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content/Math/Difference Equations and Z transform.md

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+---
+title: Difference Equations and Z transform
+draft: false
+tags:
+  - DifferenceEquation
+  - ZTransform
+---
+Let's recall a bit some basic foundations of differential equations and Laplace transform, the simplest linear ODE:
+$$
+\begin{aligned}
+\dot{x}=ax+bu
+\end{aligned}
+$$
+Laplace transform is invented to solve such a DE, take the Laplace transform, we have
+$$
+\begin{aligned}
+sX(s)=aX(s)+bU(s)
+\end{aligned}
+$$
+Rearrange we have 
+$$
+\begin{aligned}
+\frac{X(s)}{U(s)}=\frac{b}{s+a}
+\end{aligned}
+$$
+we can see that the Laplace transform helps us to ease the computation of convolution into product and then do inverse Laplace transform. 
+
+Now let's move on to difference equations
+$$
+\begin{aligned}
+x[n]=ax[n-1]+bu[n]
+\end{aligned}
+$$
+Take the z transform,
+$$
+\begin{aligned}
+X(z)=az^{-1}X(z)+bU(z)
+\end{aligned}
+$$
+Rearrange we have,
+$$
+\begin{aligned}
+\frac{X(z)}{U(z)}=\frac{b}{1-az^{-1}}
+\end{aligned}
+$$
+let $b=1-\gamma$ and $a=\gamma$, the difference equation becomes
+$$
+\begin{aligned}
+x[n]=\gamma x[n-1]+(1-\gamma) u[n]
+\end{aligned}
+$$
+
+