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)

$ℱ(\omega )=\int f(t)e^{j\omega t}dt$

CTFT Table

	Time Domain	Frequency Domain
	$x(t)={ℱ}^{-1}\{X(\omega )\}$	$X(\omega )=ℱ\{x(t)\}$
1	$X(j\omega )=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t)e^{-j\omega t}dt$	$x(t)=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}X(\omega )e^{j\omega t}d\omega$
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	$\cos {\omega }_{0}t$	$\pi [\delta (\omega +{\omega }_0)+\delta (\omega -{\omega }_0)]$
9	$\cos ({\omega }_{0}t+\theta )$	$\pi [e^{-j\theta }\delta (\omega +{\omega }_0)+e^{j\theta }\delta (\omega -{\omega }_0)]$
10	$\sin {\omega }_{0}t$	$j\pi [\delta (\omega +{\omega }_0)-\delta (\omega -{\omega }_0)]$
11	$\sin ({\omega }_{0}t+\theta )$	$j\pi [e^{-j\theta }\delta (\omega +{\omega }_0)-e^{j\theta }\delta (\omega -{\omega }_0)]$
12	$\text{rect}\left(\frac{t}{\tau }\right)$	$\tau \text{sinc}\left(\frac{\tau \omega }{2\pi }\right)$
13	$\tau \text{sinc}\left(\frac{\tau t}{2\pi }\right)$	$2\pi \text{rect}\left(\frac{\omega }{\tau }\right)$
14	$(1-\frac{2|t|}{\tau })\text{rect}\left(\frac{t}{\tau }\right)$	$\frac{\tau }{2}{\text{sinc}}^2\left(\frac{\tau \omega }{4\pi }\right)$
15	$\frac{\tau }{2}{\text{sinc}}^2\left(\frac{\tau t}{4\pi }\right)$	$2\pi (1-\frac{2|\omega |}{\tau })\text{rect}\left(\frac{\omega }{\tau }\right)$
16	$e^{-a|t|},\mathrm{\Re }\{a\}>0$	$\frac{2a}{a^2+{\omega }^2}$
Notes:	$\text{sinc}(x)=\sin (\pi x)/(\pi x)$ $\text{rect}\left(\frac{t}{\tau }\right)$ is the rectangular pulse function of width $\tau$ $u(t)$ is the Heaviside step function $\delta (t)$ is the Dirac delta function
Template:Navbar

Discrete-Time Fourier Transform (DTFT)

DTFT Table

Time domain $x[n]$ where $n\in \mathbb{Z}$	Frequency domain $X(e^{j\omega })$ where $\omega \in ℝ$	Remarks
$\frac{1}{2\pi }\underset{-\pi }{\overset{\pi }{\int }}X\left(e^{j\omega }\right)e^{j\omega n}d\omega$	$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}x[n]e^{-j\omega n}$	Definition
$x[n]=\{\begin{array}{cc}1,& |n|\le M\\ 0,& \text{otherwise}\end{array}$	$\frac{\sin \left(\omega \left(\frac{2M+1}{2}\right)\right)}{\sin \left(\frac{\omega }{2}\right)}$
${\alpha }^{n}u\left[n\right]$	$\frac{1}{1-\alpha e^{-j\omega }}$
$\delta [n]$	$1$	Here $\delta [n]$ represents the delta function which is 1 if $n=0$ and zero otherwise.
$u[n]=\{\begin{array}{cc}0& \text{for }n<0\\ 1& \text{for }n\ge 0\end{array}$	$\frac{1}{1-e^{-j\omega }}+\pi \sum _{p=-\mathrm{\infty }}^{\mathrm{\infty }}\delta (\omega -2\pi p)$
$\frac{1}{\pi n}\sin \left(Wn\right),0<W\le \pi$	$X(e^{j\omega })=\{\begin{array}{cc}1,& |\omega |\le W\\ 0,& W<|\omega |\le \pi \end{array}$	$X(e^{j\omega })$ 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	$\frac{i}{n}x[n]$	$\underset{-\pi }{\overset{\omega }{\int }}X(e^{i\vartheta })d\vartheta$
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
$x_{n-k}$	$X_k\cdot e^{-i2\pi kn/N}$
$x_n\in 𝐑$	$X_k=X_{N-k}^{*}$	Real DFT
$a^n$	$\frac{1-a^N}{1-a\cdot e^{-i2\pi k/N}}$
$(\genfrac{}{}{0ex}{}{N-1}{n})$	${(1+e^{-i2\pi k/N})}^{N-1}$

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	$\delta [n]$	$1$	$\text{all }z$
2	$\delta [n-n_0]$	$z^{-n_0}$	$z\ne 0$
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	$a^{n}u[n]$	$\frac{1}{1-a{z}^{-1}}$	$|z|>|a|$
12	$-a^{n}u[-n-1]$	$\frac{1}{1-a{z}^{-1}}$	$|z|<|a|$
13	$n{a}^{n}u[n]$	$\frac{a{z}^{-1}}{(1-a{z}^{-1}{)}^2}$	$|z|>|a|$
14	$-n{a}^{n}u[-n-1]$	$\frac{a{z}^{-1}}{(1-a{z}^{-1}{)}^2}$	$|z|<|a|$
15	$n^2{a}^{n}u[n]$	$\frac{a{z}^{-1}(1+a{z}^{-1})}{(1-a{z}^{-1}{)}^3}$	$|z|>|a|$
16	$-n^2{a}^{n}u[-n-1]$	$\frac{a{z}^{-1}(1+a{z}^{-1})}{(1-a{z}^{-1}{)}^3}$	$|z|<|a|$
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	$a^n\cos ({\omega }_{0}n)u[n]$	$\frac{1-a{z}^{-1}\cos ({\omega }_0)}{1-2a{z}^{-1}\cos ({\omega }_0)+a^2{z}^{-2}}$	$|z|>|a|$
20	$a^n\sin ({\omega }_{0}n)u[n]$	$\frac{a{z}^{-1}\sin ({\omega }_0)}{1-2a{z}^{-1}\cos ({\omega }_0)+a^2{z}^{-2}}$	$|z|>|a|$

Bilinear Transform

see [1]