Systems may be classified in several ways. Such classifications allow us to make informed decisions on the choice of a given method of analysis over others, depending on the context in which the system is considered. At the quantitative level, analog systems are usually modeled by differential equations that relate the output y(t) and input x(t) through the system parameters and the independent variable t. Using the notation y(n)(t) ≡ dny(t)
dtn , the general form of a differential equation may be written as
y(n)(t)+a1y(n−1)(t)+· · ·+an−1y(1)(t)+any(t) = b0x(m)(t)+b1x(m−1)(t)+· · ·+bm−1x(1)(t)+bmx(t) (4.5) The order n of the differential equation refers to the order of the highest derivative of the output y(t). It is customary to normalize the coefficient of the highest derivative of y(t) to 1. The coefficients ak and bk may be functions of x(t) and/or y(t) and/or t. Using the derivative operator sk ≡ dk
dtk with s0≡ 1, we may
recast this equation in operator notation as {sn+ a
1sn−1+ · · · + an−1s + an}y(t) = {b0sm+ b1sm−1+ · · · + bm−1s + bm}x(t) (4.6)
Notation: For low-order systems, we will also use the notation y′(t) ≡ d y(t)
dt , y′′(t) ≡
d2y(t)
4.2 System Classification 71
4.2.1
Linearity and Time Invariance
A linear system is one for which superposition applies and implies three constraints: 1. The system equation must involve only linear operators.
2. The system must contain no internal independent sources. 3. The system must be relaxed (with zero initial conditions).
REVIEW PANEL 4.2 What Makes a System Nonlinear?
(1) nonlinear elements or (2) nonzero initial conditions or (3) internal sources
For a linear system, scaling the input leads to an identical scaling of the output. In particular, this means zero output for zero input and a linear input-output relation passing through the origin. This is possible only if every system element obeys a similar relationship at its own terminals. Since independent sources have terminal characteristics that are constant and do not pass through the origin, a system that includes such sources is therefore nonlinear. Formally, a linear system must also be relaxed (with zero initial conditions) if superposition is to hold. We can, however, use superposition even for a system with nonzero initial conditions (or internal sources) that is otherwise linear. We treat it as a multiple-input system by including the initial conditions (or internal sources) as additional inputs. The output then equals the superposition of the outputs due to each input acting alone, and any changes in the input are related linearly to changes in the response. As a result, the response can be written as the sum of a zero-input response (due to the initial conditions alone) and the zero-state response (due to the input alone). This is the principle of decomposition, which allows us to analyze linear systems in the presence of nonzero initial conditions. Both the zero-input response and the zero-state response obey superposition individually.
REVIEW PANEL 4.3 Linearity from the Input-Output Relation
The input-output relation is a straight line passing through the origin. Examples: Input = v(t) Output = Cd v(t)
dt Input = v(t) Output = 1 L ! t −∞ v(t) dt
EXAMPLE 4.2 (Linearity from the Input-Output Relation)
The input-output relations for four systems are shown in Figure E4.2. Which systems are linear?
(a) Linear
Output Output Output Output
(d) Nonlinear (c) Nonlinear (b) Nonlinear Input Input Input Input
Figure E4.2 Input-output relations of the systems for Example 4.2
Only the first one is linear because the input-output relation is a straight line that passes through the origin. All other systems are nonlinear. The second system describes a half-wave rectifier, the third describes an internal source, and the fourth describes an operational amplifier.
72 Chapter 4 Analog Systems
4.2.2
Time-Invariant (Shift-Invariant) Systems
Time invariance (also called shift invariance) implies that the shape of the response y(t) depends only on the shape of the input x(t) and not on the time when it is applied. If the input is shifted to x(t − α), the response equals y(t − α) and is shifted by the same amount. In other words, the system does not change with time. Such a system is also called stationary or fixed.
Every element of a time-invariant system must itself be time invariant with a value that is constant with respect to time. If the element value depends on the input or output, this only makes the system nonlinear. Coefficients in a system equation that depend on the element values cannot show explicit dependence on time for time-invariant systems. In a time-varying system, the value of at least one system element is a function of time. As a result, the system equation contains time-dependent coefficients. An example is a system containing a time-varying resistor. In physical systems, aging of components often contributes to their time-varying nature.
Formally, if the operator O transforms the input x(t) to the output y(t) such that O{x(t)} = y(t), a time-invariant system requires that
If O{x(t)} = y(t) then O{x(t − t0)} = y(t − t0) (for a time-invariant operation) (4.7) REVIEW PANEL 4.4
Time Invariance from the Operational Relation
If O{x(t)} = y(t) then O{x(t − t0)} = y(t − t0) (shift input by α ⇒ shift output by α).
EXAMPLE 4.3 (Linearity and Time Invariance of Operators) (a) y(t) = x(t)x′(t) is nonlinear but time invariant.
The operation is O{ } = ({ })(d{ }
dt ). We find that
AO{x(t)} = A[x(t)x′(t)] but O{Ax(t)} = [Ax(t)][Ax′(t)] = A2x(t)x′(t). The two are not equal. O{x(t − t0)} = x(t − t0)x′(t − t0) and y(t − t0) = x(t − t0)x′(t − t0). The two are equal.
(b) y(t) = tx(t) is linear but time varying. The operation is O{ } = t{ }. We find that
AO{x(t)} = A[tx(t)] and O{Ax(t)} = t[Ax(t)]. The two are equal.
O{x(t − t0)} = t[x(t − t0)] but y(t − t0) = (t − t0)x(t − t0). The two are not equal.
(c) y(t) = x(αt) is linear but time varying. With t ⇒ αt, we see that AO{x(t)} = A[x(αt)] and O{Ax(t)} = Ax(αt). The two are equal.
To test for time invariance, we find that O{x(t − t0)} = x(αt − t0) but y(t − t0) = x[α(t − t0)]. The two are not equal, and the time-scaling operation is time varying. Figure E4.3C illustrates this for y(t) = x(2t), using a shift of t0= 2.
4.2 System Classification 73 x(t −2) y1(t) y1(t−2) y2(t) Delay 2 units
Not the same!! 1 2 6 x(t) 4 1 Time scale (compress by 2) 1 2 Time scale (compress by 2) 1 2 4 1 1 3
Figure E4.3C Illustrating time variance of the system for Example 4.3(c)
(d) y(t) = x(t − 2) is linear and time invariant. The operation t ⇒ t − 2 reveals that AO{x(t)} = A[x(t − 2)] and O{Ax(t)} = Ax(t − 2). The two are equal.
O{x(t − t0)} = x(t − t0− 2) and y(t − t0) = x(t − t0− 2). The two are equal. (e) y(t) = ex(t)x(t) is nonlinear but time invariant.
The operation is O{ } = e{ }{ } and reveals that
AO{x(t)} = Aex(t)x(t) butO{Ax(t)} = eAx(t)[Ax(t)]. The two are not equal. O{x(t − t0)} = ex(t−t0)x(t− t0) and y(t − t0) = ex(t−t0)x(t− t0). The two are equal.
4.2.3
Linear Time-Invariant Systems
The important class of linear time-invariant (LTI) systems are described by differential equations with constant coefficients. To test for linearity or time invariance of systems described by differential equations, we can formally apply the linearity or time-invariance test to each operation or recognize the nonlinear or time-varying operations by generalizing the results of the previous examples as follows.
1. Terms containing products of the input and/or output make a system equation nonlinear. A constant term also makes a system equation nonlinear.
2. Coefficients of the input or output that are explicit functions of t make a system equation time varying. Time-scaled inputs or outputs such as y(2t) also make a system equation time varying.
REVIEW PANEL 4.5
An LTI System Is Described by a Linear Constant-Coefficient Differential Equation (LCCDE) y(n)(t) + A
n−1y(n−1)(t) + · · · + A0y(t) = Bmx(m)(t) + Bm−1x(m−1)(t) + · · · + B0x(t)
All terms contain x(t) or y(t). All coefficients are constant (not functions of x(t) or y(t) or t). Notation: y(n)(t) = dny(t)
dtn , y(0)(t) = y(t) Also, y′(t) = d y(t)dt , y′′(t) = d 2y(t)
dt2
REVIEW PANEL 4.6
What Makes a System Differential Equation Nonlinear or Time Varying? It is nonlinear if any term is a constant or a nonlinear function of x(t) or y(t).
74 Chapter 4 Analog Systems
EXAMPLE 4.4 (Linearity and Time Invariance of Systems) (a) We test for the linearity and time invariance of the following systems:
1. y′(t) − 2y(t) = 4x(t). This is LTI.
2. y′′(t) − 2ty′(t) = x(t). This is linear but time varying.
3. y′(t) + 2y2(t) = 2x′(t) − x(t). This is nonlinear but time invariant. 4. y′(t) − 2y(t) = ex(t)x(t). This is nonlinear but time invariant. 5. y′(t) − 4y(t)y(2t) = x(t). This is nonlinear and time varying.
(b) What can you say about the linearity and time-invariance of the four circuits and their governing differential equations shown in the Figure E4.4B.
i2(t) + 3 − 3t Ω di(t) dt 2 +3i2(t) = v(t) di(t) dt 2 +3t i(t) = v(t) di(t) dt 2 +3i(t) + 4= v(t) di(t) dt 2 +3i(t) = v(t) 2 H i(t) v(t) + − − + 4 V 2 H i(t) v(t) + − 2 H i(t) v(t) + − 3 Ω 3 Ω 2 H i(t) v(t) + −
(a) LTI (b) Nonlinear (c) Nonlinear (d) Time varying
Figure E4.4B The circuits for Example 4.4(b)
For (a), 2i′(t) + 3i(t) = v(t). This is LTI because all the element values are constants. For (b), 2i′(t) + 3i2(t) = v(t). This is nonlinear due to the nonlinear element.
For (c), 2i′(t) + 3i(t) + 4 = v(t). This is nonlinear due to the 4-V internal source. For (d), 2i′(t) + 3ti(t) = v(t). This is time varying due to the time-varying resistor.
Implications of Linearity and Time Invariance
In most practical cases, if an input x(t) to a relaxed LTI system undergoes a linear operation, the output y(t) undergoes the same linear operation. For example, the input x′(t) results in the response y′(t). The superposition property is what makes the analysis of linear systems so much more tractable. Often, an arbitrary function may be decomposed into its simpler constituents, the response due to each analyzed separately and more effectively, and the total response found using superposition. This approach forms the basis for several methods of system analysis. Representing a signal x(t) as a weighted sum of impulses is the basis for the convolution method (Chapter 6), representing a periodic signal x(t) as a linear combination of harmonic signals is the basis for the Fourier series (Chapter 8), and representing a signal x(t) as a weighted sum of complex exponentials is the basis for Fourier and Laplace transforms (Chapters 9 and 11).
4.2 System Classification 75
4.2.4
Causal and Dynamic Systems
A causal or non-anticipating system is one whose present response does not depend on future values of the input. The system cannot anticipate future inputs in order to generate or alter the present response. Systems whose present response is affected by future inputs are termed noncausal or anticipating. A formal definition of causality requires identical inputs up to a given duration to produce identical responses over the same duration. Ideal systems (such as ideal filters) often turn out to be noncausal but also form the yardstick by which the performance of many practical systems, implemented to perform the same task, is usually assessed. A differential equation describes a noncausal system if, for example, the output terms have an argument of the form y(t) and an input term has the argument x(t + α), α > 0.
A dynamic system, or a system with memory, is characterized by differential equations. Its present response depends on both present and past inputs. The memory or past history is due to energy-storage elements that lead to the differential form. In contrast, the response of resistive circuits or circuits operating in the steady state depends only on the instantaneous value of the input, not on past or future values. Such systems are also termed instantaneous, memoryless, or static. All instantaneous systems are also causal. The system equation of an instantaneous system is algebraic, and the input and output are each of the form x(t) and y(t) (with identical scaling or shift, if any). Dynamic systems include (but are not limited to) systems described by differential equations.
REVIEW PANEL 4.7 Causal, Static, and Dynamic Systems
It is causal if the output y(t) does not depend on future inputs such as x(t + 1).
It is static if the output y(t0) depends only on the instantaneous value of the input x(t0). It is dynamic if energy storage is present, and y(t0) also depends on its own past history.
REVIEW PANEL 4.8
Noncausal and Static Systems from the System Differential Equation
It is noncausal if output terms have the form y(t) and any input term contains x(t + α), α > 0. It is static if no derivatives are present, and every term in x and y has identical arguments. EXAMPLE 4.5 (Causal and Dynamic Systems)
We investigate the following systems for causality and memory. (a) y′′(t) − 2ty′(t) = x(t). This is causal and dynamic.
(b) y(t) = x(t) + 3. This is causal and instantaneous (but nonlinear). (c) y(t) = 2(t + 1)x(t). This is causal and instantaneous (but time varying). (d) y′(t) + 2y(t) = x(t + 5). This is noncausal and dynamic.
(e) y′(t + 4) + 2y(t) = x(t + 2). This is causal and dynamic.
(f) y(t) = x(t + 2). This is noncausal and dynamic (the arguments of x and y differ).
(g) y(t) = 2x(αt) is causal and instantaneous for α = 1, causal and dynamic for α < 1, and noncausal and dynamic for α > 1. It is also time varying if α ̸= 1.
76 Chapter 4 Analog Systems