डिस्क्रीट टाइम फुरिअर ट्रान्सफार्म
गणित में डिस्क्रीट टाइम फुरिअर ट्रान्सफार्म या डीटीएफटी (discrete-time Fourier transform or DTFT), फुरिअर विश्लेषण के कई रुपों में से एक रूप है। यह अनन्त तक परिभाषित किसी अनावर्ती (नॉन्-पेरिऑडिक) डिस्क्रीट-टाइम सेक्वेंस को रूपानतरित करता है। इसे यह भी कहते हैं कि समय-डोमेन का आंकड़ा आवृत्ति-डोमेन में बदल गया। डीटीएफटी द्वारा प्राप्त आवृत्ति-डोमेन का आंकड़ा सतत (कांटिन्युअस) एवं आवर्ती होता है।
डीटीएफटी की परिभाषा संपादित करें
यदि कोई वास्तविक (real) या समिश्र (complex) संख्याओं का समुच्चय : (पूर्णांक), दिया हो तो का डीटीएफटी प्रायः इस प्रकार व्यक्त किया जाता है:
![{\displaystyle x[n],\;n\in \mathbb {Z} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/83b783e3bb4f41250a3bce26399b1885f0e2c975)
![{\displaystyle x[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b575dce543a2cf10a5a3e108204b928c2c9aaa54)
![{\displaystyle X(\omega )=\sum _{n=-\infty }^{\infty }x[n]\,e^{-i\omega n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2c9ad19546738b0bdd7606437609877a442b1ce)
व्युत्क्रम रुपान्तर (Inverse transform) संपादित करें
निम्नलिखित रुपान्तर करने पर डिस्क्रीट-टाइम सेक्वेंस फिर से प्राप्त हो जायेगा:
The integrals span one full period of the DTFT, which means that the x[n] samples are also the coefficients of a Fourier series expansion of the DTFT. Infinite limits of integration change the transform into a continuous-time Fourier transform [inverse], which produces a sequence of Dirac impulses. That is:
![{\displaystyle {\begin{aligned}\int _{-\infty }^{\infty }X_{T}(f)\cdot e^{i2\pi ft}\,df&=\int _{-\infty }^{\infty }\left(T\sum _{n=-\infty }^{\infty }x(nT)\ e^{-i2\pi fTn}\right)\cdot e^{i2\pi ft}\,df\\&=\sum _{n=-\infty }^{\infty }T\cdot x(nT)\int _{-\infty }^{\infty }e^{-i2\pi fTn}\cdot e^{i2\pi ft}\,df\\&=\sum _{n=-\infty }^{\infty }x[n]\cdot \delta (t-nT).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/decfa7832a2d257a08b027c3605593b3e80ab57f)
डीटीएफटी की सूची संपादित करें
नीचे कुछ मानक डिस्क्रीट टाइम सेक्वेंस एवं उनके डीटीएफटी रुपानतर दिये हुए हैं। इसमें प्रयुक्त प्रतीकों का अर्थ निम्नवत है:
- is an integer representing the discrete-time domain (in samples)
- is a real number in , representing continuous angular frequency (in radians per sample).
- The remainder of the transform is defined by:
- is the discrete-time unit step function
- is the normalized sinc function
- is the Dirac delta function
- is the Kronecker delta
- is the rectangle function for arbitrary real-valued t:
![{\displaystyle u[n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2911183bef1ad935d3bc8b5c1be97bac439b7f)
![{\displaystyle \delta [n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1553d02b03c2a79a43f7862ebfb8352705b0b87c)
![{\displaystyle \mathrm {rect} (t)=\sqcap (t)={\begin{cases}0&{\mbox{if }}|t|>{\frac {1}{2}}\\[3pt]{\frac {1}{2}}&{\mbox{if }}|t|={\frac {1}{2}}\\[3pt]1&{\mbox{if }}|t|<{\frac {1}{2}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3585cde90bc1dfbce7b14531690022ad0a7b3a6)
- is the triangle function for arbitrary real-valued t:
| Time domain |
Frequency domain |
Remarks |
|---|---|---|
| integer M | ||
| integer M | ||
| real number a | ||
| real number a | ||
| real number a | ||
| integer M | ||
| real number a | ||
| real number W | ||
| real numbers W, a | ||
| it works as a differentiator filter | ||
| real numbers W, a | ||
| Hilbert transform | ||
|
real numbers A, B complex C |
![{\displaystyle \delta [n-M]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2dd610c46bdeb2682e719b6a0b445c1cf1893639)
![{\displaystyle \sum _{m=-\infty }^{\infty }\delta [n-Mm]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ce55d69101c8a851f1b8f09c0492852e8e9e49)
![{\displaystyle \pi \left[\delta (\omega -a)+\delta (\omega +a)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b949fed13f25129a9f8634cb4c0aa34354b0320a)
![{\displaystyle {\frac {\pi }{i}}\left[\delta (\omega -a)-\delta (\omega +a)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b48e8676730fb37bcadaea9b1bb342d107cf2f9)
![{\displaystyle \mathrm {rect} \left[{(n-M/2) \over M}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa40f5391166dc137b30a45584c1de1a33b2f8c)
![{\displaystyle {\sin[\omega (M+1)/2] \over \sin(\omega /2)}\,e^{-i\omega M/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/883ef01f74f9697fd3a68706785bc6e702c8f961)
![{\displaystyle \operatorname {sinc} [(a+n)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3526652071fa8be34f6bccac88dc94260441a47f)
![{\displaystyle W\cdot \operatorname {sinc} [W(n+a)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4185f56193a63b426f87329e9d7c0d1ae50fced)
![{\displaystyle {\frac {W}{(n+a)}}\left\{\cos[\pi W(n+a)]-\operatorname {sinc} [W(n+a)]\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d4cd8e41d9e483f5d90da18cff3661d914ce9b)
![{\displaystyle {\frac {1}{\pi n^{2}}}[(-1)^{n}-1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4114640d8ab7e22e07fe380574400fb036941530)
![{\displaystyle {\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38083aff1d4f22b4848dfafdc988b43142f7c472)
डीटिएफटी के गुणधर्म संपादित करें
This table shows the relationships between generic discrete-time Fourier transforms. We use the following notation:
- is the convolution between two signals
- is the complex conjugate of the function x[n]
- represents the correlation between x[n] and y[n].
![{\displaystyle x[n]^{*}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e49908fe3dfbd435e478d58b4c10c47e3a609c96)
![{\displaystyle \rho _{xy}[n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/358049d59c74edde52765a758324ab97be7ee3a5)
The first column provides a description of the property, the second column shows the function in the time domain, the third column shows the spectrum in the frequency domain:
| Property | Time domain | Frequency domain | Remarks |
|---|---|---|---|
| Linearity | |||
| Shift in time | integer k | ||
| Shift in frequency (modulation) | real number a | ||
| Time reversal | |||
| Time conjugation | |||
| Time reversal & conjugation | |||
| Derivative in frequency | |||
| Integral in frequency | |||
| Convolve in time | |||
| Multiply in time | |||
| Correlation |
![{\displaystyle x[n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91384637c5188ffed9b7929f145a78fb314c4141)
![{\displaystyle ax[n]+by[n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c30c8f75093b9bafb5c5bde1392348701ce3be0)
![{\displaystyle x[n-k]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3bfc2b24164de1192ea1e17a90312a3e045911e)
![{\displaystyle x[n]e^{ian}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5f540caf14c269c45b6b5d32ceb555ce8841bae)
![{\displaystyle x[-n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61499a6c6544d338cf6ac340b27ad734c4daba05)
![{\displaystyle x[-n]^{*}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb208f439fc83acd99c3db8d63304b65890993c4)
![{\displaystyle {\frac {n}{i}}x[n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2eb71af662480e119b9bd16ff2e4c49151ab4b9)
![{\displaystyle {\frac {i}{n}}x[n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/384ec2e6f1fb841111d00c800196000a4d406964)
![{\displaystyle x[n]*y[n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3b1fd5ad225be66dfa022f43a3333b08fb3fb7)
![{\displaystyle x[n]\cdot y[n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3e208ac041ff192194184d3c708d7789f9c9178)
![{\displaystyle \rho _{xy}[n]=x[-n]^{*}*y[n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37d82c9624d4b16650c5652c000d0bb459f794c6)
सममिति के गुण (Symmetry Properties) संपादित करें
फुरिअर रुपान्तर, वास्तविक एवं काल्पनिक (real and imaginary) या सम एवं विषम (even and odd) के योग के रूप में व्यक्त की जा सकती है।
या
| Time Domain |
Frequency Domain |
|---|---|
![{\displaystyle x^{*}[n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4793a5489c4d67f5ffcc700e9ba06acd87e231c3)
![{\displaystyle x^{*}[-n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34e0bcbaa111fee79bfcf34b83f9f3b6303ddaed)
