Given a difference equation, to solve for \(y\left ( n\right ) \), do it in 2 steps: Let \(x\left ( n\right ) =\delta \left ( n\right ) \), find find \(y\left ( n\right ) \) which will be \(h\left ( n\right ) \) in this case. Second step, find \(y\left ( n\right ) \) for \(x\left ( n\right ) \) using convolution.
Example given using \(y\left ( n\right ) -ay\left ( n\right ) =x\left ( n\right ) \), solution shown assuming system is causal (i.e. \(h\left ( n\right ) =0\) for \(n<0\)) and another solution shown for anti casual (i.e. \(h\left ( n\right ) =0\) for \(n\geq 0\)). Condition for stability is decided by \(\left \vert a\right \vert \).
Mention that given a difference equation, without additional information, its solution is not unique.
Mention that if \(h\left ( n\right ) \) has finite length, then system is called FIR, else it is called IIR
Then showed how to solve a difference equation using characteristic equation. Then showed circuit description of difference equation.
Introduction to Z transform, 2 sided. Definitions and examples. Then inverse Z transform. Idea is to write the Z transform as sums of \(a_{n}z^{-n},\) then we can read the data directly since \(x\left ( n\right ) \) will be the coefficient of the \(z^{-n}\) part. i.e. \(x\left ( -1\right ) z^{1}+x\left ( 0\right ) +x\left ( 1\right ) z^{-1}+x\left ( 2\right ) z^{-2}+...\), so using long division we can find \(x\left ( n\right ) \)
End of lecture 4.