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System Algebra and Block Diagram

Z-transform converts time-domain operations such as difference and convolution into algebraic operations in z-domain. Moreover, the behavior of complex systems composed of a set of interconnected LTI systems can also be easily analyzed in z-domain. Some simple interconnections of LTI systems are listed below.

Example 0: The transfer function of a first order LTI system

\begin{displaymath}y[n]-\frac{1}{4}y[n-1]=x[n],\;\;\;\;\;Y(z)[1-\frac{1}{4}z^{-1}]=X(z) \end{displaymath}

is

\begin{displaymath}H(z)=\frac{1}{1-\frac{1}{4}z^{-1}} \end{displaymath}

Comparing this $H(z)$ with the transfer function of the feedback system, we see that a first order system can be represented as a feedback system with $H_1(z)=1$ in the forward path, and $H_2(z)$ for the product of $1/4$ and $z^{-1}$ (a delay element with input $x[n]$ and output $y[n]=x[n-1]$) in the negative feedback path.

Zdiagram1.gif

Example 1:

\begin{displaymath}H(z)=\frac{Y(z)}{X(z)}=\frac{1-2z^{-1}}{1-\frac{1}{4}z^{-1}} \end{displaymath}

This equation can be rewritten as:

\begin{displaymath}Y(z)=(1-2z^{-1}) \frac{1}{1-\frac{1}{4}z^{-1}}X(z)=(1-2z^{-1})W(z) \end{displaymath}

where

\begin{displaymath}W(z)=\frac{1}{1-\frac{1}{4}z^{-1}}X(z) \end{displaymath}

can be obtained the same way as in previous example. Once $W(z)$ and $W(z)z^{-1}$ are available, we can easily obtain $Y(z)$:

Zdiagram2.gif

Example 2: Consider a second order system with transfer function

\begin{displaymath}
H(z)=\frac{1}{1+\frac{1}{4}z^{-1}-\frac{1}{8}z^{-2}}
=\frac...
...frac{2/3}{1+\frac{1}{2}z^{-1}}+\frac{1/3}{1-\frac{1}{4}z^{-1}}
\end{displaymath}

These three expressions of this $H(z)$ correspond to three different block diagram representations of the system. The last two expressions are, respectively, the cascade and the parallel representations (same as the corresponding cases in Laplace transform), while the first one is the direct representation, as shown below. We first consider a general 2nd order system

\begin{displaymath}Y(z)+asY(z)z^{-1}+bY(z)z^{-2}=X(z),\;\;\;\;\;\mbox{or}\;\;\;\;\;\;
Y(z)=X(z)-aY(z)z^{-1}-bY(z)z^{-2} \end{displaymath}

We see that $Y(z)$ is the linear combination of the delayed versions of itself and the input $X(z)$ which can be represented as a feedback system with two feedback paths of $-az^{-1}$ and $-bz^{-2}$. In this particular system, $a=-1/4$ and $b=1/8$.

Zdiagram3.gif

Zdiagram4.gif

Zdiagram5.gif

Example 3: A second order system with transfer function

\begin{displaymath}H(z)=K \frac{1+cz^{-1}+dz^{-2}}{1+az^{-1}+bz^{-2}}
=\frac{K}{1+az^{-1}+bz^{-2}}(1+cz^{-1}+dz^{-2})
\end{displaymath}

This system can be represented as a cascade of two systems

\begin{displaymath}W(z)=H_1(z)X(z)=\frac{K}{1+az^{-1}+bz^{-2}}X(z) \end{displaymath}

and

\begin{displaymath}Y(z)=H_2(z)W(z)=(1+cz^{-1}+dz^{-2})W(z) \end{displaymath}

The first system $H_1(z)$ can be implemented by two delay elements with proper feedback paths as shown in the previous example, and the second system is a linear combination of $W(z)$, $W(z)z^{-1}$ and $W(z)z^{-2}$, all of which are available along the feedback path of the first system. The over all system can therefore be represented as shown. Obviously the block diagram of this example can be generalized to represent any system with a rational transfer function

\begin{displaymath}
H(z)=\frac{\sum_{k=0}^M b_k z^{-k}}{\sum_{k=0}^N a_k z^{-k}}\;\;\;\;(M \le N)
\end{displaymath}

Zdiagram6.gif


next up previous
Next: Unilateral Z-Transform Up: Z_Transform Previous: Second order system
Ruye Wang 2014-10-28