Английская Википедия:Discrete-time Fourier transform
Шаблон:Distinguish Шаблон:Short description Шаблон:Fourier transforms
In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of discrete 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 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 Шаблон:Slink), 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 Шаблон:Math, for all integers Шаблон:Mvar, 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 Шаблон:Math, and the Fourier series is:[1]Шаблон:Rp
The utility of this frequency domain function is rooted in the Poisson summation formula. Let Шаблон:Math be the Fourier transform of any function, Шаблон:Math, whose samples at some interval Шаблон:Mvar (seconds) are equal (or proportional) to the Шаблон:Math sequence, i.e. Шаблон:Math.[2] Then the periodic function represented by the Fourier series is a periodic summation of Шаблон:Math in terms of frequency Шаблон:Mvar in hertz (cycles/sec):Шаблон:Efn-laШаблон:Efn-ua Шаблон:NumBlk{=} \;
\sum_{k=-\infty}^{\infty} X\left(f - k/T\right).
</math>|Шаблон:EquationRef}}
The integer Шаблон:Mvar has units of cycles/sample, and Шаблон:Math is the sample-rate, Шаблон:Mvar (samples/sec). So Шаблон:Math comprises exact copies of Шаблон:Math that are shifted by multiples of Шаблон:Mvar hertz and combined by addition. For sufficiently large Шаблон:Mvar the Шаблон:Math term can be observed in the region Шаблон:Closed-closed 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 Шаблон:Math is the Fourier transform of Шаблон:Math. Therefore, an alternative definition of DTFT is:Шаблон:Efn-ua Шаблон:NumBlk
The modulated Dirac comb function is a mathematical abstraction sometimes referred to as impulse sampling.[3]
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 Шаблон:EquationNote produces the sequence in the form of a modulated Dirac comb function:
- <math>\sum_{n=-\infty}^{\infty} x[n]\cdot \delta(t-n T) = \mathcal{F}^{-1}\left \{X_{1/T}(f)\right\} \ \triangleq \int_{-\infty}^\infty X_{1/T}(f)\cdot e^{i 2 \pi f t} df.</math>
However, noting that Шаблон:Math is periodic, all the necessary information is contained within any interval of length Шаблон:Math. In both Шаблон:EquationNote and Шаблон:EquationNote, the summations over n are a Fourier series, with coefficients Шаблон:Math. The standard formulas for the Fourier coefficients are also the inverse transforms:
Шаблон:Equation box 1 \\ \displaystyle &= \frac{1}{2 \pi}\int_{2\pi} X_{2\pi}(\omega)\cdot e^{i \omega n} d\omega \quad \scriptstyle{\text{(integral over any interval of length }2\pi\textrm{)}} \end{align}</math>|Шаблон:EquationRef}} |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}}
Periodic data
When the input data sequence Шаблон:Math is Шаблон:Mvar-periodic, Шаблон:EquationNote can be computationally reduced to a discrete Fourier transform (DFT), because:
- All the available information is contained within Шаблон:Mvar samples.
- Шаблон:Math converges to zero everywhere except at integer multiples of Шаблон:Math, 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 Шаблон:Math sequence.
- The DTFT is periodic, so the maximum number of unique harmonic amplitudes is Шаблон:Math
The DFT coefficients are given by:
- <math>X[k] \triangleq \underbrace{\sum_{N} x(nT)\cdot e^{-i 2 \pi \frac{k}{N}n}}_{\text{any n-sequence of length N}},</math> and the DTFT is:
- <math>X_{1/T}(f) = \frac{1}{N} \sum_{k=-\infty}^{\infty} X[k] \cdot \delta\left(f-\frac{k}{NT}\right).</math> Шаблон:Efn-la
Substituting this expression into the inverse transform formula confirms:
- <math>\int_{\frac{1}{T}} X_{1/T}(f)\cdot e^{i 2 \pi f nT} df\ =\ \frac{1}{N} \underbrace{\sum_{N} X[k] \cdot e^{i 2 \pi \frac{n}{N}k}}_{\text{any k-sequence of length N}}\ \equiv\ x(nT), \quad n\in\mathbb{Z}\,</math> (all integers)
as expected. The inverse DFT in the line above is sometimes referred to as a Discrete Fourier series (DFS).[1]Шаблон:Rp
Шаблон:AnchorSampling the DTFT
When the DTFT is continuous, a common practice is to compute an arbitrary number of samples (Шаблон:Mvar) of one cycle of the periodic function Шаблон:Math: [1]Шаблон:Rp
- <math>
\begin{align} \underbrace{X_{1/T}\left(\frac{k}{NT}\right)}_{X_k} &= \sum_{n=-\infty}^\infty x[n]\cdot e^{-i 2\pi \frac{k}{N}n} \quad \quad k = 0, \dots, N-1 \\ &= \underbrace{\sum_{N} x_{_N}[n]\cdot e^{-i 2\pi \frac{k}{N}n},}_{\text{DFT}}\quad \scriptstyle{\text{(sum over any }n\text{-sequence of length }N)} \end{align} </math>
where <math>x_{_N}</math> is a periodic summation:
- <math>x_{_N}[n]\ \triangleq\ \sum_{m=-\infty}^{\infty} x[n-mN].</math> (see Discrete Fourier series)
The <math>x_{_N}</math> sequence is the inverse DFT. Thus, our sampling of the DTFT causes the inverse transform to become periodic. The array of Шаблон:Math values is known as a periodogram, and the parameter Шаблон:Mvar is called NFFT in the Matlab function of the same name.[4]
In order to evaluate one cycle of <math>x_{_N}</math> numerically, we require a finite-length Шаблон:Math sequence. For instance, a long sequence might be truncated by a window function of length Шаблон:Mvar resulting in three cases worthy of special mention. For notational simplicity, consider the Шаблон:Math values below to represent the values modified by the window function.
Шаблон:AnchorCase: Frequency decimation. Шаблон:Math, for some integer Шаблон:Mvar (typically 6 or 8)
A cycle of <math>x_{_N}</math> reduces to a summation of Шаблон:Mvar segments of length Шаблон:Mvar. The DFT then goes by various names, such as:
- window-presum FFT[5]
- Weight, overlap, add (WOLA)[6][7][8][9][10][11]Шаблон:Efn-uaШаблон:Efn-ua
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 Шаблон:Mvar-length DFT, the <math>x_{_N}</math> summation/overlap causes decimation in frequency,[1]Шаблон:Rp 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 Шаблон:Mvar, scalloping loss would be unacceptable. So multi-block windows are created using FIR 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 Шаблон:Mvar, the better the potential performance.
Шаблон:AnchorCase: Шаблон:Math.
When a symmetric, Шаблон:Math-length window function (<math>x</math>) 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 Шаблон:Math. To sample <math>x</math> at the same frequencies, for comparison, the DFT is computed for one cycle of the periodic summation, <math>x_{_N}.</math>Шаблон:Efn-ua
Шаблон:AnchorCase: Frequency interpolation. Шаблон:Math
In this case, the DFT simplifies to a more familiar form:
- <math>X_k = \sum_{n=0}^{N-1} x[n]\cdot e^{-i 2\pi \frac{k}{N}n}.</math>
In order to take advantage of a fast Fourier transform algorithm for computing the DFT, the summation is usually performed over all Шаблон:Mvar terms, even though Шаблон:Math of them are zeros. Therefore, the case Шаблон:Math is often referred to as zero-padding.
Spectral leakage, which increases as Шаблон:Mvar 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 Шаблон:Math 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:
- <math>x[n] = e^{i 2\pi \frac{1}{8} n},\quad </math> and <math>L=64.</math>
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: Шаблон:Math. Also visible in Fig 2 is the spectral leakage pattern of the Шаблон:Math 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
Шаблон:Main The convolution theorem for sequences is:
- <math>x * y\ =\ \scriptstyle{\rm DTFT}^{-1} \displaystyle \left[\scriptstyle{\rm DTFT} \displaystyle \{x\}\cdot \scriptstyle{\rm DTFT} \displaystyle \{y\}\right].</math>[16]Шаблон:RpШаблон:Efn-la
An important special case is the circular convolution of sequences Шаблон:Mvar and Шаблон:Mvar defined by <math>x_{_N}*y,</math> where <math>x_{_N}</math> is a periodic summation. The discrete-frequency nature of <math>\scriptstyle{\rm DTFT} \displaystyle \{x_{_N}\}</math> means that the product with the continuous function <math>\scriptstyle{\rm DTFT} \displaystyle \{y\}</math> is also discrete, which results in considerable simplification of the inverse transform:
- <math>x_{_N} * y\ =\ \scriptstyle{\rm DTFT}^{-1} \displaystyle \left[\scriptstyle{\rm DTFT} \displaystyle \{x_{_N}\}\cdot \scriptstyle{\rm DTFT} \displaystyle \{y\}\right]\ =\ \scriptstyle{\rm DFT}^{-1} \displaystyle \left[\scriptstyle{\rm DFT} \displaystyle \{x_{_N}\}\cdot \scriptstyle{\rm DFT} \displaystyle \{y_{_N}\}\right].</math>[17][1]Шаблон:Rp
For Шаблон:Mvar and Шаблон:Mvar sequences whose non-zero duration is less than or equal to Шаблон:Mvar, a final simplification is:
- <math>x_{_N} * y\ =\ \scriptstyle{\rm DFT}^{-1} \displaystyle \left[\scriptstyle{\rm DFT} \displaystyle \{x\}\cdot \scriptstyle{\rm DFT} \displaystyle \{y\}\right].</math>
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 one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform:[16]Шаблон:Rp
<math> \begin{align} \mathsf{Time\ domain} \quad &\ x \quad &= \quad & x_{_{RE}} \quad &+ \quad & x_{_{RO}} \quad &+ \quad i\ & x_{_{IE}} \quad &+ \quad &\underbrace{i\ x_{_{IO}}} \\ &\Bigg\Updownarrow\mathcal{F} & &\Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F}\\ \mathsf{Frequency\ domain} \quad &X \quad &= \quad & X_{RE} \quad &+ \quad &\overbrace{i\ X_{IO}} \quad &+ \quad i\ & X_{IE} \quad &+ \quad & X_{RO} \end{align} </math>
From this, various relationships are apparent, for example:
- The transform of a real-valued function (Шаблон:Math) is the even symmetric function Шаблон:Math. Conversely, an even-symmetric transform implies a real-valued time-domain.
- The transform of an imaginary-valued function (Шаблон:Math) is the odd symmetric function Шаблон:Math, and the converse is true.
- The transform of an even-symmetric function (Шаблон:Math) is the real-valued function Шаблон:Math, and the converse is true.
- The transform of an odd-symmetric function (Шаблон:Math) is the imaginary-valued function Шаблон:Math, and the converse is true.
Relationship to the Z-transform
<math>X_{2\pi}(\omega)</math> is a Fourier series that can also be expressed in terms of the bilateral Z-transform. I.e.:
- <math>X_{2\pi}(\omega) = \left. \widehat X(z) \, \right|_{z = e^{i \omega}} = \widehat X(e^{i \omega}),</math>
where the <math>\widehat X</math> 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:
- <math>
\begin{align} \widehat X(e^{i \omega}) &= \ X_{1/T}\left(\tfrac{\omega}{2\pi T}\right)
\ = \ \sum_{k=-\infty}^{\infty} X\left(\tfrac{\omega}{2\pi T} - k/T\right)\\
&= \sum_{k=-\infty}^{\infty} X\left(\tfrac{\omega - 2\pi k}{2\pi T} \right). \end{align} </math>
Note that when parameter Шаблон:Mvar changes, the terms of <math>X_{2\pi}(\omega)</math> remain a constant separation <math>2 \pi</math> apart, and their width scales up or down. The terms of Шаблон:Math remain a constant width and their separation Шаблон:Math scales up or down.
Table of discrete-time Fourier transforms
Some common transform pairs are shown in the table below. The following notation applies:
- <math>\omega=2 \pi f T</math> is a real number representing continuous angular frequency (in radians per sample). (<math>f</math> is in cycles/sec, and <math>T</math> is in sec/sample.) In all cases in the table, the DTFT is 2π-periodic (in <math>\omega</math>).
- <math>X_{2\pi}(\omega)</math> designates a function defined on <math>-\infty < \omega < \infty </math>.
- <math>X_o(\omega)</math> designates a function defined on <math>-\pi < \omega \le \pi</math>, and zero elsewhere. Then: <math display="block">X_{2\pi}(\omega)\ \triangleq \sum_{k=-\infty}^{\infty} X_o(\omega - 2\pi k).</math>
- <math>\delta ( \omega )</math> is the Dirac delta function
- <math>\operatorname{sinc} (t)</math> is the normalized sinc function
- <math>\operatorname{rect}\left[{n \over L}\right] \triangleq \begin{cases}
1 & |n|\leq L/2 \\ 0 & |n| > L/2 \end{cases}</math>
- <math>\operatorname{tri} (t)</math> is the triangle function
- Шаблон:Mvar is an integer representing the discrete-time domain (in samples)
- <math>u[n]</math> is the discrete-time unit step function
- <math>\delta[n]</math> is the Kronecker delta <math>\delta_{n,0}</math>
| Time domain x[n] |
Frequency domain X2π(ω) |
Remarks | Reference |
|---|---|---|---|
| <math>\delta[n]</math> | <math>X_{2\pi}(\omega) = 1</math> | [16]Шаблон:Rp | |
| <math>\delta[n-M]</math> | <math>X_{2\pi}(\omega) = e^{-i\omega M}</math> | integer <math>M</math> | |
| <math>\sum_{m = -\infty}^{\infty} \delta[n - M m] \!</math> | <math>X_{2\pi}(\omega) = \sum_{m = -\infty}^{\infty} e^{-i \omega M m} = \frac{2\pi}{M}\sum_{k = -\infty}^{\infty} \delta \left( \omega - \frac{2\pi k}{M} \right) \,</math> <math>X_o(\omega) = \frac{2\pi}{M}\sum_{k = -(M-1)/2}^{(M-1)/2} \delta \left(\omega - \frac{2\pi k}{M} \right) \,</math> odd M |
integer <math>M > 0 </math> | |
| <math>u[n]</math> | <math>X_{2\pi}(\omega) = \frac{1}{1-e^{-i \omega}} + \pi \sum_{k=-\infty}^{\infty} \delta (\omega - 2\pi k)\!</math> <math>X_o(\omega) = \frac{1}{1-e^{-i \omega}} + \pi \cdot \delta (\omega)\!</math> |
The <math>1/(1-e^{-i \omega})</math> term must be interpreted as a distribution in the sense of a Cauchy principal value around its poles at <math>\omega=2 \pi k</math>. | |
| <math>a^n u[n]</math> | <math>X_{2\pi}(\omega) = \frac{1}{1-a e^{-i \omega}}\!</math> | a| < 1 </math> | [16]Шаблон:Rp |
| <math>e^{-i a n}</math> | <math>X_o(\omega) = 2\pi\cdot \delta (\omega +a),</math> -π < a < π <math>X_{2\pi}(\omega) = 2\pi \sum_{k=-\infty}^{\infty} \delta (\omega +a -2\pi k)</math> |
real number <math>a</math> | |
| <math>\cos(a\cdot n)</math> | <math>X_o(\omega) = \pi \left[\delta \left(\omega -a\right)+\delta \left(\omega +a\right)\right],</math> <math>X_{2\pi}(\omega)\ \triangleq \sum_{k=-\infty}^{\infty} X_o(\omega - 2\pi k)</math> |
real number <math>a</math> with <math>-\pi < a < \pi</math> | |
| <math>\sin(a\cdot n)</math> | <math>X_o(\omega) = \frac{\pi}{i} \left[\delta \left(\omega -a\right)-\delta \left(\omega +a\right)\right]</math> | real number <math>a</math> with <math>-\pi < a < \pi</math> | |
| <math>\operatorname{rect} \left[ { n - M \over N } \right] \equiv \operatorname{rect} \left[ { n - M \over N-1 } \right]</math> | <math>X_o(\omega) = { \sin( N \omega / 2 ) \over \sin( \omega / 2 ) } \, e^{ -i \omega M } \!</math> | integer <math>M,</math> and odd integer <math>N</math> | |
| <math>\operatorname{sinc} ( W (n+a))</math> | <math>X_o(\omega) = \frac{1}{W} \operatorname{rect} \left( { \omega \over 2\pi W } \right) e^{ia\omega}</math> | real numbers <math>W,a</math> with <math>0 < W < 1</math> | |
| <math>\operatorname{sinc}^2(W n)\,</math> | <math>X_o(\omega) = \frac{1}{W} \operatorname{tri} \left( { \omega \over 2\pi W } \right)</math> | real number <math>W</math>, <math>0 < W < 0.5</math> | |
| <math> \begin{cases}
0 & n=0 \\ \frac{(-1)^n}{n} & \text{elsewhere} \end{cases}</math> |
<math>X_o(\omega) = j \omega</math> | it works as a differentiator filter | |
| <math>\frac{1}{(n + a)} \left\{ \cos [ \pi W (n+a)] - \operatorname{sinc} [ W (n+a)] \right\}</math> | <math>X_o(\omega) = \frac{j \omega}{W} \cdot \operatorname{rect} \left( { \omega \over \pi W } \right) e^{j a \omega}</math> | real numbers <math>W,a</math> with <math>0 < W < 1 </math> | |
| <math>\begin{cases}
\frac{\pi}{2} & n = 0 \\ \frac{(-1)^n - 1}{\pi n^2} & \text{ otherwise} \end{cases}</math> |
\omega|</math> | ||
| <math>\begin{cases}
0; & n \text{ even} \\ \frac{2}{\pi n} ; & n \text{ odd} \end{cases}</math> |
<math>X_o(\omega) = \begin{cases}
j & \omega < 0 \\ 0 & \omega = 0 \\ -j & \omega > 0 \end{cases}</math> |
Hilbert transform | |
| <math>\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]</math> | <math>X_o(\omega) = </math>Файл:Trapezoid signal.svg | real numbers <math>A,B</math> complex <math>C</math> |
Properties
This table shows some mathematical operations in the time domain and the corresponding effects in the frequency domain.
- <math>*\!</math> is the discrete convolution of two sequences
- <math>x[n]^{*}</math> is the complex conjugate of Шаблон:Math.
| Property | Time domain Шаблон:Math |
Frequency domain <math>X_{2\pi}(\omega)</math> |
Remarks | Reference |
|---|---|---|---|---|
| Linearity | <math>a\cdot x[n] + b\cdot y[n]</math> | <math>a\cdot X_{2\pi}(\omega) + b\cdot Y_{2\pi}(\omega)</math> | complex numbers <math>a,b</math> | [16]Шаблон:Rp |
| Time reversal / Frequency reversal | <math>x[-n]</math> | <math>X_{2\pi}(-\omega) \!</math> | [16]Шаблон:Rp | |
| Time conjugation | <math>x[n]^*</math> | <math>X_{2\pi}(-\omega)^* \!</math> | [16]Шаблон:Rp | |
| Time reversal & conjugation | <math>x[-n]^*</math> | <math>X_{2\pi}(\omega)^* \!</math> | [16]Шаблон:Rp | |
| Real part in time | <math>\Re{(x[n])}</math> | <math>\frac{1}{2}(X_{2\pi}(\omega) + X_{2\pi}^*(-\omega))</math> | [16]Шаблон:Rp | |
| Imaginary part in time | <math>\Im{(x[n])}</math> | <math>\frac{1}{2i}(X_{2\pi}(\omega) - X_{2\pi}^*(-\omega))</math> | [16]Шаблон:Rp | |
| Real part in frequency | <math>\frac{1}{2}(x[n]+x^*[-n])</math> | <math>\Re{(X_{2\pi}(\omega))}</math> | [16]Шаблон:Rp | |
| Imaginary part in frequency | <math>\frac{1}{2i}(x[n]-x^*[-n])</math> | <math>\Im{(X_{2\pi}(\omega))}</math> | [16]Шаблон:Rp | |
| Shift in time / Modulation in frequency | <math>x[n-k]</math> | <math>X_{2\pi}(\omega)\cdot e^{-i\omega k}</math> | integer Шаблон:Mvar | [16]Шаблон:Rp |
| Shift in frequency / Modulation in time | <math>x[n]\cdot e^{ian} \!</math> | <math>X_{2\pi}(\omega-a) \!</math> | real number <math>a</math> | [16]Шаблон:Rp |
| Decimation | <math>x[nM]</math> | <math>\frac{1}{M}\sum_{m=0}^{M-1} X_{2\pi}\left(\tfrac{\omega - 2\pi m}{M}\right) \!</math> Шаблон:Efn-ua | integer <math>M</math> | |
| Time Expansion | <math> \scriptstyle \begin{cases}
x[n/M] & n=\text{multiple of M} \\ 0 & \text{otherwise} \end{cases}</math> |
<math>X_{2\pi}(M \omega) \!</math> | integer <math>M</math> | [1]Шаблон:Rp |
| Derivative in frequency | <math>\frac{n}{i} x[n] \!</math> | <math>\frac{d X_{2\pi}(\omega)}{d \omega} \!</math> | [16]Шаблон:Rp | |
| Integration in frequency | <math> \!</math> | <math> \!</math> | ||
| Differencing in time | <math> x[n]-x[n-1] \!</math> | <math> \left( 1-e^{-i \omega} \right) X_{2\pi}(\omega) \!</math> | ||
| Summation in time | <math> \sum_{m=-\infty}^{n} x[m] \!</math> | <math> \frac{1}{\left( 1-e^{-i \omega} \right)} X_{2\pi}(\omega) + \pi X(0) \sum_{k=-\infty}^{\infty} \delta(\omega-2\pi k) \!</math> | ||
| Convolution in time / Multiplication in frequency | <math>x[n] * y[n] \!</math> | <math>X_{2\pi}(\omega) \cdot Y_{2\pi}(\omega) \!</math> | [16]Шаблон:Rp | |
| Multiplication in time / Convolution in frequency | <math>x[n] \cdot y[n] \!</math> | <math>\frac{1}{2\pi}\int_{-\pi}^{\pi}X_{2\pi}(\nu) \cdot Y_{2\pi}(\omega-\nu) d\nu \!</math> | Periodic convolution | [16]Шаблон:Rp |
| Cross correlation | <math>\rho_{xy} [n] = x[-n]^* * y[n] \!</math> | <math>R_{xy} (\omega) = X_{2\pi}(\omega)^* \cdot Y_{2\pi}(\omega) \!</math> | ||
| Parseval's theorem | <math>E_{xy} = \sum_{n=-\infty}^{\infty} {x[n] \cdot y[n]^*} \!</math> | <math>E_{xy} = \frac{1}{2\pi}\int_{-\pi}^{\pi}{X_{2\pi}(\omega) \cdot Y_{2\pi}(\omega)^* d\omega} \!</math> | [16]Шаблон:Rp |
See also
Notes
Page citations
References
Шаблон:Reflist Шаблон:Refbegin
Further reading
- ↑ 1,0 1,1 1,2 1,3 1,4 1,5 Ошибка цитирования Неверный тег
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<ref>; для сносокGumasне указан текст - ↑ Ошибка цитирования Неверный тег
<ref>; для сносокCrochiereне указан текст - ↑ Ошибка цитирования Неверный тег
<ref>; для сносокWangне указан текст - ↑ Ошибка цитирования Неверный тег
<ref>; для сносокLyonsне указан текст - ↑ 9,0 9,1 Ошибка цитирования Неверный тег
<ref>; для сносокLillingtonне указан текст - ↑ 10,0 10,1 Ошибка цитирования Неверный тег
<ref>; для сносокLillington2не указан текст - ↑ Ошибка цитирования Неверный тег
<ref>; для сносокHochgürtelне указан текст - ↑ Ошибка цитирования Неверный тег
<ref>; для сносокChennamangalamне указан текст - ↑ Ошибка цитирования Неверный тег
<ref>; для сносокDahlне указан текст - ↑ Ошибка цитирования Неверный тег
<ref>; для сносокLinне указан текст - ↑ Ошибка цитирования Неверный тег
<ref>; для сносокHarrisbookне указан текст - ↑ 16,00 16,01 16,02 16,03 16,04 16,05 16,06 16,07 16,08 16,09 16,10 16,11 16,12 16,13 16,14 16,15 16,16 16,17 Ошибка цитирования Неверный тег
<ref>; для сносокProakisне указан текст - ↑ Ошибка цитирования Неверный тег
<ref>; для сносокRabinerне указан текст
