Went back to residue integration a little. Then covered initial value theorem. This is useful to find \(x\left ( 0\right ) \) given \(X\left ( z\right ) \), where \(x\left ( 0\right ) =\lim _{z\rightarrow \infty }X\left ( z\right ) \), to find \(x\left ( 1\right ) \), the trick is to find \(x\left ( 0\right ) \) first, then if it is zero, then multiply \(X\left ( z\right ) \) by \(z\), this shifts everything back by 1, then apply the theory again to \(x\left ( 0\right ) \), which is now really \(x\left ( 1\right ) \).
Then went over a quick way to plot \(\left \vert H\left ( \omega \right ) \right \vert \) from the location of poles and zeros around the unit circle. Imagine moving around the circle from \(0\) to \(\pi \), then consider a pole to be where \(|H(\omega )|\) is very large, and a zero is where \(|H(\omega )|\) is very small, then one can draw a rough sketch of the \(|H(\omega )|\)
End of lecture.