next up previous
Next: LTI Systems Characterized by Up: Z_Transform Previous: Z-Transform of Typical Signals

Analysis of LTI Systems by Z-Transform

Due to its convolution property, the z-transform is a powerful tool to analyze LTI systems

\begin{displaymath}y[n]=h[n]*x[n] \stackrel{{\cal Z}}{\longrightarrow} Y(z)=H(z)X(z) \end{displaymath}

As discussed before, when the input is the eigenfunction of all LTI system, i.e., $x[n]=e^{sn}=z^n$, the operation on this input by the system can be found by multiplying the system's eigenvalue $H(z)$ to the input:

\begin{displaymath}y[n]={\cal O}[z^n]=h[n]*z^n=H(z) z^n \end{displaymath}

Example: The transfer function of an LTI is

\begin{displaymath}H(z)=\frac{1}{1-az^{-1}} \end{displaymath}

As shown before, without specifying the ROC, this $H(z)$ could be the z-transform of one of the two possible time signals $h[n]$.

  $\vert a\vert<1$ $\vert a\vert>1$
$\vert z\vert > \vert a\vert$ $e^{j\omega}$ inside ROC $e^{j\omega}$ outside ROC,
$h[n]=e^a^nu[n]$ causal, stable causal, unstable
$\vert z\vert<\vert a\vert$ $e^{j\omega}$ outside ROC $e^{j\omega}$ inside ROC,
$h[n]=-a^n u[-n-1]$ anti-causal, unstable anti-causal, stable


next up previous
Next: LTI Systems Characterized by Up: Z_Transform Previous: Z-Transform of Typical Signals
Ruye Wang 2014-10-28