84 lines
3.8 KiB
Markdown
84 lines
3.8 KiB
Markdown
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# Convolution
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A convolution is an operation of function, we take two functions, convolute them and get a new function.
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Definition of convolution between f and g:
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$$(f*g)(t):=\int _{0} ^t f(t-v)g(v)\, dv$$
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property 1) $f*g=g*f$
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proof:
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$f*g=\int _{0} ^t f(t-v)g(v)\, \underset{ t-v=u }{ dv }=-\int _{t} ^0 f(u)g(t-u) \, du$
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$=\int _{0} ^t g(t-u)f(u)\, du=g*f \quad \Box$
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property 2) $(f+g)*h=f*h+g*h$
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property 3) $(f*g)*h=f*(g*h)$
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property 4) $f*0=0$
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property 5) $\mathcal{L}\{f*g\}=F(s)G(s)$
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he will see us tomorrow at 10oclock. ;)
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#end of lec 19
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#start of lec 20
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lets try proving property 5:
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recall property 5: $\mathcal{L}\{f*g\}=F(s)G(s)$
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$\mathcal{L}\{f*g\}=\int _{0} ^t \left( e^{-st}\int_{0} ^t f(t-v)g(v) \, dv \right)\, dt$
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$\mathcal{L}\{f*g\}=\int _{0} ^t \left( e^{-st}\int_{0} ^\infty u(t-v)f(t-v)g(v) \, dv \right)\, dt$
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two nested integrals!
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using math 209, if both integrals exist, we can exchange the two integrals:
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$=\int _{0}^\infty ( g(v)\underbrace{ \int _{0}^\infty e^{-st}f(t-v)u(t-v)\, dt }_{ \mathcal{L}\{f(t-v)u(t-v)\}=e^{-vs}F(s) } )\, dv$
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$=F(s)\int _{0} ^\infty e^{-rs}g(v)\, dv=F(s)G(s) \quad \Box$
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This is a very useful fact. We will see how it helps us solve differential equations.
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ex:
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$$\mathcal{L}^{-1}\left\{ \frac{1}{s^2+1}\frac{1}{s^2+1} \right\}$$
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we know the inverse of 1/s^2+1 is sin(t):
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then:
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$=(\sin*\sin)(t)$
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$=\int _{0}^t \sin(t-v)\sin(v)\, dv$
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use identity: $\sin \alpha \sin \beta=\frac{1}{2}(\cos(\alpha-\beta)-\cos(\beta-\alpha)$ DOUBLE CHECK!
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$=\frac{1}{2}\int _{0} ^t (\cos(t-2v)-\cos(t))\, dv$
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$=\frac{1}{2}\left( -\frac{1}{2}\sin(t-2v)|^t_{0}-t\cos t \right)=\frac{1}{2}\left( \frac{1}{2}\sin(t)+\frac{1}{2}\sin(t)-t\cos t \right)$
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$$=\frac{1}{2}(\sin t-t\cos t)$$
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#ex
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solve the problem:
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$$y'+y-\int _{0} ^t y(v)\sin(t-v) \, dv =-\sin t,\qquad y(0)=1$$
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this is called an integral-differential equation.
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we can convert it to a differential equation by taking the derivative of both sides (wrt to dt.):
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$y''+y'-y\sin(t-v)=-\cos t$
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ew thats a gross second order linear equation. lets do it another way
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$sY-1+Y-\mathcal{L}\{(y*\sin)(t)\}=-\frac{1}{s^2+1}$
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$\left( s+1-\frac{1}{s^2+1}\right)Y(s)=1-\frac{1}{s^2+1}=\frac{s^2}{s^2+1}$
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$\frac{(s^2+1)(s+1)-1}{s^2+1}Y(s)=\frac{s^2}{s^2+1}$
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$\frac{s^3+s^2+s+\cancel{ 1 }-\cancel{ 1 }}{s^2+1}Y(s)=\frac{s^2}{s^2+1}$
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$Y(s)=\frac{s}{s^2+s+1}$
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$y(t)=\mathcal{L}^{-1}\left\{ \frac{s}{s^2+s+1} \right\}=\mathcal{L}^{-1}\{\frac{s}{\left( s+\frac{1}{2} \right)^2+\left( \frac{\sqrt{ 3 }}{2} \right)^2}\}$
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$=\mathcal{L}^{-1}\left\{ \frac{s+\frac{1}{2}-\frac{1}{2}}{\left( s+\frac{1}{2} \right)^2+\left( \frac{\sqrt{ 3 }}{2} \right)^2} \right\}$
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$=e^{-t/2}\cos\left( \frac{\sqrt{ 2 }}{2}t \right)-\frac{1}{2} \frac{2}{\sqrt{ 3 }}\mathcal{L}^{-1}\left\{ \frac{\frac{\sqrt{ 3 }}{2}}{\left( s+\frac{1}{2} \right)^2+\left( \frac{\sqrt{ 3 }}{2} \right)^2} \right\}$
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$$y(t)=e^{-t/2}\left( \cos \frac{\sqrt{ 3 }}{2}t-\frac{1}{\sqrt{ 3 }} \sin \frac{\sqrt{ 3 }}{2}t\right)$$
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this is a good algorithmic method now for solving differential equations in software, for example solving circuits.
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## Transfer function
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imagine we have the equation:
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$$ay''+by'+cy=g(t), \qquad y(0)=y_{0},\ y'(0)=y_{1}$$
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1)
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$ay''+by'+cy=g(t)$
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$y(0)=y'(0)=0$
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gives a solution $y_{*}$
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2)
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$ay''+by'+cy=0$
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$y(0)=y_{0},\ y'(0)=y_{1}$
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gives a soltuion $y_{**}=c_{1}y_{1}+c_{2}y_{2}$
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then by principle of super position:
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$y=y_{*}+y_{**}$
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solving 1) gives us:
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$as^2Y+bsY+cY=G(s)$
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$Y(s)=\frac{1}{as^2+bs+c}G(s)$ the limit approaches 0 for large s so its a legitimate Laplace transform
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let $Y(s)=H(s)G(s)$
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where $H(s)=\frac{1}{as^2+bs+c}$ and called the transfer function
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we put in $g(t)$ and we get out $Y(s)$. So it "transfers".
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$H(s)=\frac{Y(s)}{G(s)}$
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$\mathcal{L}^{-1}\{H\}=h(t)$ called the impulse response function. We will see why its called that later.
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$y_{*}(t)=(h*g)(t)$
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$y(t)=(h*g)(t)+c_{1}y_{1}+c_{2}y_{2}$
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he's finished 8 minutes early, lets go!
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#end of lec 20
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