# 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]

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

### 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:
1. ${\displaystyle {\mbox{sinc}}(x)=\sin(\pi x)/(\pi x)}$
2. ${\displaystyle {\mbox{rect}}\left({\frac {t}{\tau }}\right)}$ is the rectangular pulse function of width ${\displaystyle \tau }$
3. ${\displaystyle u(t)}$ is the Heaviside step function
4. ${\displaystyle \delta (t)}$ is the Dirac delta function

## Discrete-Time Fourier Transform (DTFT)

### 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 ${\displaystyle X(e^{j\omega })={\begin{cases}1,&|\omega |\leq W\\0,&W<|\omega |\leq \pi \end{cases}}}$ ${\displaystyle X(e^{j\omega })}$ 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].

## Discrete Fourier Transform (DFT)

### 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

### 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 ${\displaystyle a^{n}\sin(\omega _{0}n)u[n]\,}$ ${\displaystyle {\frac {az^{-1}\sin(\omega _{0})}{1-2az^{-1}\cos(\omega _{0})+a^{2}z^{-2}}}}$ ${\displaystyle |z|>|a|\,}$

see [1]