Analog systems are used to process analog signals. A description of systems relies heavily on how they respond to arbitrary or specific signals. In the time domain, many analog systems can be described by their response to arbitrary signals using differential equations. The class of linear, time-invariant systems can also be described by their impulse response, the response to an impulse input. This chapter deals with continuous-time systems and their classification, time-domain representation, and analysis based on the solution of differential equations. It also introduces the all-important concept of the impulse response, which forms a key ingredient in both system description and system analysis.
4.1
Introduction
In its broadest sense, a physical system is an interconnection of devices and elements subject to physical laws. A system that processes analog signals is referred to as an analog system or continuous-time (CT) system. The signal to be processed forms the excitation or input to the system. The processed signal is termed the response or output.
The response of any system is governed by the input and the system details. A system may of course be excited by more than one input, and this leads to the more general idea of multiple-input systems. We address only single-input, single-output systems in this text. The study of systems involves the input, the output, and the system specifications. Conceptually, we can determine any one of these in terms of the other two. System analysis implies a study of the response subject to known inputs and system formulations. Known input-output specifications, on the other hand, usually allow us to identify, or synthesize, the system. System identification or synthesis is much more difficult because many system formulations are possible for the same input-output relationship.
Most real-world systems are quite complex and almost impossible to analyze quantitatively. Of necessity, we are forced to use models or abstractions that retain the essential features of the system and simplify the analysis, while still providing meaningful results. The analysis of systems refers to the analysis of the models that in fact describe such systems, and it is customary to treat the system and its associated models synonymously. In the context of signal processing, a system that processes the input signal in some fashion is also called a filter.
4.1.1
Terminology of Systems
A system requires two separate descriptions for a complete specification, one in terms of its components or external structure and the other in terms of its energy level or internal state. The state of a system is described by a set of state variables that allow us to establish the energy level of the system at any instant. 68
4.1 Introduction 69 Such variables may represent physical quantities or may have no physical significance whatever. Their choice is governed primarily by what the analysis requires. For example, capacitor voltages and inductor currents are often used as state variables since they provide an instant measure of the system energy. Any inputs applied to the system result in a change in the energy or state of the system. All physical systems are, by convention, referenced to a zero-energy state (variously called the ground state, the rest state, the relaxed state, or the zero state) at t = −∞.
The behavior of a system is governed not only by the input but also by the state of the system at the instant at which the input is applied. The initial values of the state variables define the initial conditions or initial state. This initial state, which must be known before we can establish the complete system response, embodies the past history of the system. It allows us to predict the future response due to any input regardless of how the initial state was arrived at.
4.1.2
Operators
Any equation is based on a set of operations. An operator is a rule or a set of directions—a recipe if you will—that shows us how to transform one function to another. For example, the derivative operator s ≡ d
dt transforms a function x(t) to y(t) = s{x(t)} or d x(t)
dt . If an operator or a rule of operation is represented by the symbol O, the equation
O{x(t)} = y(t) (4.1)
implies that if the function x(t) is treated exactly as the operator O requires, we obtain the function y(t). For example, the operation O{ } = 4 d
dt{ } + 6 says that to get y(t), we must take the derivative of x(t), multiply by 4 and then add 6 to the result 4 d
dt{x(t)} + 6 = 4dxdt + 6 = y(t).
If an operation on the sum of two functions is equivalent to the sum of operations applied to each separately, the operator is said to be additive. In other words,
O{x1(t) + x2(t)} = O{x1(t)} + O{x2(t)} (for an additive operation) (4.2) If an operation on Kx(t) is equivalent to K times the linear operation on x(t) where K is a scalar, the operator is said to be homogeneous. In other words,
O{Kx(t)} = KO{x(t)} (for a homogeneous operation) (4.3)
Together, the two describe the principle of superposition. An operator O is termed a linear operator if it is both additive and homogeneous. In other words,
O{Ax1(t) + Bx2(t)} = AO{x1(t)} + BO{x2(t)} (for a linear operation) (4.4) If an operation performed on a linear combination of x1(t) and x2(t) produces the same results as a linear combination of operations on x1(t) and x2(t) separately, the operation is linear. If not, it is nonlinear. Linearity thus implies superposition. An important concept that forms the basis for the study of linear systems is that the superposition of linear operators is also linear.
REVIEW PANEL 4.1
A Linear Operator Obeys Superposition: aO{x1(t)} + bO{x2(t)} = O{ax1(t) + bx2(t)} Superposition implies both homogeneity and additivity.
Homogeneity: O{ax(t)} = aO{x(t)} Additivity: O{x1(t)} + O{x2(t)} = O{x1(t) + x2(t)}
Testing an Operator for Linearity: If an operator fails either the additive or the homogeneity test, it is nonlinear. In all but a few (usually contrived) cases, if an operator passes either the additive or the homogeneity test, it is linear (meaning that it will also pass the other). In other words, only one test, additivity or homogeneity, suffices to confirm linearity (or lack thereof) in most cases.
70 Chapter 4 Analog Systems
EXAMPLE 4.1 (Testing for Linear Operations) (a) Consider the operator O{ } = log{ }.
Since log(Kx) ̸= K log x, the log operator is nonlinear because it is not homogeneous. (b) Consider the operator O{x} = C{x} + D.
If we test for homogeneity, we find that O{Kx} = KCx + D but KO{x} = KCx + KD. The operator is thus nonlinear. Only if D = 0 is the operation linear.
(c) Consider the squaring operator O{ } = { }2, which transforms x(t) to x2(t). We find AO{x(t)} = Ax2(t) but O{Ax(t)} = [Ax(t)]2= A2x2(t).
Since the two are not equal, the squaring operator is also nonlinear. (d) Consider the derivative operator O{ } =d{ }
dt , which transforms x(t) to x′(t). We find that AO{x(t)} = Ax′(t) and O{Ax(t)} = x′(At) = Ax′(t).
The two are equal, and the derivative operator is homogeneous and thus linear.
Of course, to be absolutely certain, we could use the full force of the linearity relation to obtain O{Ax1(t) + Bx2(t)} = dtd [Ax1(t) + Bx2(t)] and AO{x1(t)} + AO{x2(t)} = Ax′1(t) + Bx′2(t). The two results are equal, and we thus confirm the linearity of the derivative operator.