Digital Signal Processing/Transforms

This page is going to list some of the transforms from the book, explain their uses, and list 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	Fourier Domain
$x(t)=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}X(j\omega )e^{j\omega t}d\omega$	$X(j\omega )=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t)e^{-j\omega t}dt$
$1$	$2\pi \delta (\omega )$
$-0.5+u(t)$	$\frac{1}{j\omega }$
$\delta (t)$	$1$
$\delta (t-c)$	$e^{-j\omega c}$
$u(t)$	$\pi \delta (\omega )+\frac{1}{j\omega }$
$e^{-bt}u(t)$	$\frac{1}{j\omega +b}$
$\cos {\omega }_{0}t$	$\pi [\delta (\omega +{\omega }_0)+\delta (\omega -{\omega }_0)]$
$\cos ({\omega }_{0}t+\theta )$	$\pi [e^{-j\theta }\delta (\omega +{\omega }_0)+e^{j\theta }\delta (\omega -{\omega }_0)]$
$\sin {\omega }_{0}t$	$j\pi [\delta (\omega +{\omega }_0)-\delta (\omega -{\omega }_0)]$
$\sin ({\omega }_{0}t+\theta )$	$j\pi [e^{-j\theta }\delta (\omega +{\omega }_0)-e^{j\theta }\delta (\omega -{\omega }_0)]$
$rect(\frac{t}{\tau })$	$\tau sinc\frac{\tau \omega }{2\pi }$
$\tau sinc\frac{\tau t}{2\pi }$	$2\pi p_{\tau }(\omega )$
$(1-\frac{2|t|}{\tau })p_{\tau }(t)$	$\frac{\tau }{2}sin{c}^2\frac{\tau \omega }{4\pi }$
$\frac{\tau }{2}sin{c}^2(\frac{\tau t}{4\pi })$	$2\pi (1-\frac{2|\omega |}{\tau })p_{\tau }(\omega )$

Note: $sinc(x)=\sin (x)/x$ ; $p_{\tau }(t)$ is the rectangular pulse function of width $\tau$

Discrete-Time Fourier Transform (DTFT)

DTFT Table

Time domain $x[n]$	Frequency domain $X(\omega )$	Remarks
$\delta [n]$	$1$
$\delta [n-k]$	$e^{-ik\omega }$	integer k
$u[n]$	$\frac{1}{1-e^{-i\omega }}$
$e^{-an}$	$\delta (\omega +a)$	real number a
$\cos (an)$	$\frac{1}{2}[\delta (\omega -a)+\delta (\omega +a)]$	real number a
$\sin (an)$	$\frac{1}{2i}[\delta (\omega -a)-\delta (\omega +a)]$	real number a
$\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\left[\frac{(n-M/2)}{M}\right]$	$\frac{\sin [\omega (M+1)/2]}{\sin (\omega /2)}{e}^{-i\omega M/2}$	integer M
$\mathrm{sinc}[(a+n)]$	$e^{ia\omega }$	real number a
$W\cdot {\mathrm{sinc}}^2(Wn)$	$\frac{1}{2\pi W}\cdot \mathrm{tri}\left(\frac{\omega }{2\pi W}\right)$	real number W
$W\cdot \mathrm{sinc}[W(n+a)]$	$\mathrm{rect}\left(\frac{\omega }{2\pi W}\right)\cdot e^{ja\omega }$	real numbers W, a $0<W\le 1$
$\frac{W}{(n+a)}\{\cos [\pi W(n+a)]-\mathrm{sinc}[W(n+a)]\}$	$j\omega \cdot \mathrm{rect}\left(\frac{\omega }{\pi W}\right)e^{ja\omega }$	real numbers W, a
$\frac{1}{\pi n^2}[(-1{)}^n-1]$	$|\omega |$
$\frac{C(A+B)}{2\pi }\cdot \mathrm{sinc}\left[\frac{A-B}{2\pi }n\right]\cdot \mathrm{sinc}\left[\frac{A+B}{2\pi }n\right]$		real numbers A, B complex C

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

	Signal, $x[n]$	Z-transform, $X(z)$	ROC
1	$\delta [n]$	$1$	$\text{all }z$
2	$\delta [n-n_0]$	$\frac{1}{z^{n_0}}$	$\text{all }z$
3	$u[n]$	$\frac{z}{z-1}$	$|z|>1$
4	$a^{n}u[n]$	$\frac{1}{1-a{z}^{-1}}$	$|z|>|a|$
5	$n{a}^{n}u[n]$	$\frac{a{z}^{-1}}{(1-a{z}^{-1}{)}^2}$	$|z|>|a|$
6	$-a^{n}u[-n-1]$	$\frac{1}{1-a{z}^{-1}}$	$|z|<|a|$
7	$-n{a}^{n}u[-n-1]$	$\frac{a{z}^{-1}}{(1-a{z}^{-1}{)}^2}$	$|z|<|a|$
8	$\cos ({\omega }_{0}n)u[n]$	$\frac{1-z^{-1}\cos ({\omega }_0)}{1-2z^{-1}\cos ({\omega }_0)+z^{-2}}$	$|z|>1$
9	$\sin ({\omega }_{0}n)u[n]$	$\frac{z^{-1}\sin ({\omega }_0)}{1-2z^{-1}\cos ({\omega }_0)+z^{-2}}$	$|z|>1$
10	$a^n\cos ({\omega }_{0}n)u[n]$	$\frac{1-a{z}^{-1}\cos ({\omega }_0)}{1-2a{z}^{-1}\cos ({\omega }_0)+a^2{z}^{-2}}$	$|z|>|a|$
11	$a^n\sin ({\omega }_{0}n)u[n]$	$\frac{a{z}^{-1}\sin ({\omega }_0)}{1-2a{z}^{-1}\cos ({\omega }_0)+a^2{z}^{-2}}$	$|z|>|a|$