Digital Signal Processing/Transforms

This page lists some of the transforms from the book, explains their uses, and lists some transform pairs of common functions.

Continuous-Time Fourier Transform (CTFT)

[CTFT]

${\mathcal {F}}(\omega )=\int f(t)e^{j\omega t}dt$

CTFT Table

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

Discrete-Time Fourier Transform (DTFT)

DTFT Table

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

DTFT Properties

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

Where:

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

Discrete Fourier Transform (DFT)

DFT Table

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

Z-Transform

Z-Transform Table

Here:

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

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

Bilinear Transform

see [1]