118 C H A P T E R 2: Continuous-Time Systems
synthesis short intervals of speech are modeled as the output of linear time-invariant models.
Finally, it will be seen that the LTI model is not appropriate to represent communication systems;
rather, nonlinear or time-varying systems are more appropriate.
n Convolution integral, causality, and stability—The output of a LTI system due to any signal is
obtained by means of the generic signal representation obtained in Chapter 1. The response due
to an impulse, together with the linearity and time-invariance of the system, gives the output as
an integral. This convolution integral, although difficult to compute, even in simple cases, has
significant theoretical value. It allows us not only to determine the response of the system for very
general cases, but also provides a way to characterize causal and stable systems. Causality relates
to the cause and effect of the input and the output of the system, giving us the conditions for
real-time processing while stability characterizes useful systems. These two conditions are of great
practical significance.2.2 System Concept
Although we view asystemas a mathematical transformation of an input signal (or signals) into an
output signal (or signals), it is important to understand that such transformation results from an
idealized model of the physical device or process we are interested in.For instance, in the interconnection of physical resistors, capacitors, and inductors, the model
idealizes how to deal with the resistors, capacitors, and inductors. In this simple RLC circuit, we
would ignore, for instance, stray inductive and capacitive effects and the effect of temperature on
the resistors. The resistance, capacitance, and impedance would be assumed localized in the physical
devices and the wires would not have resistance, inductance, or capacitance. We would then use the
circuits laws to obtain a differential equation to characterize the interconnection. A wire that in the
RLC circuit model connects two elements, in a transmission line a similar wire is modeled as having
capacitance, inductance, and resistance distributed over the line to realize the way the voltages travel
over it. In practice, the model and the mathematical representation are not unique.A system can be considered a connection of subsystems. Thinking of the RLC circuit as a system, for
instance, the resistor, the capacitor, the inductor, and the source are the subsystems.In engineering, the models are typically developed in the different areas. There will be, however,
analogs as it is the case between mechanical and electrical systems. In such cases, the mathematical
equations are similar, or even identical, but their significance is very different.2.2.1 System Classification................................................................
According to general characteristics attributed to systems, they can be classified as follows:n Static or dynamic systems—A dynamic system has the capability of storing energy, or remembering
its state, while a static system does not. A battery connected to resistors is a static system, while
the same battery connected to resistors, capacitors, and inductors constitutes a dynamic system.
The main difference is the capability of capacitors and inductors to store energy, to remember the
state of the device, that resistors do not have.