Discretetime Fourier transform
In mathematics, the discretetime Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values.
Fourier transforms 

The DTFT is often used to analyze samples of a continuous function. The term discretetime 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 § 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 discretetime Fourier transform of a discrete sequence of real or complex numbers x[n], for all integers n, is a Fourier series, which produces a periodic function of a frequency variable. When the frequency variable, ω, has normalized units of radians/sample, the periodicity is 2π, and the Fourier series is:[1]^{: p.147 }

(Eq.1)
The utility of this frequency domain function is rooted in the Poisson summation formula. Let X(f) be the Fourier transform of any function, x(t), whose samples at some interval T (seconds) are equal (or proportional) to the x[n] sequence, i.e. T⋅x(nT) = x[n].[2] Then the periodic function represented by the Fourier series is a periodic summation of X(f) in terms of frequency f in hertz (cycles/sec):[loweralpha 1][upperalpha 1]

(Eq.2) 
The integer k has units of cycles/sample, and 1/T is the samplerate, f_{s} (samples/sec). So X_{1/T}(f) comprises exact copies of X(f) that are shifted by multiples of f_{s} hertz and combined by addition. For sufficiently large f_{s} the k = 0 term can be observed in the region [−f_{s}/2, f_{s}/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 e^{−i2πfTn} is the Fourier transform of δ(t − nT). Therefore, an alternative definition of DTFT is:[upperalpha 2]

(Eq.3) 
The modulated Dirac comb function is a mathematical abstraction sometimes referred to as impulse sampling.[4]
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:
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:

(Eq.4)
Periodic data
When the input data sequence x[n] is Nperiodic, Eq.2 can be computationally reduced to a discrete Fourier transform (DFT), because:
 All the available information is contained within N samples.
 X_{1/T}(f) converges to zero everywhere except at integer multiples of 1/(NT), known as harmonic frequencies. At those frequencies, the DTFT diverges at different frequencydependent rates. And those rates are given by the DFT of one cycle of the x[n] sequence.
 The DTFT is periodic, so the maximum number of unique harmonic amplitudes is (1/T) / (1/(NT)) = N
The DFT coefficients are given by:
 and the DTFT is:
Substituting this expression into the inverse transform formula confirms:
as expected. The inverse DFT in the line above is sometimes referred to as a Discrete Fourier series (DFS).[1]^{: p 542 }
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}: [1]^{: pp 557–559 & 703 }
where is a periodic summation:
 (see Discrete Fourier series)
The sequence is the inverse DFT. Thus, our sampling of the DTFT causes the inverse transform to become periodic. The array of X_{k}^{2} values is known as a periodogram, and the parameter N is called NFFT in the Matlab function of the same name.[5]
In order to evaluate one cycle of numerically, we require a finitelength x[n] sequence. For instance, a long sequence might be truncated by a window function of length L resulting in three cases worthy of special mention. For notational simplicity, consider the x[n] values below to represent the values modified by the window function.
Case: Frequency decimation. L = N ⋅ I, for some integer I (typically 6 or 8)
A cycle of reduces to a summation of I segments of length N. The DFT then goes by various names, such as:
 windowpresum FFT[6]
 Weight, overlap, add (WOLA)[7][8][9][10][11][12][upperalpha 3][upperalpha 4]
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 Llength DFT, the summation/overlap causes decimation in frequency,[1]^{: p.558 } leaving only DTFT samples least affected by spectral leakage. That is usually a priority when implementing an FFT filterbank (channelizer). With a conventional window function of length L, scalloping loss would be unacceptable. So multiblock windows are created using FIR filter design tools.[15][16] 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.
Case: L = N+1.
When a symmetric, Llength window function () is truncated by 1 coefficient it is called periodic or DFTeven. The truncation affects the DTFT. A DFT of the truncated sequence samples the DTFT at frequency intervals of 1/N. To sample at the same frequencies, for comparison, the DFT is computed for one cycle of the periodic summation, [upperalpha 5]
Case: Frequency interpolation. L ≤ N
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 N terms, even though N − L of them are zeros. Therefore, the case L < N is often referred to as zeropadding.
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 zeropadding 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: f = 1/8 = 0.125. Also visible in Fig 2 is the spectral leakage pattern of the L = 64 rectangular window. The illusion in Fig 3 is a result of sampling the DTFT at just its zerocrossings. Rather than the DTFT of a finitelength 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 DFTeven Hann window).
Convolution
The convolution theorem for sequences is:
 [18]^{: p.297 }[loweralpha 3]
An important special case is the circular convolution of sequences x and y defined by where is a periodic summation. The discretefrequency nature of means that the product with the continuous function is also discrete, which results in considerable simplification of the inverse transform:
For x and y sequences whose nonzero duration is less than or equal to N, a final simplification is:
The significance of this result is explained at Circular convolution 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 onetoone mapping between the four components of a complex time function and the four components of its complex frequency transform:[18]^{: p.291 }
From this, various relationships are apparent, for example:
 The transform of a realvalued function (x_{RE}+ x_{RO}) is the even symmetric function X_{RE}+ i X_{IO}. Conversely, an evensymmetric transform implies a realvalued timedomain.
 The transform of an imaginaryvalued function (i x_{IE}+ i x_{IO}) is the odd symmetric function X_{RO}+ i X_{IE}, and the converse is true.
 The transform of an evensymmetric function (x_{RE}+ i x_{IO}) is the realvalued function X_{RE}+ X_{RO}, and the converse is true.
 The transform of an oddsymmetric function (x_{RO}+ i x_{IE}) is the imaginaryvalued function i X_{IE}+ i X_{IO}, and the converse is true.
Relationship to the Ztransform
is a Fourier series that can also be expressed in terms of the bilateral Ztransform. I.e.:
where the notation distinguishes the Ztransform from the Fourier transform. Therefore, we can also express a portion of the Ztransform in terms of the Fourier transform:
Note that when parameter T changes, the terms of remain a constant separation apart, and their width scales up or down. The terms of X_{1/T}(f) remain a constant width and their separation 1/T scales up or down.
Table of discretetime 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 2πperiodic (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
 n is an integer representing the discretetime domain (in samples)
 is the discretetime unit step function
 is the Kronecker delta
Time domain x[n] 
Frequency domain X_{2π}(ω) 
Remarks  Reference 

[18]^{: p.305 }  
integer  
odd M 
integer  
The term must be interpreted as a distribution in the sense of a Cauchy principal value around its poles at .  
[18]^{: p.305 }  
π < a < π 
real number  
real number with  
real number with  
integer and odd integer  
real numbers with  
real number ,  
it works as a differentiator filter  
real numbers with  
Hilbert transform  
real numbers complex 
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
 is the complex conjugate of x[n].
Property  Time domain x[n] 
Frequency domain 
Remarks  Reference 

Linearity  complex numbers  [18]^{: p.294 }  
Time reversal / Frequency reversal  [18]^{: p.297 }  
Time conjugation  [18]^{: p.291 }  
Time reversal & conjugation  [18]^{: p.291 }  
Real part in time  [18]^{: p.291 }  
Imaginary part in time  [18]^{: p.291 }  
Real part in frequency  [18]^{: p.291 }  
Imaginary part in frequency  [18]^{: p.291 }  
Shift in time / Modulation in frequency  integer k  [18]^{: p.296 }  
Shift in frequency / Modulation in time  real number  [18]^{: p.300 }  
Decimation  [upperalpha 6]  integer  
Time Expansion  integer  [1]^{: p.172 }  
Derivative in frequency  [18]^{: p.303 }  
Integration in frequency  
Differencing in time  
Summation in time  
Convolution in time / Multiplication in frequency  [18]^{: p.297 }  
Multiplication in time / Convolution in frequency  Periodic convolution  [18]^{: p.302 }  
Cross correlation  
Parseval's theorem  [18]^{: p.302 } 
Notes
 When the dependency on T is unimportant, a common practice is to replace it with Then f has units of (cycles/sample), called normalized frequency.
 In fact Eq.2 is often justified as follows:[1]^{: p.143 }
 WOLA should not be confused with the Overlapadd method of piecewise convolution.
 WOLA example: File:WOLA channelizer example.png
 An example is figure Sampling the DTFT. The realvalued DFT samples are a result of DFTeven symmetry[17]^{: p.52 }
 This expression is derived as follows:[1]^{: p.168 }
Page citations
References

Oppenheim, Alan V.; Schafer, Ronald W.; Buck, John R. (1999). "4.2, 8.4". Discretetime signal processing (2nd ed.). Upper Saddle River, N.J.: Prentice Hall. ISBN 0137549202.
samples of the Fourier transform of an aperiodic sequence x[n] can be thought of as DFS coefficients of a periodic sequence obtained through summing periodic replicas of x[n].
url=https://d1.amobbs.com/bbs_upload782111/files_24/ourdev_523225.pdf  Ahmed, N.; Rao, K.R. (July 10, 1975). Orthogonal Transforms for Digital Signal Processing (1 ed.). Berlin Heidelberg New York: SpringerVerlag. doi:10.1007/9783642454509. ISBN 9783540065562.

Prandoni, Paolo; Vetterli, Martin (2008). Signal Processing for Communications (PDF) (1 ed.). Boca Raton, FL: CRC Press. pp. 72, 76. ISBN 9781420070460. Retrieved 4 October 2020.
the DFS coefficients for the periodized signal are a discrete set of values for its DTFT
 Rao, R. (2008). Signals and Systems. PrenticeHall Of India Pvt. Limited. ISBN 9788120338593.
 "Periodogram power spectral density estimate  MATLAB periodogram".

Gumas, Charles Constantine (July 1997). "Windowpresum FFT achieves highdynamic range, resolution". Personal Engineering & Instrumentation News: 58–64. Archived from the original on 20010210.
{{cite journal}}
: CS1 maint: bot: original URL status unknown (link)  Crochiere, R.E.; Rabiner, L.R. (1983). "7.2". Multirate Digital Signal Processing. Englewood Cliffs, NJ: PrenticeHall. pp. 313–326. ISBN 0136051626.
 Wang, Hong; Lu, Youxin; Wang, Xuegang (16 October 2006). "Channelized Receiver with WOLA Filterbank". 2006 CIE International Conference on Radar. Shanghai, China: IEEE: 1–3. doi:10.1109/ICR.2006.343463. ISBN 0780395824.
 Lyons, Richard G. (June 2008). "DSP Tricks: Building a practical spectrum analyzer". EE Times. Retrieved 20200220. Note however, that it contains a link labeled weighted overlapadd structure which incorrectly goes to Overlapadd method.

Lillington, John (March 2003). "Comparison of Wideband Channelisation Architectures" (PDF). Dallas: International Signal Processing Conference. p. 4 (fig 7). Archived from the original (PDF) on 20190308. Retrieved 20200906.
The "Weight Overlap and Add" or WOLA or its subset the "Polyphase DFT", is becoming more established and is certainly very efficient where large, high quality filter banks are required.

Lillington, John. "A Review of Filter Bank Techniques  RF and Digital" (PDF). armms.org. Isle of Wight, UK: Libra Design Associates Ltd. p. 11. Retrieved 20200906.
Fortunately, there is a much more elegant solution, as shown in Figure 20 below, known as the Polyphase or WOLA (Weight, Overlap and Add) FFT.

Hochgürtel, Stefan (2013). "Efficient implementations of highresolution wideband FFTspectrometers and their application to an APEX Galactic Center line survey" (PDF). hss.ulb.unibonn.de. Bonn: Rhenish Friedrich Wilhelms University of Bonn. pp. 26–27. Retrieved 20200906.
To perform Mfold WOLA for an Npoint DFT, M·N real input samples a_{j} first multiplied by a window function w_{j} of same size
 Chennamangalam, Jayanth (20161018). "The Polyphase Filter Bank Technique". CASPER Group. Retrieved 20161030.
 Dahl, Jason F. (20030206). Time Aliasing Methods of Spectrum Estimation (Ph.D.). Brigham Young University. Retrieved 20161031.
 Lin, YuanPei; Vaidyanathan, P.P. (June 1998). "A Kaiser Window Approach for the Design of Prototype Filters of Cosine Modulated Filterbanks" (PDF). IEEE Signal Processing Letters. 5 (6): 132–134. Bibcode:1998ISPL....5..132L. doi:10.1109/97.681427. Retrieved 20170316.
 Harris, Frederic J. (20040524). "9". Multirate Signal Processing for Communication Systems. Upper Saddle River, NJ: Prentice Hall PTR. pp. 226–253. ISBN 0131465112.
 Harris, Fredric J. (Jan 1978). "On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform" (PDF). Proceedings of the IEEE. 66 (1): 51–83. Bibcode:1978IEEEP..66...51H. CiteSeerX 10.1.1.649.9880. doi:10.1109/PROC.1978.10837.
 Proakis, John G.; Manolakis, Dimitri G. (1996). Digital Signal Processing: Principles, Algorithms and Applications (3 ed.). New Jersey: PrenticeHall International. Bibcode:1996dspp.book.....P. ISBN 9780133942897. sAcfAQAAIAAJ.
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Further reading
 Porat, Boaz (1996). A Course in Digital Signal Processing. John Wiley and Sons. pp. 27–29 and 104–105. ISBN 0471149616.
 Siebert, William M. (1986). Circuits, Signals, and Systems. MIT Electrical Engineering and Computer Science Series. Cambridge, MA: MIT Press. ISBN 0262690950.
 Lyons, Richard G. (2010). Understanding Digital Signal Processing (3rd ed.). Prentice Hall. ISBN 9780137027415.