# 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 | ${\mbox{sinc}}(x)=\sin(\pi x)/(\pi x)$ ${\mbox{rect}}\left({\frac {t}{\tau }}\right)$ is the rectangular pulse function of width $\tau$ $u(t)$ is the Heaviside step function $\delta (t)$ is the Dirac delta function |
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## 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 |
| $x_{n-k}\,$ | $X_{k}\cdot e^{-i2\pi kn/N}$ | Shift theorem |
| $x_{n}\in \mathbf {R}$ | $X_{k}=X_{N-k}^{*}\,$ | 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