Signals and Systems - Electrical Engineering

2.3 LTI Continuous-Time Systems 119

n Lumped- or distributed-parameter systems—This classification relates as to how the elements of the

system are viewed. In the case of the RLC circuit, the resistance, capacitance, and inductance

are localized so that these physical elements are modeled as lumped elements. In the case of

a transmission line resistance, capacitance and inductance are modeled as distributed over the

length of the line.

n Passive or active systems—A system is passive if it is not able to deliver energy to the outside world.

Constant resistors, capacitors, and inductors are passive elements. An operational amplifier is an

active system.

Dynamic systems with lumped parameters, such as the RLC circuit, are typically represented by ordi-

nary differential equations, while distributed-parameter dynamic systems like the transmission line

are represented by partial differential equations. In the case of lumped systems only the time varia-

tion is of interest, while in the case of distributed systems we are interested in both time and space

variations of the signals. In this book we consider only dynamic systems with lumped parameters,

possibly changing with time, with a single input and a single output.

A further classification of systems is obtained by considering the types of signals present at the input

and the output of the system.

Whenever the input(s) and output(s) are both continuous time, discrete time, or digital, the corresponding

systems arecontinuous time, discrete time, ordigital, respectively. It is also possible to havehybridsystems

when the input(s) and output(s) are not of the same type.

Of the systems presented in Chapter 0, the CD player is a hybrid system as it has a digital input (the

bits stored on the disc) and an analog output (the acoustic signal put out by the player). The SDR

system, on the other hand, can be considered to have an analog input (in the transmitter) and an

analog output (at the receiver), making it an analog system, but having hybrid subsystems.

2.3 LTI Continuous-Time Systems

A continuous-time system is a system in which the signals at its input and output are continuous-time signals.

Mathematically we represent it as a transformationSthat converts an input signalx(t)into an output signal

y(t)=S[x(t)](see Figure 2.1):

x(t) ⇒ y(t)=S[x(t)] (2.1)

Input Output

FIGURE 2.1
SystemSwith inputx(t)and
outputy(t). S

Input

x(t)

Output

y(t)=S[x(t)]