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Next: Evaluation of Fourier Transform Up: Z_Transform Previous: Analysis of LTI Systems

LTI Systems Characterized by LCCDEs

The first order difference is defined as

\begin{displaymath}dx[n]=x[n+1]-x[n],\;\;\;\;\;\mbox{or}\;\;\;\;\;\;dx[n]=x[n]-x[n-1] \end{displaymath}

The second order difference is defined as

\begin{displaymath}d^2 x[n]=dx[n]-dx[n-1]=x[n]-x[n-1]-x[n-1]+x[n-2]=x[n]-2x[n-1]+x[n-2] \end{displaymath}

In general, the kth order difference will need to involve $x[n-k]$.

Similar to an LTI continuous system which can be described by an LCCDE (differential equation), a discrete LTI system can be described by an LCCDE (difference equation):

\begin{displaymath}\sum_{k=0}^N a_k y[n-k]=\sum_{k=0}^M b_k x[n-k] \end{displaymath}

where $x[n]$ and $y[n]$ are the discrete input and output, respectively. After taking the z-transform on both sides of the LCCDE, we get an algebraic equation in the $z$ domain:

\begin{displaymath}Y(z)[\sum_{k=0}^N a_k z^{-k}]=X(z)[\sum_{k=0}^M b_k z^{-k}] \end{displaymath}

and its transfer function is rational:

\begin{displaymath}
H(z)=\frac{Y(z)}{X(z)}=\frac{\sum_{k=0}^M b_k z^{-k}}{\sum_{...
...-z_{z_k})}{\prod_{k=1}^N (z-z_{z_k})}
=K \; \frac{N(z)}{D(z)}
\end{displaymath}

where $K=b_M/a_N$ is a coefficient and $z_{z_k}, (k=1,2,
\cdots, M)$ are the roots of the numerator polynomial and $z_{p_k}, (k=1,2, \cdots, N)$ are the roots of the denominator polynomial. Note that just as the LCCDE alone does not completely specify the relationship between $x[n]$ and $y[n]$ (additional information such as the initial conditions is needed), the transfer function $H(z)$ does not completely specify the system. For example, the same $H(z)$ with different ROCs will represent different systems (e.g., causal or anti-causal).

Example: The input and output of an LTI system are related by

\begin{displaymath}y[n]-\frac{1}{2}y[n-1]=x[n]+\frac{1}{3}x[n-1] \end{displaymath}

Note that without further information such as the initial condition, this equation does not uniquely specify $y[n]$ when $x[n]$ is given. Taking z-transform of this equation and using the time shifting property, we get

\begin{displaymath}Y(z)-\frac{1}{2}z^{-1}Y(z)=X(z)+\frac{1}{3}z^{-1}X(z) \end{displaymath}

and the transfer function can be obtained

\begin{displaymath}
H(z)=\frac{Y(z)}{X(z)}=\frac{1+\frac{1}{3}z^{-1}}{1-\frac{1}{2}z^{-1}}
=\frac{1}{1-\frac{1}{2}z^{-1}}(1+\frac{1}{3}z^{-1}) \end{displaymath}

Note that the causality and stability of the system is not provided by this equation, unless the ROC of this $H(z)$ is specified. Consider these two possible ROCs:


next up previous
Next: Evaluation of Fourier Transform Up: Z_Transform Previous: Analysis of LTI Systems
Ruye Wang 2014-10-28