Discrete-time Fourier transform

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

${\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}s[n]\cdot \delta (t-nT)={\mathcal{F}}^{-1}\left\{S_{1/T}(f)\right\}\triangleq {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{S}_{1/T}(f)\cdot e^{i2\pi ft}df.}$

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

Periodic data

When the input data sequence ${\displaystyle s[n]}$ is ${\displaystyle N}$-periodic, Eq.2 can be computationally reduced to a discrete Fourier transform (DFT), because:

• All the available information is contained within ${\displaystyle N}$ samples.
• ${\displaystyle S_{1/T}(f)}$ converges to zero everywhere except at integer multiples of ${\displaystyle 1/(NT),}$ known as harmonic frequencies. At those frequencies, the DTFT diverges at different frequency-dependent rates. And those rates are given by the DFT of one cycle of the ${\displaystyle s[n]}$ sequence.
• The DTFT is periodic, so the maximum number of unique harmonic amplitudes is ${\displaystyle (1/T)/(1/(NT))=N.}$

The DFT of one cycle of the ${\displaystyle s[n]}$ sequence is:

${\displaystyle S[k]\triangleq \underset{\text{any n-sequence of length N}}{\underbrace{\sum _{N}s[n]\cdot e^{-i2\pi \frac{k}{N}n}}},k\in \mathbf{Z}.}$

And ${\displaystyle s[n]}$ can be expressed in terms of the inverse transform, which is sometimes referred to as a discrete Fourier series (DFS):

${\displaystyle s[n]=\frac{1}{N}\underset{\text{any k-sequence of length N}}{\underbrace{\sum _{N}S[k]\cdot e^{i2\pi \frac{k}{N}n}}},n\in \mathbf{Z}.}$

With these definitions, we can demonstrate the relationship between the DTFT and the DFT:

${\displaystyle \begin{array}{cc}{S}_{1/T}(f)& \triangleq \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}s[n]\cdot e^{-i2\pi fnT}\\ & =\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left[\frac{1}{N}\sum _{k=0}^{N-1}S[k]\cdot e^{i2\pi \frac{k}{N}n}\right]\cdot e^{-i2\pi fnT}\\ & =\frac{1}{N}\sum _{k=0}^{N-1}S[k]\underset{\mathrm{DTFT}\left(e^{i2\pi \frac{k}{N}n}\right)}{\underbrace{\left[\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{i2\pi \frac{k}{N}n}\cdot e^{-i2\pi fnT}\right]}}\\ & =\frac{1}{N}\sum _{k=0}^{N-1}S[k]\cdot \frac{1}{T}\sum _{M=-\mathrm{\infty }}^{\mathrm{\infty }}\delta \left(f-\frac{k}{NT}-\frac{M}{T}\right)\end{array}}$

Due to the ${\displaystyle N}$-periodicity of both functions of ${\displaystyle k,}$ this can be simplified to:

${\displaystyle S_{1/T}(f)=\frac{1}{NT}\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}S[k]\cdot \delta \left(f-\frac{k}{NT}\right),}$

which satisfies the inverse transform requirement:

${\displaystyle \begin{array}{cc}s[n]& =T{\int }_0^{\frac{1}{T}}{S}_{1/T}(f)\cdot e^{i2\pi fnT}df\\ & =\frac{1}{N}\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}S[k]\underset{\text{zero for }k\notin [0,N-1]}{\underbrace{{\int }_0^{\frac{1}{T}}\delta \left(f-\frac{k}{NT}\right)e^{i2\pi fnT}df}}\\ & =\frac{1}{N}\sum _{k=0}^{N-1}S[k]{\int }_0^{\frac{1}{T}}\delta \left(f-\frac{k}{NT}\right)e^{i2\pi fnT}df\\ & =\frac{1}{N}\sum _{k=0}^{N-1}S[k]\cdot e^{i2\pi \frac{k}{NT}nT}\\ & =\frac{1}{N}\sum _{k=0}^{N-1}S[k]\cdot e^{i2\pi \frac{k}{N}n}\end{array}}$

Sampling the DTFT

When the DTFT is continuous, a common practice is to compute an arbitrary number of samples ${\displaystyle (N)}$ of one cycle of the periodic function ${\displaystyle S_{1/T}}$:

${\displaystyle \begin{array}{cc}\underset{S_k}{\underbrace{S_{1/T}\left(\frac{k}{NT}\right)}}& =\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}s[n]\cdot e^{-i2\pi \frac{k}{N}n}k=0,\dots ,N-1\\ & =\underset{\text{DFT}}{\underbrace{\sum _N{s}_{{}_N}[n]\cdot e^{-i2\pi \frac{k}{N}n},}}\text{(sum over any }n\text{-sequence of length }N)\end{array}}$

where ${\displaystyle s_{{}_N}}$ is a periodic summation:

${\displaystyle s_{{}_N}[n]\triangleq \sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}s[n-mN].}$ (see Discrete Fourier series)

The ${\displaystyle s_{{}_N}}$ sequence is the inverse DFT. Thus, our sampling of the DTFT causes the inverse transform to become periodic. The array of ${\displaystyle |S_k{|}^2}$ values is known as a periodogram, and the parameter ${\displaystyle N}$ is called NFFT in the Matlab function of the same name.

In order to evaluate one cycle of ${\displaystyle s_{{}_N}}$ numerically, we require a finite-length ${\displaystyle s[n]}$ sequence. For instance, a long sequence might be truncated by a window function of length ${\displaystyle L}$ resulting in three cases worthy of special mention. For notational simplicity, consider the ${\displaystyle s[n]}$ values below to represent the values modified by the window function.

Case: Frequency decimation. ${\displaystyle L=N\cdot I,}$ for some integer ${\displaystyle I}$ (typically 6 or 8)

A cycle of ${\displaystyle s_{{}_N}}$ reduces to a summation of ${\displaystyle I}$ segments of length ${\displaystyle N.}$ The DFT then goes by various names, such as:

• window-presum FFT
• Weight, overlap, add (WOLA)
• polyphase DFT
• polyphase filter bank
• multiple block windowing and time-aliasing.

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 ${\displaystyle L}$-length DFT, the ${\displaystyle s_{{}_N}}$ 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 (channelizer). With a conventional window function of length ${\displaystyle L,}$ 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 ${\displaystyle I,}$ the better the potential performance.

Case: ${\displaystyle L=N+1}$

When a symmetric, ${\displaystyle L}$-length window function (${\displaystyle s}$) is truncated by 1 coefficient it is called periodic or DFT-even. That is a common practice, but the truncation affects the DTFT (spectral leakage) by a small amount. It is at least of academic interest to characterize that effect. An ${\displaystyle N}$-length DFT of the truncated window produces frequency samples at intervals of ${\displaystyle 1/N,}$ instead of ${\displaystyle 1/L.}$ The samples are real-valued, but their values do not exactly match the DTFT of the symmetric window. The periodic summation, ${\displaystyle s_{{}_N},}$ along with an ${\displaystyle N}$-length DFT, can also be used to sample the DTFT at intervals of ${\displaystyle 1/N.}$ Those samples are also real-valued and do exactly match the DTFT (example: File:Sampling the Discrete-time Fourier transform.svg). To use the full symmetric window for spectral analysis at the ${\displaystyle 1/N}$ spacing, one would combine the ${\displaystyle n=0}$ and ${\displaystyle n=N}$ data samples (by addition, because the symmetrical window weights them equally) and then apply the truncated symmetric window and the ${\displaystyle N}$-length DFT.

Case: Frequency interpolation. ${\displaystyle L\le N}$

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

${\displaystyle S_k=\sum _{n=0}^{N-1}s[n]\cdot e^{-i2\pi \frac{k}{N}n}.}$

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

Spectral leakage, which increases as ${\displaystyle L}$ 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 ${\displaystyle s[n]}$ 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:

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

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: ${\displaystyle f=1/8=0.125}$. Also visible in Fig 2 is the spectral leakage pattern of the ${\displaystyle L=64}$ 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 Hann window would produce a similar result, except the peak would be widened to 3 samples (see DFT-even Hann window).

Convolution

The convolution theorem for sequences is:

${\displaystyle s\ast y={\mathrm{D}\mathrm{T}\mathrm{F}\mathrm{T}}^{-1}{\displaystyle \left[\mathrm{D}\mathrm{T}\mathrm{F}\mathrm{T}{\displaystyle \{s\}\cdot \mathrm{D}\mathrm{T}\mathrm{F}\mathrm{T}{\displaystyle \{y\}}}\right].}}$

An important special case is the circular convolution of sequences s and y defined by ${\displaystyle s_{{}_N}\ast y,}$ where ${\displaystyle s_{{}_N}}$ is a periodic summation. The discrete-frequency nature of ${\displaystyle \mathrm{D}\mathrm{T}\mathrm{F}\mathrm{T}{\displaystyle \{s_{{}_N}\}}}$ means that the product with the continuous function ${\displaystyle \mathrm{D}\mathrm{T}\mathrm{F}\mathrm{T}{\displaystyle \{y\}}}$ is also discrete, which results in considerable simplification of the inverse transform:

${\displaystyle s_{{}_N}\ast y={\mathrm{D}\mathrm{T}\mathrm{F}\mathrm{T}}^{-1}{\displaystyle \left[\mathrm{D}\mathrm{T}\mathrm{F}\mathrm{T}{\displaystyle \{s_{{}_N}\}\cdot \mathrm{D}\mathrm{T}\mathrm{F}\mathrm{T}{\displaystyle \{y\}}}\right]={\mathrm{D}\mathrm{F}\mathrm{T}}^{-1}{\displaystyle \left[\mathrm{D}\mathrm{F}\mathrm{T}{\displaystyle \{s_{{}_N}\}\cdot \mathrm{D}\mathrm{F}\mathrm{T}{\displaystyle \{y_{{}_N}\}}}\right].}}}$

For s and y sequences whose non-zero duration is less than or equal to N, a final simplification is:

${\displaystyle s_{{}_N}\ast y={\mathrm{D}\mathrm{F}\mathrm{T}}^{-1}{\displaystyle \left[\mathrm{D}\mathrm{F}\mathrm{T}{\displaystyle \{s\}\cdot \mathrm{D}\mathrm{F}\mathrm{T}{\displaystyle \{y\}}}\right].}}$

The significance of this result is explained at circular convolution and fast convolution algorithms.

Relationship to the Z-transform

${\displaystyle S_{2\pi }(\omega )}$ is a Fourier series that can also be expressed in terms of the bilateral Z-transform. I.e.:

${\displaystyle S_{2\pi }(\omega )={S_z(z)|}_{z=e^{i\omega }}=S_z(e^{i\omega }),}$

where the ${\displaystyle S_z}$ 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:

${\displaystyle \begin{array}{cc}{S}_z(e^{i\omega })& =S_{1/T}\left(\frac{\omega }{2\pi T}\right)=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}S\left(\frac{\omega }{2\pi T}-k/T\right)\\ & =\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}S\left(\frac{\omega -2\pi k}{2\pi T}\right).\end{array}}$

Note that when parameter T changes, the terms of ${\displaystyle S_{2\pi }(\omega )}$ remain a constant separation ${\displaystyle 2\pi }$ apart, and their width scales up or down. The terms of S_1/T(f) remain a constant width and their separation 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:

• ${\displaystyle \omega =2\pi fT}$ is a real number representing continuous angular frequency (in radians per sample). (${\displaystyle f}$ is in cycles/sec, and ${\displaystyle T}$ is in sec/sample.) In all cases in the table, the DTFT is 2π-periodic (in ${\displaystyle \omega }$).
• ${\displaystyle S_{2\pi }(\omega )}$ designates a function defined on ${\displaystyle -\mathrm{\infty }<\omega <\mathrm{\infty }}$.
• ${\displaystyle S_o(\omega )}$ designates a function defined on ${\displaystyle -\pi <\omega \le \pi }$, and zero elsewhere. Then: $${\displaystyle S_{2\pi }(\omega )\triangleq \sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{S}_o(\omega -2\pi k).}$$
• ${\displaystyle \delta (\omega )}$ is the Dirac delta function
• ${\displaystyle \mathrm{sinc}(t)}$ is the normalized sinc function
• ${\displaystyle \mathrm{rect}\left[\frac{n}{L}\right]\triangleq \{\begin{array}{ll}1& |n|\le L/2\\ 0& |n|>L/2\end{array}}$
• ${\displaystyle \mathrm{tri}(t)}$ is the triangle function
• n is an integer representing the discrete-time domain (in samples)
• ${\displaystyle u[n]}$ is the discrete-time unit step function
• ${\displaystyle \delta [n]}$ is the Kronecker delta ${\displaystyle {\delta }_{n,0}}$

Properties

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

• ${\displaystyle \ast }$ is the discrete convolution of two sequences
• ${\displaystyle s^{\ast }[n]}$ is the complex conjugate of ${\displaystyle s[n].}$

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