Inmathematics, thediscrete-time Fourier transform (DTFT) is a form ofFourier analysis that is applicable to a sequence of discrete values.

The DTFT is often used to analyze samples of acontinuous function. The termdiscrete-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 aperiodic summation of thecontinuous Fourier transform of the original continuous function. In simpler terms, when you take the DTFT of regularly-spaced samples of a continuous signal, you get repeating (and possibly overlapping) copies of the signal's frequency spectrum, spaced at intervals corresponding to the sampling frequency. Under certain theoretical conditions, described by thesampling 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 thediscrete 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 reconstructs the original sampled data sequence, while the inverse DFT produces a periodic summation of the original sequence. Thefast Fourier transform (FFT) is an algorithm for computing one cycle of the DFT, and its inverse produces one cycle of the inverse DFT.

Let${\displaystyle s(t)}$ be a continuous function in the time domain.We begin with a common definition of the continuousFourier transform,where${\displaystyle f}$ represents frequency in hertz and${\displaystyle t}$ represents time in seconds:

${\displaystyle S(f)\triangleq {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}s(t)\cdot e^{-i2\pi ft}dt.}$

We can reduce the integral into a summation by sampling${\displaystyle s(t)}$ at intervals of${\displaystyle T}$ seconds(seeFourier transform § Numerical integration of a series of ordered pairs).Specifically, we can replace${\displaystyle s(t)}$ with a discrete sequence of its samples,${\displaystyle s(nT)}$, for integer values of${\displaystyle n}$,and replace the differential element${\displaystyle dt}$ with the sampling period${\displaystyle T}$.Thus, we obtain one formulation for the discrete-time Fourier transform (DTFT):

${\displaystyle S_{1/T}(f)\triangleq \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\underset{s[n]}{\underbrace{T\cdot s(nT)}}{e}^{-i2\pi fTn}.}$

ThisFourier series (in frequency) is a continuousperiodic function, whose periodicity is the sampling frequency${\displaystyle 1/T}$.The subscript${\displaystyle 1/T}$ distinguishes it from the continuous Fourier transform${\displaystyle S(f)}$,and from the angular frequency form of the DTFT.The latter is obtained by defining an angular frequency variable,${\displaystyle \omega \triangleq 2\pi fT}$ (which hasnormalized units ofradians/sample), giving us a periodic function of angular frequency, with periodicity${\displaystyle 2\pi }$:^[a]

${\displaystyle S_{2\pi }(\omega )=S_{1/T}\left(\frac{\omega }{2\pi T}\right)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}s[n]\cdot e^{-i\omega n}.}$

Eq.1

The utility of the DTFT is rooted in thePoisson summation formula, which tells us that the periodic function represented by the Fourier series is a periodic summation of the continuous Fourier transform:^[b]

Poisson summation

${\displaystyle S_{1/T}(f)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}s[n]\cdot e^{-i2\pi fTn}=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}S\left(f-k/T\right).}$

Eq.2

The components of the periodic summation are centered at integer values (denoted by${\displaystyle k}$) of anormalized frequency (cycles per sample). Ordinary/physical frequency (cycles per second) is the product of${\displaystyle k}$ and the sample-rate,${\displaystyle f_s=1/T.}$   For sufficiently large${\displaystyle f_s,}$ the${\displaystyle k=0}$ term can be observed in the region${\displaystyle [-f_s/2,f_s/2]}$ with little or no distortion (aliasing) from the other terms. Fig.1 depicts an example where${\displaystyle 1/T}$ is not large enough to prevent aliasing.

We also note that${\displaystyle e^{-i2\pi fTn}}$ is the Fourier transform of${\displaystyle \delta (t-nT).}$ Therefore, an alternative definition of DTFT is:^[A]

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

Eq.3

The modulatedDirac comb function is a mathematical abstraction sometimes referred to asimpulse sampling.^[3]

An operation that recovers the discrete data sequence from the DTFT function is called aninverse DTFT. For instance, the inverse continuous Fourier transform of both sides ofEq.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 bothEq.1 andEq.2, the summations over${\displaystyle n}$ are aFourier series, with coefficients${\displaystyle s[n].}$  The standard formulas for the Fourier coefficients are also the inverse transforms:

Eq.4

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 asharmonic 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 aDiscrete Fourier series (DFS):^{[1]: p 542}

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

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

     ^[c][B]

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:

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${\displaystyle S_{1/T}}$: ^{[1]: pp 557–559 & 703  [2]: p 76}

where${\displaystyle s_{{}_N}}$ is aperiodic summation:

${\displaystyle s_{{}_N}[n]\triangleq \sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}s[n-mN].}$     (seeDiscrete 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 aperiodogram, and the parameter${\displaystyle N}$ is called NFFT in the Matlab function of the same name.^[4]

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 awindow 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^[5]
• Weight, overlap, add (WOLA)^{[6][7][8][9][10][11][C][D]}

• polyphase DFT^[9][10]
• polyphase filter bank^[12]
• multiple block windowing andtime-aliasing.^[13]

Recall that decimation of sampled data in one domain (time or frequency) produces overlap (sometimes known asaliasing) in the other, and vice versa. Compared to an${\displaystyle L}$-length DFT, the${\displaystyle s_{{}_N}}$ summation/overlap causes decimation in frequency,^[1]: p.558 leaving only DTFT samples least affected byspectral leakage. That is usually a priority when implementing an FFTfilter-bank (channelizer). With a conventional window function of length${\displaystyle L,}$scalloping loss would be unacceptable. So multi-block windows are created usingFIR filter design tools.^[14][15]  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,${\displaystyle L}$-lengthwindow function (${\displaystyle s}$) is truncated by 1 coefficient it is calledperiodic orDFT-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,^[16]: p.52  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 is often referred to aszero-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 usezero-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 inFig 2 is the spectral leakage pattern of the${\displaystyle L=64}$ rectangular window. The illusion inFig 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. AHann window would produce a similar result, except the peak would be widened to 3 samples (seeDFT-even Hann window).

Theconvolution theorem for sequences is:

${\displaystyle s\ast y={\mathrm{D}\mathrm{T}\mathrm{F}\mathrm{T}}^{-1}{\displaystyle \left[\mathrm{D}\mathrm{T}\mathrm{F}\mathrm{T}{\displaystyle \{s\}\cdot \mathrm{D}\mathrm{T}\mathrm{F}\mathrm{T}{\displaystyle \{y\}}}\right].}}$^{[17]: p.297 [d]}

An important special case is thecircular convolution of sequencess andy 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:

${\displaystyle s_{{}_N}\ast y={\mathrm{D}\mathrm{T}\mathrm{F}\mathrm{T}}^{-1}{\displaystyle \left[\mathrm{D}\mathrm{T}\mathrm{F}\mathrm{T}{\displaystyle \{s_{{}_N}\}\cdot \mathrm{D}\mathrm{T}\mathrm{F}\mathrm{T}{\displaystyle \{y\}}}\right]={\mathrm{D}\mathrm{F}\mathrm{T}}^{-1}{\displaystyle \left[\mathrm{D}\mathrm{F}\mathrm{T}{\displaystyle \{s_{{}_N}\}\cdot \mathrm{D}\mathrm{F}\mathrm{T}{\displaystyle \{y_{{}_N}\}}}\right].}}}$^{[18][1]: p.548}

Fors andy sequences whose non-zero duration is less than or equal toN, 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 atCircular convolution andFast convolution algorithms.

${\displaystyle S_{2\pi }(\omega )}$ is aFourier series that can also be expressed in terms of the bilateralZ-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:

Note that when parameterT 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 ofS_1/T(f) remain a constant width and their separation1/T scales up or down.

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

• ${\displaystyle \omega =2\pi fT}$ is areal 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.
• ${\displaystyle S_o(\omega )}$ designates a function defined on, 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 theDirac delta function
• ${\displaystyle \mathrm{sinc}(t)}$ is the normalizedsinc function
• 
• ${\displaystyle \mathrm{tri}(t)}$ is thetriangle function
• n is an integer representing the discrete-time domain (in samples)
• ${\displaystyle u[n]}$ is the discrete-timeunit step function
• ${\displaystyle \delta [n]}$ is theKronecker delta${\displaystyle {\delta }_{n,0}}$

Time domain s[n]	Frequency domain S2π(ω)	Remarks	Reference
${\displaystyle \delta [n]}$	${\displaystyle S_{2\pi }(\omega )=1}$		[17]: p.305
${\displaystyle \delta [n-M]}$	${\displaystyle S_{2\pi }(\omega )=e^{-i\omega M}}$	integer${\displaystyle M}$
${\displaystyle \sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\delta [n-Mm]}$	${\displaystyle S_{2\pi }(\omega )=\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-i\omega Mm}=\frac{2\pi }{M}\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}\delta \left(\omega -\frac{2\pi k}{M}\right)}$ ${\displaystyle S_o(\omega )=\frac{2\pi }{M}\sum _{k=-(M-1)/2}^{(M-1)/2}\delta \left(\omega -\frac{2\pi k}{M}\right)}$     oddM ${\displaystyle S_o(\omega )=\frac{2\pi }{M}\sum _{k=-M/2+1}^{M/2}\delta \left(\omega -\frac{2\pi k}{M}\right)}$     evenM	integer${\displaystyle M>0}$
${\displaystyle u[n]}$	${\displaystyle S_{2\pi }(\omega )=\frac{1}{1-e^{-i\omega }}+\pi \sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}\delta (\omega -2\pi k)}$ ${\displaystyle S_o(\omega )=\frac{1}{1-e^{-i\omega }}+\pi \cdot \delta (\omega )}$	The${\displaystyle 1/(1-e^{-i\omega })}$ term must be interpreted as adistribution in the sense of aCauchy principal value around itspoles at${\displaystyle \omega =2\pi k}$.
${\displaystyle a^{n}u[n]}$	${\displaystyle S_{2\pi }(\omega )=\frac{1}{1-a{e}^{-i\omega }}}$		[17]: p.305
${\displaystyle e^{-ian}}$	${\displaystyle S_o(\omega )=2\pi \cdot \delta (\omega +a),}$     -π < a < π ${\displaystyle S_{2\pi }(\omega )=2\pi \sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}\delta (\omega +a-2\pi k)}$	real number${\displaystyle a}$
${\displaystyle \cos (a\cdot n)}$	${\displaystyle S_o(\omega )=\pi \left[\delta \left(\omega -a\right)+\delta \left(\omega +a\right)\right],}$ ${\displaystyle S_{2\pi }(\omega )\triangleq \sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{S}_o(\omega -2\pi k)}$	real number${\displaystyle a}$ with
${\displaystyle \sin (a\cdot n)}$	${\displaystyle S_o(\omega )=\frac{\pi }{i}\left[\delta \left(\omega -a\right)-\delta \left(\omega +a\right)\right]}$	real number${\displaystyle a}$ with
${\displaystyle \mathrm{rect}\left[\frac{n-M}{N}\right]\equiv \mathrm{rect}\left[\frac{n-M}{N-1}\right]}$	${\displaystyle S_o(\omega )=\frac{\sin (N\omega /2)}{\sin (\omega /2)}{e}^{-i\omega M}}$	integer${\displaystyle M,}$ andodd integer${\displaystyle N}$
${\displaystyle \mathrm{sinc}(W(n+a))}$	${\displaystyle S_o(\omega )=\frac{1}{W}\mathrm{rect}\left(\frac{\omega }{2\pi W}\right)e^{ia\omega }}$	real numbers${\displaystyle W,a}$ with
${\displaystyle {\mathrm{sinc}}^2(Wn)}$	${\displaystyle S_o(\omega )=\frac{1}{W}\mathrm{tri}\left(\frac{\omega }{2\pi W}\right)}$	real number${\displaystyle W}$,
	${\displaystyle S_o(\omega )=j\omega }$	it works as adifferentiator filter
${\displaystyle \frac{1}{(n+a)}\left\{\cos [\pi W(n+a)]-\mathrm{sinc}[W(n+a)]\right\}}$	${\displaystyle S_o(\omega )=\frac{j\omega }{W}\cdot \mathrm{rect}\left(\frac{\omega }{\pi W}\right)e^{ja\omega }}$	real numbers${\displaystyle W,a}$ with
	${\displaystyle S_o(\omega )=|\omega |}$
		Hilbert transform
${\displaystyle \frac{C(A+B)}{2\pi }\cdot \mathrm{sinc}\left[\frac{A-B}{2\pi }n\right]\cdot \mathrm{sinc}\left[\frac{A+B}{2\pi }n\right]}$	${\displaystyle S_o(\omega )=}$	real numbers${\displaystyle A,B}$ complex${\displaystyle C}$

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

• ${\displaystyle \ast }$ is thediscrete convolution of two sequences
• ${\displaystyle s^{\ast }[n]}$ is thecomplex conjugate of${\displaystyle s[n].}$

Property	Time domain s[n]	Frequency domain ${\displaystyle S_{2\pi }(\omega )}$	Remarks	Reference
Linearity	${\displaystyle a\cdot s[n]+b\cdot y[n]}$	${\displaystyle a\cdot S_{2\pi }(\omega )+b\cdot Y_{2\pi }(\omega )}$	complex numbers${\displaystyle a,b}$	[17]: p.294
Time reversal / Frequency reversal	${\displaystyle s[-n]}$	${\displaystyle S_{2\pi }(-\omega )}$		[17]: p.297
Time conjugation	${\displaystyle s^{\ast }[n]}$	${\displaystyle S_{2\pi }^{\ast }(-\omega )}$		[17]: p.291
Time reversal & conjugation	${\displaystyle s^{\ast }[-n]}$	${\displaystyle S_{2\pi }^{\ast }(\omega )}$		[17]: p.291
Real part in time	${\displaystyle \mathrm{Re}(s[n])}$	${\displaystyle \frac{1}{2}(S_{2\pi }(\omega )+S_{2\pi }^{\ast }(-\omega ))}$		[17]: p.291
Imaginary part in time	${\displaystyle \mathrm{Im}(s[n])}$	${\displaystyle \frac{1}{2i}(S_{2\pi }(\omega )-S_{2\pi }^{\ast }(-\omega ))}$		[17]: p.291
Real part in frequency	${\displaystyle \frac{1}{2}(s[n]+s^{\ast }[-n])}$	${\displaystyle \mathrm{Re}(S_{2\pi }(\omega ))}$		[17]: p.291
Imaginary part in frequency	${\displaystyle \frac{1}{2i}(s[n]-s^{\ast }[-n])}$	${\displaystyle \mathrm{Im}(S_{2\pi }(\omega ))}$		[17]: p.291
Shift in time / Modulation in frequency	${\displaystyle s[n-k]}$	${\displaystyle S_{2\pi }(\omega )\cdot e^{-i\omega k}}$	integerk	[17]: p.296
Shift in frequency / Modulation in time	${\displaystyle s[n]\cdot e^{ian}}$	${\displaystyle S_{2\pi }(\omega -a)}$	real number${\displaystyle a}$	[17]: p.300
Decimation	${\displaystyle s[nM]}$	${\displaystyle \frac{1}{M}\sum _{m=0}^{M-1}{S}_{2\pi }\left(\frac{\omega -2\pi m}{M}\right)}$ [E]	integer${\displaystyle M}$
Time Expansion		${\displaystyle S_{2\pi }(M\omega )}$	integer${\displaystyle M}$	[1]: p.172
Derivative in frequency	${\displaystyle \frac{n}{i}s[n]}$	${\displaystyle \frac{d{S}_{2\pi }(\omega )}{d\omega }}$		[17]: p.303
Integration in frequency	${\displaystyle }$	${\displaystyle }$
Differencing in time	${\displaystyle s[n]-s[n-1]}$	${\displaystyle \left(1-e^{-i\omega }\right)S_{2\pi }(\omega )}$
Summation in time	${\displaystyle \sum _{m=-\mathrm{\infty }}^{n}s[m]}$	${\displaystyle \frac{1}{\left(1-e^{-i\omega }\right)}{S}_{2\pi }(\omega )+\pi S(0)\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}\delta (\omega -2\pi k)}$
Convolution in time / Multiplication in frequency	${\displaystyle s[n]\ast y[n]}$	${\displaystyle S_{2\pi }(\omega )\cdot Y_{2\pi }(\omega )}$		[17]: p.297
Multiplication in time / Convolution in frequency	${\displaystyle s[n]\cdot y[n]}$	${\displaystyle \frac{1}{2\pi }{\int }_{-\pi }^{\pi }{S}_{2\pi }(\nu )\cdot Y_{2\pi }(\omega -\nu )d\nu }$	Periodic convolution	[17]: p.302
Cross correlation	${\displaystyle {\rho }_{sy}[n]=s^{\ast }[-n]\ast y[n]}$	${\displaystyle R_{sy}(\omega )=S_{2\pi }^{\ast }(\omega )\cdot Y_{2\pi }(\omega )}$
Parseval's theorem	${\displaystyle E_{sy}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}s[n]\cdot y^{\ast }[n]}$	${\displaystyle E_{sy}=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{S}_{2\pi }(\omega )\cdot Y_{2\pi }^{\ast }(\omega )d\omega }$		[17]: p.302

• Least-squares spectral analysis
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