CTFT Table

	Time Domain	Frequency Domain
	${\displaystyle x(t)={\mathcal {F}}^{-1}\left\{X(\omega )\right\}}$	${\displaystyle X(\omega )={\mathcal {F}}\left\{x(t)\right\}}$
1	${\displaystyle X(j\omega )=\int _{-\infty }^{\infty }x(t)e^{-j\omega t}dt}$	${\displaystyle x(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\omega )e^{j\omega t}d\omega }$
2	${\displaystyle 1\,}$	${\displaystyle 2\pi \delta (\omega )\,}$
3	${\displaystyle -0.5+u(t)\,}$	${\displaystyle {\frac {1}{j\omega }}\,}$
4	${\displaystyle \delta (t)\,}$	${\displaystyle 1\,}$
5	${\displaystyle \delta (t-c)\,}$	${\displaystyle e^{-j\omega c}\,}$
6	${\displaystyle u(t)\,}$	${\displaystyle \pi \delta (\omega )+{\frac {1}{j\omega }}\,}$
7	${\displaystyle e^{-bt}u(t)\,(b>0)}$	${\displaystyle {\frac {1}{j\omega +b}}\,}$
8	${\displaystyle \cos \omega _{0}t\,}$	${\displaystyle \pi \left[\delta (\omega +\omega _{0})+\delta (\omega -\omega _{0})\right]\,}$
9	${\displaystyle \cos(\omega _{0}t+\theta )\,}$	${\displaystyle \pi \left[e^{-j\theta }\delta (\omega +\omega _{0})+e^{j\theta }\delta (\omega -\omega _{0})\right]\,}$
10	${\displaystyle \sin \omega _{0}t\,}$	${\displaystyle j\pi \left[\delta (\omega +\omega _{0})-\delta (\omega -\omega _{0})\right]\,}$
11	${\displaystyle \sin(\omega _{0}t+\theta )\,}$	${\displaystyle j\pi \left[e^{-j\theta }\delta (\omega +\omega _{0})-e^{j\theta }\delta (\omega -\omega _{0})\right]\,}$
12	${\displaystyle {\mbox{rect}}\left({\frac {t}{\tau }}\right)\,}$	${\displaystyle \tau {\mbox{sinc}}\left({\frac {\tau \omega }{2\pi }}\right)\,}$
13	${\displaystyle \tau {\mbox{sinc}}\left({\frac {\tau t}{2\pi }}\right)\,}$	${\displaystyle 2\pi {\mbox{rect}}\left({\frac {\omega }{\tau }}\right)\,}$
14	${\displaystyle \left(1-{\frac {2|t|}{\tau }}\right){\mbox{rect}}\left({\frac {t}{\tau }}\right)\,}$	${\displaystyle {\frac {\tau }{2}}{\mbox{sinc}}^{2}\left({\frac {\tau \omega }{4\pi }}\right)\,}$
15	${\displaystyle {\frac {\tau }{2}}{\mbox{sinc}}^{2}\left({\frac {\tau t}{4\pi }}\right)\,}$	${\displaystyle 2\pi \left(1-{\frac {2|\omega |}{\tau }}\right){\mbox{rect}}\left({\frac {\omega }{\tau }}\right)\,}$
16	${\displaystyle e^{-a|t|},\Re \{a\}>0\,}$	${\displaystyle {\frac {2a}{a^{2}+\omega ^{2}}}\,}$
Notes:	${\displaystyle {\mbox{sinc}}(x)=\sin(\pi x)/(\pi x)}$ ${\displaystyle {\mbox{rect}}\left({\frac {t}{\tau }}\right)}$ is the rectangular pulse function of width ${\displaystyle \tau }$ ${\displaystyle u(t)}$ is the Heaviside step function ${\displaystyle \delta (t)}$ is the Dirac delta function

DTFT Table

Time domain ${\displaystyle x[n]\,}$ where ${\displaystyle n\in \mathbb {Z} }$	Frequency domain ${\displaystyle X(e^{j\omega })}$ where ${\displaystyle \omega \in \mathbb {R} }$	Remarks
${\displaystyle {\frac {1}{2\pi }}\int _{-\pi }^{\pi }{X\left(e^{j\omega }\right)}e^{j\omega n}d\omega }$	${\displaystyle \sum _{n=-\infty }^{\infty }{x[n]e^{-j\omega n}}}$	Definition
${\displaystyle x[n]={\begin{cases}1,&|n|\leq M\\0,&{\text{otherwise}}\end{cases}}}$	${\displaystyle {\frac {\sin \left(\omega \left({\frac {2M+1}{2}}\right)\right)}{\sin \left({\frac {\omega }{2}}\right)}}}$
${\displaystyle \alpha ^{n}u\left[n\right]}$	${\displaystyle {\frac {1}{1-\alpha e^{-j\omega }}}}$
${\displaystyle \delta [n]}$	${\displaystyle 1\!}$	Here ${\displaystyle \delta [n]}$ represents the delta function which is 1 if ${\displaystyle n=0}$ and zero otherwise.
${\displaystyle u[n]={\begin{cases}0&{\text{for }}n<0\\1&{\text{for }}n\geq 0\end{cases}}}$	${\displaystyle {\frac {1}{1-e^{-j\omega }}}+\pi \sum _{p=-\infty }^{\infty }{\delta \left(\omega -2\pi p\right)}}$
${\displaystyle {\frac {1}{\pi n}}\sin \left(Wn\right),\;\;\;\;0<W\leq \pi }$	${\displaystyle X(e^{j\omega })={\begin{cases}1,&|\omega |\leq W\\0,&W<|\omega |\leq \pi \end{cases}}}$	${\displaystyle X(e^{j\omega })}$ is 2π periodic
${\displaystyle (n+1)\alpha ^{n}u\left[n\right]}$	${\displaystyle {\frac {1}{(1-\alpha e^{-j\omega })^{2}}}}$

DTFT Properties

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	${\displaystyle x[n]e^{ian}\!}$	${\displaystyle X(e^{i(\omega -a)})\!}$	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	${\displaystyle \rho _{xy}[n]=x[-n]^{*}*y[n]\!}$	${\displaystyle R_{xy}(\omega )=X(e^{i\omega })^{*}\cdot Y(e^{i\omega })\!}$

Where:

• ${\displaystyle *\!}$ 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].

DFT Table

Time-Domain x[n]	Frequency Domain X[k]	Notes
${\displaystyle x_{n}\equiv {\frac {1}{N}}\sum _{k=0}^{N-1}X_{k}\cdot e^{i2\pi kn/N}}$	${\displaystyle X_{k}\equiv \sum _{n=0}^{N-1}x_{n}\cdot e^{-i2\pi kn/N}}$	DFT Definition
${\displaystyle x_{n}\cdot e^{i2\pi kn/N}\,}$	${\displaystyle X_{n-k}\,}$	Shift theorem
${\displaystyle x_{n-k}\,}$	${\displaystyle X_{k}\cdot e^{-i2\pi kn/N}}$
${\displaystyle x_{n}\in \mathbf {R} }$	${\displaystyle X_{k}=X_{N-k}^{*}\,}$	Real DFT
${\displaystyle a^{n}\,}$	${\displaystyle {\frac {1-a^{N}}{1-a\cdot e^{-i2\pi k/N}}}}$
${\displaystyle {N-1 \choose n}\,}$	${\displaystyle \left(1+e^{-i2\pi k/N}\right)^{N-1}\,}$

Z-Transform Table

Here:

• ${\displaystyle u[n]=1}$ for ${\displaystyle n>=0}$, ${\displaystyle u[n]=0}$ for ${\displaystyle n<0}$
• ${\displaystyle \delta [n]=1}$ for ${\displaystyle n=0}$, ${\displaystyle \delta [n]=0}$ otherwise

	Signal, ${\displaystyle x[n]}$	Z-transform, ${\displaystyle X(z)}$	ROC
1	${\displaystyle \delta [n]\,}$	${\displaystyle 1\,}$	${\displaystyle {\mbox{all }}z\,}$
2	${\displaystyle \delta [n-n_{0}]\,}$	${\displaystyle z^{-n_{0}}\,}$	${\displaystyle z\neq 0\,}$
3	${\displaystyle u[n]\,}$	${\displaystyle {\frac {1}{1-z^{-1}}}}$	${\displaystyle |z|>1\,}$
4	${\displaystyle -u[-n-1]\,}$	${\displaystyle {\frac {1}{1-z^{-1}}}}$	${\displaystyle |z|<1\,}$
5	${\displaystyle nu[n]\,}$	${\displaystyle {\frac {z^{-1}}{(1-z^{-1})^{2}}}}$	${\displaystyle |z|>1\,}$
6	${\displaystyle -nu[-n-1]\,}$	${\displaystyle {\frac {z^{-1}}{(1-z^{-1})^{2}}}}$	${\displaystyle |z|<1\,}$
7	${\displaystyle n^{2}u[n]\,}$	${\displaystyle {\frac {z^{-1}(1+z^{-1})}{(1-z^{-1})^{3}}}}$	${\displaystyle |z|>1\,}$
8	${\displaystyle -n^{2}u[-n-1]\,}$	${\displaystyle {\frac {z^{-1}(1+z^{-1})}{(1-z^{-1})^{3}}}}$	${\displaystyle |z|<1\,}$
9	${\displaystyle n^{3}u[n]\,}$	${\displaystyle {\frac {z^{-1}(1+4z^{-1}+z^{-2})}{(1-z^{-1})^{4}}}}$	${\displaystyle |z|>1\,}$
10	${\displaystyle -n^{3}u[-n-1]\,}$	${\displaystyle {\frac {z^{-1}(1+4z^{-1}+z^{-2})}{(1-z^{-1})^{4}}}}$	${\displaystyle |z|<1\,}$
11	${\displaystyle a^{n}u[n]\,}$	${\displaystyle {\frac {1}{1-az^{-1}}}}$	${\displaystyle |z|>|a|\,}$
12	${\displaystyle -a^{n}u[-n-1]\,}$	${\displaystyle {\frac {1}{1-az^{-1}}}}$	${\displaystyle |z|<|a|\,}$
13	${\displaystyle na^{n}u[n]\,}$	${\displaystyle {\frac {az^{-1}}{(1-az^{-1})^{2}}}}$	${\displaystyle |z|>|a|\,}$
14	${\displaystyle -na^{n}u[-n-1]\,}$	${\displaystyle {\frac {az^{-1}}{(1-az^{-1})^{2}}}}$	${\displaystyle |z|<|a|\,}$
15	${\displaystyle n^{2}a^{n}u[n]\,}$	${\displaystyle {\frac {az^{-1}(1+az^{-1})}{(1-az^{-1})^{3}}}}$	${\displaystyle |z|>|a|\,}$
16	${\displaystyle -n^{2}a^{n}u[-n-1]\,}$	${\displaystyle {\frac {az^{-1}(1+az^{-1})}{(1-az^{-1})^{3}}}}$	${\displaystyle |z|<|a|\,}$
17	${\displaystyle \cos(\omega _{0}n)u[n]\,}$	${\displaystyle {\frac {1-z^{-1}\cos(\omega _{0})}{1-2z^{-1}\cos(\omega _{0})+z^{-2}}}}$	${\displaystyle |z|>1\,}$
18	${\displaystyle \sin(\omega _{0}n)u[n]\,}$	${\displaystyle {\frac {z^{-1}\sin(\omega _{0})}{1-2z^{-1}\cos(\omega _{0})+z^{-2}}}}$	${\displaystyle |z|>1\,}$
19	${\displaystyle a^{n}\cos(\omega _{0}n)u[n]\,}$	${\displaystyle {\frac {1-az^{-1}\cos(\omega _{0})}{1-2az^{-1}\cos(\omega _{0})+a^{2}z^{-2}}}}$	${\displaystyle |z|>|a|\,}$
20	${\displaystyle a^{n}\sin(\omega _{0}n)u[n]\,}$	${\displaystyle {\frac {az^{-1}\sin(\omega _{0})}{1-2az^{-1}\cos(\omega _{0})+a^{2}z^{-2}}}}$	${\displaystyle |z|>|a|\,}$