Laplace transform converts many time-domain operations such as differentiation, integration, convolution, time shifting into algebraic operations in s-domain. Moreover, the behavior of complex systems composed of a set of interconnected LTI systems can also be easily analyzed in s-domain. We first consider some simple interconnections of LTI systems.

**Parallel combination:**If the system is composed of two LTI systems with and connected in parallel:

where is the overall impulse response:

**Series combination:**If the system is composed of two LTI systems with and connected in series:

where is the overall impulse response:

**Feedback system:**This is a feedback system 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 so that , it is easy to solve the algebraic equation in s-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 1: ** A first order LTI system

which can be represented in the block diagram shown below:

Alternatively, the system can be described in s-domain by its transfer function:

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 (an integrator implementable by an operational amplifier) in the forward path, and (a feedback coefficient) in the negative feedback path.

**Example 2: ** Consider a second order system with transfer function

These three expressions of correspond to three different block diagram representations of the system. The last two expressions are, respectively, the

Alternatively, the first expression, a *direct form*, can also be used.
To do so, we first consider a general
, i.e.,

Given , we can first obtain by an integrator , and then obtain the output from by another integrator . We see that this system can be represented as a feedback system with two negative feedback paths of from and from .

**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 integrators 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 forward path of the first system. The over all system can therefore by represented as shown below.

Obviously the block diagram of this example can be generalized to represent
any system with a rational transfer function:

If , can be separated into several terms (by long-division) which can be individually implemented and then combined to generate the overall output .