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।

यदि कोई वास्तविक (real) या समिश्र (complex) संख्याओं का समुच्चय : ${\displaystyle x[n],\;n\in \mathbb {Z} }$ (पूर्णांक), दिया हो तो ${\displaystyle x[n]\,}$ का डीटीएफटी प्रायः इस प्रकार व्यक्त किया जाता है:

${\displaystyle X(\omega )=\sum _{n=-\infty }^{\infty }x[n]\,e^{-i\omega n}.}$

निम्नलिखित रुपान्तर करने पर डिस्क्रीट-टाइम सेक्वेंस फिर से प्राप्त हो जायेगा:

${\displaystyle x[n]\,}$	${\displaystyle ={\frac {1}{2\pi }}\int _{-\pi }^{\pi }X(\omega )\cdot e^{i\omega n}\,d\omega }$
	${\displaystyle =T\int _{-{\frac {1}{2T}}}^{\frac {1}{2T}}X_{T}(f)\cdot e^{i2\pi fnT}\,df.}$

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:

${\displaystyle {\begin{aligned}\int _{-\infty }^{\infty }X_{T}(f)\cdot e^{i2\pi ft}\,df&=\int _{-\infty }^{\infty }\left(T\sum _{n=-\infty }^{\infty }x(nT)\ e^{-i2\pi fTn}\right)\cdot e^{i2\pi ft}\,df\\&=\sum _{n=-\infty }^{\infty }T\cdot x(nT)\int _{-\infty }^{\infty }e^{-i2\pi fTn}\cdot e^{i2\pi ft}\,df\\&=\sum _{n=-\infty }^{\infty }x[n]\cdot \delta (t-nT).\end{aligned}}}$

नीचे कुछ मानक डिस्क्रीट टाइम सेक्वेंस एवं उनके डीटीएफटी रुपानतर दिये हुए हैं। इसमें प्रयुक्त प्रतीकों का अर्थ निम्नवत है:

• ${\displaystyle n\!}$ is an integer representing the discrete-time domain (in samples)
• ${\displaystyle \omega \!}$ is a real number in ${\displaystyle (-\pi ,\ \pi )}$, representing continuous angular frequency (in radians per sample).
  • The remainder of the transform ${\displaystyle (|\omega |>\pi \,)}$ is defined by: ${\displaystyle X(\omega +2\pi k)=X(\omega )\,}$
• ${\displaystyle u[n]\!}$ is the discrete-time unit step function
• ${\displaystyle \operatorname {sinc} (t)\!}$ is the normalized sinc function
• ${\displaystyle \delta (\omega )\!}$ is the Dirac delta function
• ${\displaystyle \delta [n]\!}$ is the Kronecker delta ${\displaystyle \delta _{n,0}\!}$
• ${\displaystyle \operatorname {rect} (t)}$ is the rectangle function for arbitrary real-valued t:

${\displaystyle \mathrm {rect} (t)=\sqcap (t)={\begin{cases}0&{\mbox{if }}|t|>{\frac {1}{2}}\\[3pt]{\frac {1}{2}}&{\mbox{if }}|t|={\frac {1}{2}}\\[3pt]1&{\mbox{if }}|t|<{\frac {1}{2}}\end{cases}}}$

• ${\displaystyle \operatorname {tri} (t)}$ is the triangle function for arbitrary real-valued t:

${\displaystyle \operatorname {tri} (t)=\land (t)={\begin{cases}1+t;&-1\leq t\leq 0\\1-t;&0<t\leq 1\\0&{\mbox{otherwise}}\end{cases}}}$

Time domain ${\displaystyle x[n]\,}$	Frequency domain ${\displaystyle X(\omega )\,}$	Remarks
${\displaystyle \delta [n]\!}$	${\displaystyle 1\!}$
${\displaystyle \delta [n-M]\!}$	${\displaystyle e^{-i\omega M}\!}$	integer M
${\displaystyle \sum _{m=-\infty }^{\infty }\delta [n-Mm]\,}$	${\displaystyle \sum _{m=-\infty }^{\infty }e^{-i\omega Mm}={\frac {1}{M}}\sum _{k=-\infty }^{\infty }\delta \left({\frac {\omega }{2\pi }}-{\frac {k}{M}}\right)\,}$	integer M
${\displaystyle u[n]\!}$	${\displaystyle {\frac {1}{1-e^{-i\omega }}}\!}$
${\displaystyle e^{-ian}\!}$	${\displaystyle 2\pi \delta (\omega +a)\,}$	real number a
${\displaystyle \cos(an)\!}$	${\displaystyle \pi \left[\delta (\omega -a)+\delta (\omega +a)\right]}$	real number a
${\displaystyle \sin(an)\!}$	${\displaystyle {\frac {\pi }{i}}\left[\delta (\omega -a)-\delta (\omega +a)\right]}$	real number a
${\displaystyle \mathrm {rect} \left[{(n-M/2) \over M}\right]}$	${\displaystyle {\sin[\omega (M+1)/2] \over \sin(\omega /2)}\,e^{-i\omega M/2}}$	integer M
${\displaystyle \operatorname {sinc} [(a+n)]}$	${\displaystyle e^{ia\omega }\!}$	real number a
${\displaystyle W\cdot \operatorname {sinc} ^{2}(Wn)\,}$	${\displaystyle \operatorname {tri} \left({\omega \over 2\pi W}\right)}$	real number W ${\displaystyle 0<W\leq 0.5}$
${\displaystyle W\cdot \operatorname {sinc} [W(n+a)]}$	${\displaystyle \operatorname {rect} \left({\omega \over 2\pi W}\right)\cdot e^{ja\omega }}$	real numbers W, a ${\displaystyle 0<W\leq 1}$
${\displaystyle {\begin{cases}0&n=0\\{\frac {(-1)^{n}}{n}}&{\mbox{elsewhere}}\end{cases}}}$	${\displaystyle j\omega }$	it works as a differentiator filter
${\displaystyle {\frac {W}{(n+a)}}\left\{\cos[\pi W(n+a)]-\operatorname {sinc} [W(n+a)]\right\}}$	${\displaystyle j\omega \cdot \operatorname {rect} \left({\omega \over \pi W}\right)e^{ja\omega }}$	real numbers W, a ${\displaystyle 0<W\leq 1}$
${\displaystyle {\frac {1}{\pi n^{2}}}[(-1)^{n}-1]}$	${\displaystyle |\omega |\!}$
${\displaystyle {\begin{cases}0;&n{\mbox{ odd}}\\{\frac {2}{\pi n}};&n{\mbox{ even}}\end{cases}}}$	${\displaystyle {\begin{cases}j&\omega <0\\0&\omega =0\\-j&\omega >0\end{cases}}}$	Hilbert transform
${\displaystyle {\frac {C(A+B)}{2\pi }}\cdot \operatorname {sinc} \left[{\frac {A-B}{2\pi }}n\right]\cdot \operatorname {sinc} \left[{\frac {A+B}{2\pi }}n\right]}$		real numbers A, B complex C

This table shows the relationships between generic discrete-time Fourier transforms. We use the following notation:

• ${\displaystyle *\!}$ is the convolution between two signals
• ${\displaystyle x[n]^{*}\!}$ is the complex conjugate of the function x[n]
• ${\displaystyle \rho _{xy}[n]\!}$ represents the correlation between x[n] and y[n].

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 ${\displaystyle x[n]\!}$	Frequency domain ${\displaystyle X(\omega )\!}$	Remarks
Linearity	${\displaystyle ax[n]+by[n]\!}$	${\displaystyle aX(e^{i\omega })+bY(e^{i\omega })\!}$
Shift in time	${\displaystyle x[n-k]\!}$	${\displaystyle X(e^{i\omega })e^{-i\omega k}\!}$	integer k
Shift in frequency (modulation)	${\displaystyle x[n]e^{ian}\!}$	${\displaystyle X(e^{i(\omega -a)})\!}$	real number a
Time reversal	${\displaystyle x[-n]\!}$	${\displaystyle X(e^{-i\omega })\!}$
Time conjugation	${\displaystyle x[n]^{*}\!}$	${\displaystyle X(e^{-i\omega })^{*}\!}$
Time reversal & conjugation	${\displaystyle x[-n]^{*}\!}$	${\displaystyle X(e^{i\omega })^{*}\!}$
Derivative in frequency	${\displaystyle {\frac {n}{i}}x[n]\!}$	${\displaystyle {\frac {dX(e^{i\omega })}{d\omega }}\!}$
Integral in frequency	${\displaystyle {\frac {i}{n}}x[n]\!}$	${\displaystyle \int _{-\pi }^{\omega }X(e^{i\vartheta })d\vartheta \!}$
Convolve in time	${\displaystyle x[n]*y[n]\!}$	${\displaystyle X(e^{i\omega })\cdot Y(e^{i\omega })\!}$
Multiply in time	${\displaystyle x[n]\cdot y[n]\!}$	${\displaystyle {\frac {1}{2\pi }}X(e^{i\omega })*Y(e^{i\omega })\!}$
Correlation	${\displaystyle \rho _{xy}[n]=x[-n]^{*}*y[n]\!}$	${\displaystyle R_{xy}(\omega )=X(e^{i\omega })^{*}\cdot Y(e^{i\omega })\!}$

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।
${\displaystyle X(e^{i\omega })=X_{R}(e^{i\omega })+iX_{I}(e^{i\omega })\!}$
या
${\displaystyle X(e^{i\omega })=X_{E}(e^{i\omega })+X_{O}(e^{i\omega })\!}$

Time Domain ${\displaystyle x[n]\!}$	Frequency Domain ${\displaystyle X(e^{i\omega })\!}$
${\displaystyle x^{*}[n]\!}$	${\displaystyle X^{*}(e^{-i\omega })\!}$
${\displaystyle x^{*}[-n]\!}$	${\displaystyle X^{*}(e^{i\omega })\!}$