Engineering Tables/DTFT Transform Table

From Wikibooks, open books for an open world
< Engineering Tables
Jump to: navigation, search


The information in the table below may be inaccurate - The DTFT transfers into the periodic frequency domain. The signals shown in the table are not periodic, and hence they are obviously wrong. The table seems to list ordinary Fourier transforms for angular frequency, which coincidentally also uses the greek letter ω, although with a different meaning.

Time domain
 x(n) \, where n\in\Z
Frequency domain
  X(\omega) \, where \omega\in [-\pi,\pi)
Remarks
\delta (n) \! 1 \! Here \delta (n) \! represents the delta function
which is 1 if n=0 and zero otherwise.
\delta (n - k) \! e^{-i k \omega} \! for some k \in \mathbb{Z}
u(n)=\begin{cases}0 &\text{for } n\\ 1 &\text{for } n\geq 0\end{cases} \frac{1}{1-e^{-i \omega}} \!
e^{-ian} \!    \delta (\omega + a)  \, For any real number  a \,. Here \delta(\omega) is a periodic Dirac delta function. That is
\textstyle \delta(\omega)=\delta(\omega+2\pi), \textstyle\int_{-\pi}^\pi\delta(\omega)\,d\omega=1 and informally \delta(\omega)=0 if \omega\neq 0
\cos (a n) \! \frac{1}{2} \Big( \delta (\omega - a) + \delta (\omega + a) \Big) a \in \mathbb{R}
\sin (a n) \! \frac{1}{2 i} \Big( \delta (\omega - a) - \delta ( \omega + a) \Big) a \in \mathbb{R}
 \mathrm{rect} \left( { ( n - M/2 ) \over M  } \right)  { \sin\Big( \omega (M+1) / 2 \Big) \over \sin( \omega / 2 ) } \,  e^{ -i \omega M / 2 } M \in \mathbb{Z}
\operatorname{sinc} (a + n) e^{i a \omega} \! a \in \mathbb{R}
W\cdot \operatorname{sinc}^2(W n)\, { 1 \over 2\pi W}\cdot \operatorname{tri} \left( { \omega \over 2\pi W } \right) W \in \mathbb{Z}
W\cdot \operatorname{sinc} \Big( W (n + a)\Big) \operatorname{rect} \left( { \omega \over 2\pi W } \right) \cdot e^{j a \omega} a, W \in \mathbb{R}

0 < W \le 1

\frac{W}{(n + a)} \Big[ \cos \big( \pi W (n+a)\big) - \operatorname{sinc} \big( W (n+a)\big) \Big] j \omega \cdot \operatorname{rect} \left( { \omega \over \pi W } \right) e^{j a \omega} a, W \in \mathbb{R}
\frac{1}{\pi n^2} \Big[(-1)^n - 1\Big] | \omega | \!
\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) Trapezoid signal.svg A, B \in \mathbb{R}

C \in \mathbb{C}