MATH201/content/Convolution (lec 19-20).md

# Convolution
A convolution is an operation of function, we take two functions, convolute them and get a new function.
Definition of convolution between f and g:
$$(f*g)(t):=\int _{0} ^t f(t-v)g(v)\, dv$$
property 1) $f*g=g*f$
proof:
$f*g=\int _{0} ^t f(t-v)g(v)\, \underset{ t-v=u }{ dv }=-\int _{t} ^0 f(u)g(t-u) \, du$
$=\int _{0} ^t g(t-u)f(u)\, du=g*f \quad \Box$

property 2) $(f+g)*h=f*h+g*h$
property 3) $(f*g)*h=f*(g*h)$
property 4) $f*0=0$
property 5) $\mathcal{L}\{f*g\}=F(s)G(s)$
#end of lec 19 #start of lec 20
lets try proving property 5:
recall property 5: $\mathcal{L}\{f*g\}=F(s)G(s)$
$\mathcal{L}\{f*g\}=\int _{0} ^t \left( e^{-st}\int_{0} ^t f(t-v)g(v) \, dv \right)\, dt$
$\mathcal{L}\{f*g\}=\int _{0} ^t \left( e^{-st}\int_{0} ^\infty u(t-v)f(t-v)g(v) \, dv \right)\, dt$
two nested integrals!
using math 209, if both integrals exist, we can exchange the two integrals:
$=\int _{0}^\infty ( g(v)\underbrace{ \int _{0}^\infty e^{-st}f(t-v)u(t-v)\, dt }_{ \mathcal{L}\{f(t-v)u(t-v)\}=e^{-vs}F(s) } )\, dv$
$=F(s)\int _{0} ^\infty e^{-rs}g(v)\, dv=F(s)G(s) \quad \Box$
This is a very useful fact. We will see how it helps us solve differential equations.

ex:
$$\mathcal{L}^{-1}\left\{ \frac{1}{s^2+1}\frac{1}{s^2+1} \right\}$$
we know the inverse LT of $\frac{1}{s^2+1}$ is $\sin(t)$:
then, using property 5: $\mathcal{L}\{f*g\}=F(s)G(s)$
$f*g=\mathcal{L}^{-1}\{F(s)G(s)\}=(\sin*\sin)(t)$
$\mathcal{L}^{-1}\{\frac{1}{s^2+1} \frac{1}{s^2+1}\}=(\sin*\sin)(t)$
using the definition of convolution:
$\mathcal{L}^{-1}\left\{ \frac{1}{s^2+1} \frac{1}{s^2+1} \right\}=\int _{0}^t \sin(t-v)\sin(v)\, dv$
use trig identity: $\sin \alpha \sin \beta=\frac{1}{2}(\cos(\alpha-\beta)-\cos(\beta+\alpha)$
$=\frac{1}{2}\int _{0} ^t (\cos(t-2v)-\cos(t))\, dv$
$=\frac{1}{2}\left( -\frac{1}{2}\sin(t-2v)|^t_{0}-t\cos t \right)=\frac{1}{2}\left( \frac{1}{2}\sin(t)+\frac{1}{2}\sin(t)-t\cos t \right)$
$$=\frac{1}{2}(\sin t-t\cos t)$$
#ex 
solve the problem:
$$y'+y-\int _{0} ^t y(v)\sin(t-v) \, dv =-\sin t,\qquad y(0)=1$$
this is called an integral-differential equation.
we can convert it to a differential equation by taking the derivative of both sides (wrt to dt.):
$y''+y'-y\sin(t-v)=-\cos t$
ew thats a gross second order linear equation. lets solve using laplace.

$sY-1+Y-\mathcal{L}\{(y*\sin)(t)\}=-\frac{1}{s^2+1}$
$\left( s+1-\frac{1}{s^2+1}\right)Y(s)=1-\frac{1}{s^2+1}=\frac{s^2}{s^2+1}$
$\frac{(s^2+1)(s+1)-1}{s^2+1}Y(s)=\frac{s^2}{s^2+1}$
$\frac{s^3+s^2+s+\cancel{ 1 }-\cancel{ 1 }}{s^2+1}Y(s)=\frac{s^2}{s^2+1}$
$Y(s)=\frac{s}{s^2+s+1}$
$y(t)=\mathcal{L}^{-1}\left\{ \frac{s}{s^2+s+1} \right\}=\mathcal{L}^{-1}\{\frac{s}{\left( s+\frac{1}{2} \right)^2+\left( \frac{\sqrt{ 3 }}{2} \right)^2}\}$
$=\mathcal{L}^{-1}\left\{ \frac{s+\frac{1}{2}-\frac{1}{2}}{\left( s+\frac{1}{2} \right)^2+\left( \frac{\sqrt{ 3 }}{2} \right)^2} \right\}$
$=e^{-t/2}\cos\left( \frac{\sqrt{ 2 }}{2}t \right)-\frac{1}{2} \frac{2}{\sqrt{ 3 }}\mathcal{L}^{-1}\left\{ \frac{\frac{\sqrt{ 3 }}{2}}{\left( s+\frac{1}{2} \right)^2+\left( \frac{\sqrt{ 3 }}{2} \right)^2} \right\}$
$$y(t)=e^{-t/2}\left( \cos \frac{\sqrt{ 3 }}{2}t-\frac{1}{\sqrt{ 3 }} \sin \frac{\sqrt{ 3 }}{2}t\right)$$
this is a good algorithmic method now for solving differential equations in software, for example solving circuits.

## Transfer function
imagine we have the equation:
$$ay''+by'+cy=g(t), \qquad y(0)=y_{0},\ y'(0)=y_{1}$$
1)
$ay''+by'+cy=g(t)$
$y(0)=y'(0)=0$
gives a solution $y_{*}$
2)
$ay''+by'+cy=0$
$y(0)=y_{0},\ y'(0)=y_{1}$
gives a soltuion $y_{**}=c_{1}y_{1}+c_{2}y_{2}$

then by principle of super position:
$y=y_{*}+y_{**}$

solving 1) gives us:
$as^2Y+bsY+cY=G(s)$
$Y(s)=\frac{1}{as^2+bs+c}G(s)$ the limit approaches 0 for large s so its a legitimate Laplace transform
let $Y(s)=H(s)G(s)$
where $H(s)=\frac{1}{as^2+bs+c}$ and called the transfer function

we put in $g(t)$ and we get out $Y(s)$. So it "transfers".
$H(s)=\frac{Y(s)}{G(s)}$
$\mathcal{L}^{-1}\{H\}=h(t)$ called the impulse response function. We will see why its called that later.
$y_{*}(t)=(h*g)(t)$
$y(t)=(h*g)(t)+c_{1}y_{1}+c_{2}y_{2}$

he's finished 8 minutes early, lets go!
#end of lec 20