diff --git a/README.html b/README.html index 33961cd..5752366 100644 --- a/README.html +++ b/README.html @@ -200,8 +200,8 @@ along with this program. If not, see http - - + + @@ -242,34 +242,6 @@ along with this program. If not, see http - - - - - - - - - - - - - - - - - - - - - - - - - - - - @@ -299,6 +271,90 @@ along with this program. If not, see http
Time FunctionFourier TransformTime domain x(t)x(t)Frequency domain X(f)X(f)
exp⁡(−j2πft0)\exp(-j2\pi f t_0)
g(t−a)g(t-a)exp⁡(−j2πfa)G(f)shift property\exp(-j2\pi fa)G(f)\quad\text{shift property}
g(bt)g(bt)G(f/b)∣b∣scaling property\frac{G(f/b)}{|b|}\quad\text{scaling property}
g(bt−a)g(bt-a)1∣b∣exp⁡(−j2πa(f/b))⋅G(f/b)shift and scale\frac{1}{|b|}\exp(-j2\pi a(f/b))\cdot G(f/b)\quad\text{shift and scale}
ddtg(t)\frac{d}{dt}g(t)j2πfG(f)differentiation propertyj2\pi fG(f)\quad\text{differentiation property}
G(t)G(t)g(−f)duality propertyg(-f)\quad\text{duality property}
g(t)h(t)g(t)h(t)G(f)∗H(f)G(f)*H(f)
g(t)∗h(t)g(t)*h(t)G(f)H(f)G(f)H(f)
exp⁡(j2πfct)\exp(j2\pi f_c t) δ(f−fc)\delta(f - f_c)
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Time domain x(t)x(t)Frequency domain X(f)X(f)Property
g(t−a)g(t-a)exp⁡(−j2πfa)G(f)\exp(-j2\pi fa)G(f)Time shifting
exp⁡(−j2πfct)g(t)\exp(-j2\pi f_c t)g(t)G(f−fc)G(f-f_c)Frequency shifting
g(bt)g(bt)G(f/b)∣b∣\frac{G(f/b)}{|b|}Time scaling
g(bt−a)g(bt-a)1∣b∣exp⁡(−j2πa(f/b))⋅G(f/b)\frac{1}{|b|}\exp(-j2\pi a(f/b))\cdot G(f/b)Time scaling and shifting
ddtg(t)\frac{d}{dt}g(t)j2πfG(f)j2\pi fG(f)\quadDifferentiation wrt time
g∗(t)g^*(t)G∗(−f)G^*(-f)Conjugate functions
G(t)G(t)g(−f)g(-f)Duality
∫−∞tg(τ)dτ\int_{-\infty}^t g(\tau)d\tau1j2πfG(f)+G(0)2δ(f)\frac{1}{j2\pi f}G(f)+\frac{G(0)}{2}\delta(f)Integration wrt time
g(t)h(t)g(t)h(t)G(f)∗H(f)G(f)*H(f)Time multiplication
g(t)∗h(t)g(t)*h(t)G(f)H(f)G(f)H(f)Time convolution
ag(t)+bh(t)ag(t)+bh(t)aG(f)+bH(f)aG(f)+bH(f)Linearity a,ba,b constants
+ + + + + + + + + + + + + + + + + +
DescriptionProperty
g(0)=∫−∞∞G(f)dfg(0)=\int_{-\infty}^\infty G(f)dfArea under G(f)G(f)
G(0)=∫−∞∞G(t)dtG(0)=\int_{-\infty}^\infty G(t)dtArea under g(t)g(t)

u(t)={1,t>012,t=00,t<0Unit Step Functionsgn(t)={+1,t>00,t=0−1,t<0Signum Functionsinc(2Wt)=sin⁡(2πWt)2πWtsinc Functionrect(t)=Π(t)={1,−0.5<t<0.50,∣t∣>0.5Rectangular/Gate Functiong(t)∗h(t)=(g∗h)(t)=∫∞∞g(τ)h(t−τ)dτConvolution\begin{align*} u(t) &= \begin{cases} 1, & t > 0 \\ \frac{1}{2}, & t = 0 \\ 0, & t < 0 \end{cases}&\text{Unit Step Function}\\ \text{sgn}(t) &= \begin{cases} +1, & t > 0 \\ 0, & t = 0 \\ -1, & t < 0 \end{cases}&\text{Signum Function}\\ @@ -2072,6 +2128,7 @@ M1001 80h400000v40h-400000z"/>

Mutual information

+

Mutual information

Amount of entropy decrease of xx after observation by yy.

I(x;y)=H(x)−H(x∣y)=H(y)−H(y∣x)\begin{align*} I(x;y) &= H(x)-H(x|y)=H(y)-H(y|x)\\ diff --git a/README.md b/README.md index 47b14eb..fdb3ff6 100644 --- a/README.md +++ b/README.md @@ -72,7 +72,7 @@ along with this program. If not, see . ## 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 . | $\delta(t)$ | $1$ | | $1$ | $\delta(f)$ | | $\delta(t - t_0)$ | $\exp(-j2\pi f t_0)$ | -| $g(t-a)$ | $\exp(-j2\pi fa)G(f)\quad\text{shift property}$ | -| $g(bt)$ | $\frac{G(f/b)}{\|b\|}\quad\text{scaling property}$ | -| $g(bt-a)$ | $\frac{1}{\|b\|}\exp(-j2\pi a(f/b))\cdot G(f/b)\quad\text{shift and scale}$ | -| $\frac{d}{dt}g(t)$ | $j2\pi fG(f)\quad\text{differentiation property}$ | -| $G(t)$ | $g(-f)\quad\text{duality property}$ | -| $g(t)h(t)$ | $G(f)*H(f)$ | -| $g(t)*h(t)$ | $G(f)H(f)$ | | $\exp(j2\pi f_c t)$ | $\delta(f - f_c)$ | | $\cos(2\pi f_c t)$ | $\frac{1}{2}[\delta(f - f_c) + \delta(f + f_c)]$ | | $\sin(2\pi f_c t)$ | $\frac{1}{2j} [\delta(f - f_c) - \delta(f + f_c)]$ | @@ -98,6 +91,25 @@ along with this program. If not, see . | $u(t)$ | $\frac{1}{2} \delta(f) + \frac{1}{j2\pi f}$ | | $\sum_{n=-\infty}^{\infty} \delta(t - nT_0)$ | $\frac{1}{T_0} \sum_{n=-\infty}^{\infty} \delta\left(f - \frac{n}{T_0}\right)=f_0 \sum_{n=-\infty}^{\infty} \delta\left(f - n f_0\right)$ | +| 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 diff --git a/README.pdf b/README.pdf index 40eb3ad..b4d6b8c 100644 Binary files a/README.pdf and b/README.pdf differ diff --git a/images/MutualInformation.drawio.svg b/images/MutualInformation.drawio.svg new file mode 100644 index 0000000..a159569 --- /dev/null +++ b/images/MutualInformation.drawio.svg @@ -0,0 +1,4 @@ + + + +

 \ No newline at end of file