peter/IDIOTS-GUIDE-TO-ELEC4402-communication-systems
1
0
Fork 0
mirror of https://github.com/peter-tanner/IDIOTS-GUIDE-TO-ELEC4402-communication-systems.git synced 2024-11-30 13:40:16 +08:00
Code Issues Actions Packages Projects Releases Wiki Activity

Add proper fourier transform properties list, add mutual information

Browse Source 
graphic
This commit is contained in:
Peter 2024-10-31 02:59:06 +08:00
parent d48ff87455
commit dbb68ed60a
4 changed files with 113 additions and 38 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

117
README.html
View File

File diff suppressed because one or more lines are too long

30
README.md
View File

@ -72,7 +72,7 @@ along with this program. If not, see <http://www.gnu.org/licenses/>.
## Fourier transform identities
| **Time Function** | **Fourier Transform** |
| Time domain $x(t)$ | Frequency domain $X(f)$ |
| --------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------- |
| $\text{rect}\left(\frac{t}{T}\right)\quad\Pi\left(\frac{t}{T}\right)$ | $T \text{sinc}(fT)$ |
| $\text{sinc}(2Wt)$ | $\frac{1}{2W}\text{rect}\left(\frac{f}{2W}\right)\quad\frac{1}{2W}\Pi\left(\frac{f}{2W}\right)$ |
@ -83,13 +83,6 @@ along with this program. If not, see <http://www.gnu.org/licenses/>.
| $\delta(t)$ | $1$ |
| $1$ | $\delta(f)$ |
| $\delta(t - t_0)$ | $\exp(-j2\pi f t_0)$ |
| $g(t-a)$ | $\exp(-j2\pi fa)G(f)\quad\text{shift property}$ |
| $g(bt)$ | $\frac{G(f/b)}{\|b\|}\quad\text{scaling property}$ |
| $g(bt-a)$ | $\frac{1}{\|b\|}\exp(-j2\pi a(f/b))\cdot G(f/b)\quad\text{shift and scale}$ |
| $\frac{d}{dt}g(t)$ | $j2\pi fG(f)\quad\text{differentiation property}$ |
| $G(t)$ | $g(-f)\quad\text{duality property}$ |
| $g(t)h(t)$ | $G(f)*H(f)$ |
| $g(t)*h(t)$ | $G(f)H(f)$ |
| $\exp(j2\pi f_c t)$ | $\delta(f - f_c)$ |
| $\cos(2\pi f_c t)$ | $\frac{1}{2}[\delta(f - f_c) + \delta(f + f_c)]$ |
| $\sin(2\pi f_c t)$ | $\frac{1}{2j} [\delta(f - f_c) - \delta(f + f_c)]$ |
@ -98,6 +91,25 @@ along with this program. If not, see <http://www.gnu.org/licenses/>.
| $u(t)$ | $\frac{1}{2} \delta(f) + \frac{1}{j2\pi f}$ |
| $\sum_{n=-\infty}^{\infty} \delta(t - nT_0)$ | $\frac{1}{T_0} \sum_{n=-\infty}^{\infty} \delta\left(f - \frac{n}{T_0}\right)=f_0 \sum_{n=-\infty}^{\infty} \delta\left(f - n f_0\right)$ |
| Time domain $x(t)$ | Frequency domain $X(f)$ | Property |
| ------------------------------- | ------------------------------------------------ | ------------------------- |
| $g(t-a)$ | $\exp(-j2\pi fa)G(f)$ | Time shifting |
| $\exp(-j2\pi f_c t)g(t)$ | $G(f-f_c)$ | Frequency shifting |
| $g(bt)$ | $\frac{G(f/b)}{\|b\|}$ | Time scaling |
| $g(bt-a)$ | $\frac{1}{\|b\|}\exp(-j2\pi a(f/b))\cdot G(f/b)$ | Time scaling and shifting |
| $\frac{d}{dt}g(t)$ | $j2\pi fG(f)\quad$ | Differentiation wrt time |
| $g^*(t)$ | $G^*(-f)$ | Conjugate functions |
| $G(t)$ | $g(-f)$ | Duality |
| $\int_{-\infty}^t g(\tau)d\tau$ | $\frac{1}{j2\pi f}G(f)+\frac{G(0)}{2}\delta(f)$ | Integration wrt time |
| $g(t)h(t)$ | $G(f)*H(f)$ | Time multiplication |
| $g(t)*h(t)$ | $G(f)H(f)$ | Time convolution |
| $ag(t)+bh(t)$ | $aG(f)+bH(f)$ | Linearity $a,b$ constants |
| Description | Property |
| ----------------------------------- | ----------------- |
| $g(0)=\int_{-\infty}^\infty G(f)df$ | Area under $G(f)$ |
| $G(0)=\int_{-\infty}^\infty G(t)dt$ | Area under $g(t)$ |
```math
\begin{align*}
u(t) &= \begin{cases} 1, & t > 0 \\ \frac{1}{2}, & t = 0 \\ 0, & t < 0 \end{cases}&\text{Unit Step Function}\\
@ -1015,6 +1027,8 @@ TODO: Cut out if not required
### Mutual information
![Mutual information](images/MutualInformation.drawio.svg)
Amount of entropy decrease of $x$ after observation by $y$.
```math

BIN
README.pdf
View File

Binary file not shown.

4
images/MutualInformation.drawio.svg Normal file
View File

File diff suppressed because one or more lines are too long
Side by Side

After

Width:  |  Height:  |  Size: 80 KiB