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Add proper fourier transform properties list, add mutual information

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Peter 2024-10-31 02:59:06 +08:00
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@ -72,7 +72,7 @@ along with this program. If not, see <http://www.gnu.org/licenses/>.
## Fourier transform identities ## Fourier transform identities
| **Time Function** | **Fourier Transform** | | Time domain $x(t)$ | Frequency domain $X(f)$ |
| --------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------- | | --------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------- |
| $\text{rect}\left(\frac{t}{T}\right)\quad\Pi\left(\frac{t}{T}\right)$ | $T \text{sinc}(fT)$ | | $\text{rect}\left(\frac{t}{T}\right)\quad\Pi\left(\frac{t}{T}\right)$ | $T \text{sinc}(fT)$ |
| $\text{sinc}(2Wt)$ | $\frac{1}{2W}\text{rect}\left(\frac{f}{2W}\right)\quad\frac{1}{2W}\Pi\left(\frac{f}{2W}\right)$ | | $\text{sinc}(2Wt)$ | $\frac{1}{2W}\text{rect}\left(\frac{f}{2W}\right)\quad\frac{1}{2W}\Pi\left(\frac{f}{2W}\right)$ |
@ -83,13 +83,6 @@ along with this program. If not, see <http://www.gnu.org/licenses/>.
| $\delta(t)$ | $1$ | | $\delta(t)$ | $1$ |
| $1$ | $\delta(f)$ | | $1$ | $\delta(f)$ |
| $\delta(t - t_0)$ | $\exp(-j2\pi f t_0)$ | | $\delta(t - t_0)$ | $\exp(-j2\pi f t_0)$ |
| $g(t-a)$ | $\exp(-j2\pi fa)G(f)\quad\text{shift property}$ |
| $g(bt)$ | $\frac{G(f/b)}{\|b\|}\quad\text{scaling property}$ |
| $g(bt-a)$ | $\frac{1}{\|b\|}\exp(-j2\pi a(f/b))\cdot G(f/b)\quad\text{shift and scale}$ |
| $\frac{d}{dt}g(t)$ | $j2\pi fG(f)\quad\text{differentiation property}$ |
| $G(t)$ | $g(-f)\quad\text{duality property}$ |
| $g(t)h(t)$ | $G(f)*H(f)$ |
| $g(t)*h(t)$ | $G(f)H(f)$ |
| $\exp(j2\pi f_c t)$ | $\delta(f - f_c)$ | | $\exp(j2\pi f_c t)$ | $\delta(f - f_c)$ |
| $\cos(2\pi f_c t)$ | $\frac{1}{2}[\delta(f - f_c) + \delta(f + f_c)]$ | | $\cos(2\pi f_c t)$ | $\frac{1}{2}[\delta(f - f_c) + \delta(f + f_c)]$ |
| $\sin(2\pi f_c t)$ | $\frac{1}{2j} [\delta(f - f_c) - \delta(f + f_c)]$ | | $\sin(2\pi f_c t)$ | $\frac{1}{2j} [\delta(f - f_c) - \delta(f + f_c)]$ |
@ -98,6 +91,25 @@ along with this program. If not, see <http://www.gnu.org/licenses/>.
| $u(t)$ | $\frac{1}{2} \delta(f) + \frac{1}{j2\pi f}$ | | $u(t)$ | $\frac{1}{2} \delta(f) + \frac{1}{j2\pi f}$ |
| $\sum_{n=-\infty}^{\infty} \delta(t - nT_0)$ | $\frac{1}{T_0} \sum_{n=-\infty}^{\infty} \delta\left(f - \frac{n}{T_0}\right)=f_0 \sum_{n=-\infty}^{\infty} \delta\left(f - n f_0\right)$ | | $\sum_{n=-\infty}^{\infty} \delta(t - nT_0)$ | $\frac{1}{T_0} \sum_{n=-\infty}^{\infty} \delta\left(f - \frac{n}{T_0}\right)=f_0 \sum_{n=-\infty}^{\infty} \delta\left(f - n f_0\right)$ |
| 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 ```math
\begin{align*} \begin{align*}
u(t) &= \begin{cases} 1, & t > 0 \\ \frac{1}{2}, & t = 0 \\ 0, & t < 0 \end{cases}&\text{Unit Step Function}\\ 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
![Mutual information](images/MutualInformation.drawio.svg)
Amount of entropy decrease of $x$ after observation by $y$. Amount of entropy decrease of $x$ after observation by $y$.
```math ```math

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