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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.

• Parallel systems: If the system is composed of two LTI systems with and connected in parallel, its impulse response is

or in s-domain

• Serial or cascade system: If the system is composed of two LTI systems with and connected in series, its impulse response is

or in s-domain

• Feedback system: If the system is composed of an LTI system with in a forward path and another LTI system in a feedback path, its output can be implicitly found in time domain

or in s-domain

While it is difficult to solve the equation in time domain to find an explicit expression for output , it is easy to solve the algebraic equation in z-domain to find

and the transfer function can be obtained

The feedback could be either positive or negative. For the latter, there will be a negative sign in front of and of the feedback path so that and

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

is

Comparing this with the transfer function of the feedback system, we see that a first order system can be represented as a feedback system with in the forward path, and for the product of and (a delay element with input and output ) in the negative feedback path.

Example 1:

This equation can be rewritten as:

where

can be obtained the same way as in previous example. Once and are available, we can easily obtain :

Example 2: Consider a second order system with transfer function

These three expressions of this 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

We see that is the linear combination of the delayed versions of itself and the input which can be represented as a feedback system with two feedback paths of and . In this particular system, and .

Example 3: A second order system with transfer function

This system can be represented as a cascade of two systems

and

The first system 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 , and , 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

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