In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
In signal processing, any periodic function s_P(t) with period ''P'' can be represented by a summation of an infinite number of instances of an aperiodic function s(t), that are offset by integer multiples of ''P''. This representation is called p ...
of the
continuous Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of the original continuous function. Under certain theoretical conditions, described by the
sampling theorem
Sampling may refer to:
*Sampling (signal processing), converting a continuous signal into a discrete signal
* Sampling (graphics), converting continuous colors into discrete color components
*Sampling (music), the reuse of a sound recording in ano ...
, 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
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
(DFT) (see ), 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 sequence of real or complex numbers , for all
integers
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
, is a
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
, which produces a periodic function of a frequency variable. When the frequency variable, ω, has
normalized units of ''radians/sample'', the periodicity is , and the Fourier series is:
[
The utility of this frequency domain function is rooted in the ]Poisson summation formula
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of ...
. Let be the Fourier transform of any function, , whose samples at some interval (''seconds'') are equal (or proportional) to the sequence, i.e. .[ Then the periodic function represented by the Fourier series is a periodic summation of in terms of frequency in ]hertz
The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that on ...
(''cycles/sec''):
The integer has units of ''cycles/sample'', and is the sample-rate, (''samples/sec''). So comprises exact copies of that are shifted by multiples of hertz and combined by addition. For sufficiently large the term can be observed in the region with little or no distortion (aliasing
In signal processing and related disciplines, aliasing is an effect that causes different signals to become indistinguishable (or ''aliases'' of one another) when sampled. It also often refers to the distortion or artifact that results when a ...
) 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 is the Fourier transform of . Therefore, an alternative definition of DTFT is:
The modulated Dirac comb
In mathematics, a Dirac comb (also known as shah function, impulse train or sampling function) is a periodic function with the formula
\operatorname_(t) \ := \sum_^ \delta(t - k T)
for some given period T. Here ''t'' is a real variable and th ...
function is a mathematical abstraction sometimes referred to as ''impulse sampling''.[
]
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 produces the sequence in the form of a modulated Dirac comb function:
:
However, noting that is periodic, all the necessary information is contained within any interval of length . In both and , the summations over n are a Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
, with coefficients . The standard formulas for the Fourier coefficients are also the inverse transforms:
Periodic data
When the input data sequence is -periodic, can be computationally reduced to a discrete Fourier transform (DFT), because:
* All the available information is contained within samples.
* converges to zero everywhere except at integer multiples of , 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 sequence.
* The DTFT is periodic, so the maximum number of unique harmonic amplitudes is
The DFT coefficients are given by:
: and the DTFT is:
:
Substituting this expression into the inverse transform formula confirms:
: ( all integers)
as expected. The inverse DFT in the line above is sometimes referred to as a Discrete Fourier series In digital signal processing, the term Discrete Fourier series (DFS) is any periodic discrete-time signal comprising harmonically-related (i.e. ''Fourier'') discrete real sinusoids or discrete complex exponentials, combined by a weighted summation. ...
(DFS).[
]
Sampling the DTFT
When the DTFT is continuous, a common practice is to compute an arbitrary number of samples () of one cycle of the periodic function : [
:
where is a ]periodic summation
In signal processing, any periodic function s_P(t) with period ''P'' can be represented by a summation of an infinite number of instances of an aperiodic function s(t), that are offset by integer multiples of ''P''. This representation is called p ...
:
: (see Discrete Fourier series In digital signal processing, the term Discrete Fourier series (DFS) is any periodic discrete-time signal comprising harmonically-related (i.e. ''Fourier'') discrete real sinusoids or discrete complex exponentials, combined by a weighted summation. ...
)
The sequence is the inverse DFT. Thus, our sampling of the DTFT causes the inverse transform to become periodic. The array of values is known as a ''periodogram In signal processing, a periodogram is an estimate of the spectral density of a signal. The term was coined by Arthur Schuster in 1898. Today, the periodogram is a component of more sophisticated methods (see spectral estimation). It is the most ...
'', and the parameter is called NFFT in the Matlab function of the same name.[
In order to evaluate one cycle of numerically, we require a finite-length sequence. For instance, a long sequence might be truncated by a ]window function
In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the int ...
of length resulting in three cases worthy of special mention. For notational simplicity, consider the values below to represent the values modified by the window function.
Case: Frequency decimation. , for some integer (typically 6 or 8)
A cycle of reduces to a summation of segments of length . 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 signal processing and related disciplines, aliasing is an effect that causes different signals to become indistinguishable (or ''aliases'' of one another) when sampled. It also often refers to the distortion or artifact that results when a ...
) in the other, and vice versa. Compared to an -length DFT, the summation/overlap causes decimation in frequency,[ leaving only DTFT samples least affected by ]spectral leakage
The Fourier transform of a function of time, s(t), is a complex-valued function of frequency, S(f), often referred to as a frequency spectrum. Any linear time-invariant operation on s(t) produces a new spectrum of the form H(f)•S(f), which cha ...
. That is usually a priority when implementing an FFT filter-bank (channelizer). With a conventional window function of length , scalloping loss would be unacceptable. So multi-block windows are created using FIR filter
In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of ''finite'' duration, because it settles to zero in finite time. This is in contrast to infinite impulse ...
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 , the better the potential performance.
Case: .
When a symmetric, -length ]window function
In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the int ...
() is truncated by 1 coefficient it is called ''periodic'' or ''DFT-even''. The truncation affects the DTFT. A DFT of the truncated sequence samples the DTFT at frequency intervals of . To sample at the same frequencies, for comparison, the DFT is computed for one cycle of the periodic summation,
Case: Frequency interpolation.
In this case, the DFT simplifies to a more familiar form:
:
In order to take advantage of a fast Fourier transform algorithm for computing the DFT, the summation is usually performed over all terms, even though of them are zeros. Therefore, the case is often referred to as zero-padding.
Spectral leakage, which increases as 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 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:
: and
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: . Also visible in Fig 2 is the spectral leakage pattern of the 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
The Hann function is named after the Austrian meteorologist Julius von Hann. It is a window function used to perform Hann smoothing. The function, with length L and amplitude 1/L, is given by:
:
w_0(x) \triangleq \left\.
For digital sign ...
would produce a similar result, except the peak would be widened to 3 samples (se
DFT-even Hann window
.
Convolution
The convolution theorem
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g ...
for sequences is:
:[
An important special case is the ]circular convolution Circular convolution, also known as cyclic convolution, is a special case of periodic convolution, which is the convolution of two periodic functions that have the same period. Periodic convolution arises, for example, in the context of the discre ...
of sequences and defined by where is a periodic summation. The discrete-frequency nature of means that the product with the continuous function is also discrete, which results in considerable simplification of the inverse transform:
:[
For and sequences whose non-zero duration is less than or equal to , a final simplification is:
:
The significance of this result is explained at ]Circular convolution Circular convolution, also known as cyclic convolution, is a special case of periodic convolution, which is the convolution of two periodic functions that have the same period. Periodic convolution arises, for example, in the context of the discre ...
and Fast convolution algorithms.
Symmetry properties
When the real and imaginary parts of a complex function are decomposed into their 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:[
From this, various relationships are apparent, for example:
*The transform of a real-valued function () is the even symmetric function . Conversely, an even-symmetric transform implies a real-valued time-domain.
*The transform of an imaginary-valued function () is the odd symmetric function , and the converse is true.
*The transform of an even-symmetric function () is the real-valued function , and the converse is true.
*The transform of an odd-symmetric function () is the imaginary-valued function , and the converse is true.
]
Relationship to the Z-transform
is a Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
that can also be expressed in terms of the bilateral Z-transform
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation.
It can be considered as a discrete-tim ...
. I.e.:
:
where the 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:
:
Note that when parameter changes, the terms of remain a constant separation apart, and their width scales up or down. The terms of remain a constant width and their separation scales up or down.
Table of discrete-time Fourier transforms
Some common transform pairs are shown in the table below. The following notation applies:
* is a real number representing continuous angular frequency (in radians per sample). ( is in cycles/sec, and is in sec/sample.) In all cases in the table, the DTFT is 2π-periodic (in ).
* designates a function defined on .
* designates a function defined on , and zero elsewhere. Then:
* is the Dirac delta function
* is the normalized sinc function
*
* is the triangle function
A triangular function (also known as a triangle function, hat function, or tent function) is a function whose graph takes the shape of a triangle. Often this is an isosceles triangle of height 1 and base 2 in which case it is referred to as ''th ...
* is an integer representing the discrete-time domain (in samples)
*