and  $f_{s}=1/T\,$  is the sampling rate (samples per second).  Then the DTFT provides an approximation of the continuous-time Fourier transform:

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

To understand this, consider the Poisson summation formula, which indicates that a periodic summation of function X(f) can be constructed from the samples of function x(t).  The result is:

$X_{1/T}(f)\ {\stackrel {\mathrm {def} }{=}}\sum _{k=-\infty }^{\infty }X\left(f-k/T\right)\equiv \sum _{n=-\infty }^{\infty }\underbrace {T\cdot x(nT)} _{x[n]}\ e^{-i2\pi fTn}\ =\ {\mathcal {F}}\left\{\sum _{n=-\infty }^{\infty }x[n]\cdot \delta (t-nT)\right\}$

(Eq.2)

$\scriptstyle X_{1/T}(f)\,$ comprises exact copies of $\scriptstyle X(f)\,$ that are shifted by multiples of ƒ_s and combined by addition.  For sufficiently large ƒ_s,  the k=0 term can be observed in the region [−ƒ_s/2, ƒ_s/2] with little or no distortion (aliasing) from the other terms.

With the following definition of a normalized frequency, the expressions in Eq.2 and  Eq.1 are identical:  $\omega =2\pi \textstyle {\frac {f}{f_{s}}}=2\pi fT.$

Since $f\,$ represents ordinary frequency (cycles per second) and $f_{s}\,$ has units of samples per second, the units of $f/f_{s}\,$ are cycles per sample. It is common notational practice to replace this ratio with a single variable, which represents actual frequencies as multiples (usually fractional) of the sample rate. In terms of that normalization, the periodicty of the DTFT is 1.   ω is also a normalized frequency, with units of radians per sample.   And in terms of ω, the periodicity is 2π.

Periodic data

When the input data sequence x[n] is N-periodic, Eq.2 can be computationally reduced to a discrete Fourier transform (DFT) by expanding the periodic comb function into a Fourier series:

$\sum _{n=-\infty }^{\infty }x[n]\cdot \delta (t-nT)=\underbrace {\sum _{k=-\infty }^{\infty }X[k]\cdot e^{i2\pi {\frac {k}{NT}}t}} _{Fourier\ series}\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad \underbrace {\sum _{k=-\infty }^{\infty }X[k]\ \cdot \ \delta \left(f-{\frac {k}{NT}}\right)} _{DTFT\ of\ a\ periodic\ sequence},$

which also shows that periodicity in the time domain causes the DTFT to become discontinuous, and in fact it diverges (to a comb function) at the harmonic frequencies. Nevertheless, we can evaluate the coefficients that modulate the comb, and the integral formula conveniently reduces to an N-term summation:

${\begin{aligned}X[k]\ &{\stackrel {\text{def}}{=}}\ {\frac {1}{NT}}\int _{NT}\left[\sum _{n=-\infty }^{\infty }x[n]\cdot \delta (t-nT)\right]e^{-i2\pi {\frac {k}{NT}}t}dt\quad \scriptstyle {(integral\ over\ any\ interval\ of\ length\ NT)}\displaystyle \\&={\frac {1}{NT}}\sum _{n=-\infty }^{\infty }x[n]\cdot \int _{NT}\delta (t-nT)\cdot e^{-i2\pi {\frac {k}{NT}}t}dt\\&={\frac {1}{NT}}\underbrace {\sum _{N}x[n]\cdot e^{-i2\pi {\frac {k}{N}}n}} _{DFT}={\frac {1}{N}}\underbrace {\sum _{N}x(nT)\cdot e^{-i2\pi {\frac {k}{N}}n}} _{DFT},\quad \scriptstyle {(sum\ over\ any\ n-sequence\ of\ length\ N)}\end{aligned}}$

which is also an N-periodic sequence (in k); i.e. there is no need to evaluate more than N coefficients.

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 Eq.2 produces the sequence in the form of a modulated Dirac comb function:

$\sum _{n=-\infty }^{\infty }x[n]\cdot \delta (t-nT)={\mathcal {F}}^{-1}\left\{X_{1/T}(f)\right\}\ {\stackrel {\mathrm {def} }{=}}\int _{-\infty }^{\infty }X_{1/T}(f)\cdot e^{i2\pi ft}df.$

However, noting that $X_{1/T}(f)$ is periodic, all the necessary information is contained within any interval of length 1/T.  In both Eq.1 and Eq.2, the summations over n are a Fourier series, with coefficients x[n].  The standard formulas for the Fourier coefficients are also the inverse transforms:

${\begin{aligned}x[n]&=T\int _{\frac {1}{T}}X_{1/T}(f)\cdot e^{i2\pi fnT}df\quad \scriptstyle {(integral\ over\ any\ interval\ of\ length\ 1/T)}\\\displaystyle &={\frac {1}{2\pi }}\int _{2\pi }X(\omega )\cdot e^{i\omega n}d\omega .\quad \scriptstyle {(integral\ over\ any\ interval\ of\ length\ 2\pi )}\end{aligned}}$

Sampling the DTFT

When the DTFT is continuous, a common practice is to compute an arbitrary number of samples (N) of one cycle of the periodic function X_1/T :

${\begin{aligned}\underbrace {X_{1/T}\left({\frac {k}{NT}}\right)} _{X_{k}}&=\sum _{n=-\infty }^{\infty }x[n]\cdot e^{-i2\pi {\frac {kn}{N}}}\quad \quad k=0,\dots ,N-1\\&=\underbrace {\sum _{N}x_{N}[n]\cdot e^{-i2\pi {\frac {kn}{N}}},} _{DFT}\quad \scriptstyle {(sum\ over\ any\ n-sequence\ of\ length\ N)}\end{aligned}}$

where x_N is a periodic summation:

$x_{N}[n]\ {\stackrel {\text{def}}{=}}\ \sum _{m=-\infty }^{\infty }x[n-mN].$

The $x_{N}\,$ sequence is the inverse DFT. Thus, our sampling of the DTFT causes the inverse transform to become periodic.

In order to evaluate one cycle of $x_{N}\,$ numerically, we require a finite-length x[n] sequence. For instance, a long sequence might be truncated by a window function of length $L,\,$ resulting in two cases worthy of special mention: L ≤ N and L = I•N, for some integer I (typically 6 or 8). For notational simplicity, consider the x[n] values below to represent the modified values.

When $L=I\cdot N,\,$  a cycle of $x_{N}\,$  reduces to a summation of I blocks of length N. This goes by various names, such as multi-block windowing and window presum-DFT .  A good way to understand/motivate the technique is to recall that decimation of sampled data in one domain (time or frequency) produces aliasing in the other, and vice versa. The x_N summation is mathematically equivalent to aliasing, leading to decimation in frequency, leaving only DTFT samples least affected by spectral leakage. That is usually a priority when implementing an FFT filter-bank (channelizer). With a conventional window function of length L, scalloping loss would be unacceptable. So multi-block windows are created using FIR filter design tools. 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 $I,\,$ the better the potential performance. We note that the same results can be obtained by computing and decimating an L-length DFT, but that is not computationally efficient.

When $L\leq N,\,$  the DFT is usually written in this more familiar form:

$X_{k}=\sum _{n=0}^{N-1}x[n]\cdot e^{-i2\pi {\frac {kn}{N}}}.$

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

Spectral leakage, which increases as 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 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:

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

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

Convolution

The Convolution theorem for sequences is:

$x*y\ =\ \scriptstyle {DTFT}^{-1}\displaystyle {\big [}\ \scriptstyle {DTFT}\displaystyle \{x\}\cdot \ \scriptstyle {DTFT}\displaystyle \{y\}\ {\big ]}.$

An important special case is the circular convolution of sequences  $x\,$ and $y\,$ defined by  $x_{N}*y,\,$  where $x_{N}\,$  is a periodic summation.  The discrete-frequency nature of $\scriptstyle {DTFT}\displaystyle \{x_{N}\}$ "selects" only discrete values from the continuous function $\scriptstyle {DTFT}\displaystyle \{y\},$ which results in considerable simplification of the inverse transform.  As shown at Convolution_theorem#Functions_of_a_discrete_variable..._sequences:

${\begin{aligned}x_{N}*y\ &=\ \scriptstyle {DTFT}^{-1}\displaystyle {\big [}\ \scriptstyle {DTFT}\displaystyle \{x_{N}\}\cdot \ \scriptstyle {DTFT}\displaystyle \{y\}\ {\big ]}\\&=\ \scriptstyle {DFT}^{-1}\displaystyle {\big [}\ \scriptstyle {DFT}\displaystyle \{x_{N}\}\cdot \ \scriptstyle {DFT}\displaystyle \{y_{N}\}\ {\big ]}.\end{aligned}}$

For $x\,$ and $y\,$ sequences whose non-zero duration is ≤ N, a final simplification is:

$x_{N}*y\ =\ \scriptstyle {DFT}^{-1}\displaystyle {\big [}\ \scriptstyle {DFT}\displaystyle \{x\}\cdot \ \scriptstyle {DFT}\displaystyle \{y\}\ {\big ]}.$

The significance of this result is expounded at Circular convolution and Fast convolution algorithms.

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.

Alternative notation

A popular notation for the DTFT is $X(e^{i\omega }),\,$ with several desireable features:

1. highlights the periodicity property, and
2. helps distinguish between the DTFT and the underlying Fourier transform of $\scriptstyle x(t)$; that is, $\scriptstyle X(f)$ (or $\scriptstyle X(\omega )$), 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 (periodic summation), line Eq.2. So the notation $X(\omega )\,$ is also commonly used, as in the table below.

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 normalized sinc function
• $\delta (\omega )\!$ 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
$\delta [n]\!$	$1\!$
$\delta [n-M]\!$	$e^{-i\omega M}\!$	integer M
$\sum _{m=-\infty }^{\infty }\delta [n-Mm]\!$	$\sum _{m=-\infty }^{\infty }e^{-i\omega Mm}={\frac {1}{M}}\sum _{k=-\infty }^{\infty }\delta \left({\frac {\omega }{2\pi }}-{\frac {k}{M}}\right)\,$	integer M
$u[n]\!$	${\frac {1}{1-e^{-i\omega }}}+\sum \limits _{k=-\infty }^{\infty }{\pi \delta (\omega -2\pi k)}\!$	The $1/(1-e^{-i\omega })$ term must be interpreted as a distribution in the sense of a Cauchy principal value around its poles at $\omega =2\pi k$.
$a^{n}u[n]\!$	${\frac {1}{1-ae^{-i\omega }}}\!$	$0<|a|<1$
$e^{-ian}\!$	$2\pi \delta (\omega +a)\,$	real number a
$\cos(an)\!$	$\pi \left[\delta (\omega -a)+\delta (\omega +a)\right]$	real number a
$\sin(an)\!$	${\frac {\pi }{i}}\left[\delta (\omega -a)-\delta (\omega +a)\right]$	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
$\operatorname {sinc} [(a+n)]$	$e^{ia\omega }\!$	real number a
$W\cdot \operatorname {sinc} ^{2}(Wn)\,$	$\operatorname {tri} \left({\omega \over 2\pi W}\right)$	real number W $0<W\leq 0.5$
$W\cdot \operatorname {sinc} (Wn)$	$\operatorname {rect} \left({\omega \over 2\pi W}\right)$	real numbers W $0<W\leq 1$
${\begin{cases}0&n=0\\{\frac {(-1)^{n}}{n}}&{\mbox{elsewhere}}\end{cases}}$	$j\omega$	it works as a differentiator filter
${\frac {W}{(n+a)}}\left\{\cos[\pi W(n+a)]-\operatorname {sinc} [W(n+a)]\right\}$	$j\omega \cdot \operatorname {rect} \left({\omega \over \pi W}\right)e^{ja\omega }$	real numbers W, a $0<W\leq 1$
${\frac {1}{\pi n^{2}}}[(-1)^{n}-1]\!$	$|\omega |\!$
${\begin{cases}0;&n{\mbox{ even}}\\{\frac {2}{\pi n}};&n{\mbox{ odd}}\end{cases}}$	${\begin{cases}j&\omega <0\\0&\omega =0\\-j&\omega >0\end{cases}}$	Hilbert transform
${\frac {C(A+B)}{2\pi }}\cdot \operatorname {sinc} \left[{\frac {A-B}{2\pi }}n\right]\cdot \operatorname {sinc} \left[{\frac {A+B}{2\pi }}n\right]$		real numbers A, B complex C

Properties

This table shows some mathematical operations in the time-domain and the corresponding effects in the frequency domain.

• $*\!$ is the discrete convolution of two sequences
• $x[n]^{*}\!$ is the complex conjugate of $x[n]\!$

Property	Time domain $x[n]\!$	Frequency domain $X(\omega )\!$	Remarks
Linearity	$ax[n]+by[n]\!$	$aX(e^{i\omega })+bY(e^{i\omega })\!$
Shift in time	$x[n-k]\!$	$X(e^{i\omega })e^{-i\omega k}\!$	integer k
Shift in frequency (modulation)	$x[n]e^{ian}\!$	$X(e^{i(\omega -a)})\!$	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	$x[n]\cdot y[n]\!$	${\frac {1}{2\pi }}\int _{-\pi }^{\pi }{X(e^{i\vartheta })\cdot Y(e^{i(\omega -\vartheta )})d\vartheta }\!$	Periodic convolution
Cross correlation	$\rho _{xy}[n]=x[-n]^{*}*y[n]\!$	$R_{xy}(\omega )=X(e^{i\omega })^{*}\cdot Y(e^{i\omega })\!$
Parseval's theorem	$E=\sum _{n=-\infty }^{\infty }{x[n]\cdot y^{*}[n]}\!$	$E={\frac {1}{2\pi }}\int _{-\pi }^{\pi }{X(e^{i\omega })\cdot Y^{*}(e^{i\omega })d\omega }\!$

Symmetry Properties

The Fourier Transform can be decomposed into a real and imaginary part or into an even and odd part.
$X(e^{i\omega })=X_{R}(e^{i\omega })+iX_{I}(e^{i\omega })\!$
or
$X(e^{i\omega })=X_{E}(e^{i\omega })+X_{O}(e^{i\omega })\!$

Time Domain $x[n]\!$	Frequency Domain $X(e^{i\omega })\!$
$x^{*}[n]\!$	$X^{*}(e^{-i\omega })\!$
$x^{*}[-n]\!$	$X^{*}(e^{i\omega })\!$

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. 27–29 and 104–105. ISBN 0-471-14961-6.