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C H A P T E R
8
TIME-VARYING SIGNALS
Circuits that work with time-dependent signals allow us to do operations
such as integration and differentiation with respect to time. The nervous
system makes incredibly effective use of the fact that there are numerous
events happening in the world that are functions of time; that they are
functions of time is the important thing. If sound, for example, did not
vary with time, it would not be sound. Your ear responds to derivative
signals; the time variation of the signal contains all the information your
ear needs. The steady (DC) part carries no information whatsoever.
People doing vision research take a picture out of the New York Times
or Life magazine and throw it at a computer. They try to get the computer to determine whether there is a tree in that picture. That is a hard
job; it is so hard that researchers are still arguing about how to do it, and
will be for years. It is one of those classic problems in vision.
The reason such problems are intractable is that they start with only a
small fraction of the information your eye uses to do the same job. Most
of the input to our sensory systems is actively generated by our body
movements. In fact, it is fair to view all sensory information as being
generated either by the movement of objects in the world around us, or
by our movements relative to those objects.
The retina is an excellent example of time-domain processing. The
retina does not recognize trees at all. Except for the tiny fovea in the
middle (with which we see acutely and which we use for activities such
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as reading), the vast majority of the area of the retina is devoted to the periphery.
Peripheral vision is not concerned with the precise representation of shapes. It is
overwhelmingly concerned with motion. For that reason, most of the bandwidth
in the optic nerve is used for time-derivative information.
The reason for this preoccupation with motion is that animals that noticed
something was jumping on them were able to jump out of the way, and they
did not get eaten. The ones that did not notice leaping predators got eaten.
So the ones that have survived have good peripheral vision. Peripheral vision is
almost totally concerned with detecting time derivatives, because nobody ever
got jumped on by something that was not moving.
An example of information generated by our natural body movements is
the visual cues about depth. For objects more than about 6 feet away from us,
most depth information comes from motion rather than from binocular disparity.
If we are in the woods, it is important to know which tree is nearest to us. We
move our bodies back and forth, and the shape (tree) that moves most, relative
to the background, is the closest. We use this technique to detect depth out
to great distances. People who navigate in the woods do it regularly without
being conscious of what they are doing. Motion cues are built into our sense
of the three-dimensionality of space. The derivative machinery in our peripheral vision gives us a great deal of information about what is in the world. It
is only when we use our attentive system, when we fasten on tiny details and
try to figure out what they are, that we use the shape-determining machinery in
the fovea.
In this and the next few chapters, we will discuss time-varying signals and the
kind of time-domain processing that comes naturally in silicon. In Chapter 9, we
will do integration with respect to time. Then, we will be in a position to discuss
differentiation with respect to time in Chapter 10. At that point, we will build
several important system components.
LINEAR SYSTEMS
By definition, a linear system is one for which the size of the output is
proportional to the size of the input. Because all real systems have upper limits
on the size of the signals they can handle, no such system is linear over an
arbitrary range of input level. Many systems, however, are close enough to linear
that the viewpoint that comes from linear systems provides a basis from which
we can analyze the nonlinearity.
For example, many people listen to their car radios while they drive. The
radio-receiver systems in all cars are equipped with automatic gain control.
Automatic gain control is a technique for maintaining a steady output amplitude,
independent of the input signal strength, so that, for example, the music does
not fade out when we drive under an overpass. In a linear system, if we cut the
size of the input in half, then the response at the output also will be cut in
half. So, by its very nature, a radio that has automatic gain control is a highly
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nonlinear system. On the other hand, when we receive music from the local
radio station, the program is broadcast without a lot of distortion. Absence of
distortion is one characteristic of a linear system. The people who designed the
radio were clever. They did not ask how loud the music was when they built
the automatic gain control; they asked how much signal they were getting from
the radio station—and they decreased the gain proportionally. So, if the signal
strength stays relatively constant over the period of oscillation of the sound
waveform, the receiver acts like a linear system even though it is, in another
sense, extremely nonlinear.
This mixed linear and nonlinear behavior is characteristic of many living
systems. Your ear, for example, acts in many ways like the radio receiver: It has
an automatic-gain-control system that allows it to work over a factor of 1012
in sound amplitude. In spite of this tremendous dynamic range, a particular
waveform is perceived as being essentially the same at any amplitude; a very
faint voice is just as intelligible as a very loud voice. The same comments hold
for the visual system.
In summary, although many sensory systems are decidedly nonlinear overall,
the way they are actually used allows much of their behavior to be represented,
at least to first order, by linear-system terminology. We use this terminology
as a language in which we can discuss any system that deals with signals, even
nonlinear systems. The language is a useful underlying representation.
THE RESISTOR–CAPACITOR (RC) CIRCUIT
We will illustrate the linear-system viewpoint with the specific electrical
circuit shown in Figure 8.1. We write Kirchhoff’s law for the node labeled V :
dV
V
= −C
R
dt
(8.1)
Any current that goes through the resistor necessarily originates from charge
stored on the capacitor; there is nowhere else from which that current can come.
The current that flows through the resistor is V /R, and the current through the
capacitor is −C dV /dt.
V
R
C
FIGURE 8.1 Schematic diagram of the
simple resistor–capacitor (RC) circuit. Because there is no input, any current through
the resistor must come from charge stored
on the capacitor.
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We can rewrite Equation 8.1 as
V + RC
dV
=0
dt
(8.2)
Notice that the first term has the units of voltage and the second term contains
dV /dt, so the units would not be correct if the term in front of dV /dt did not
have the units of time. The RC combination is called the time constant of the
circuit, τ , which has units of time. We can develop an intuition for this relation
from the hydraulic analogy in Figure 2.1 (p. 14). There the water tank represents
a capacitor, and the friction in the pipe represents the resistor. The time it takes
to empty the tank is proportional to the volume of water in the tank. It is also
inversely proportional to the flow rate through the pipe. Because the flow rate
is inversely proportional to the friction in the pipe, the emptying time will be
proportional to the pipe resistance times the tank capacity.
We thus rewrite Equation 8.2 as
V +τ
dV
=0
dt
(8.3)
There is only one function that has a derivative equal to itself: the exponential.
So the solution to Equation 8.3 is
V = V0 e−t/τ
(8.4)
Because this is a linear system, the constant V0 can be any size; it is just the
voltage on the capacitor at t = 0. The circuit behaves in the same way whether
we start with 1 volt or 10 volts.
Equation 8.3 is a linear differential equation because it contains no terms
such as V 2 or V dV /dt. It is time invariant because none of the coefficients are
explicit functions of t. It is a homogeneous equation because its driving term
on the right-hand side is zero. It is a first-order equation because it contains no
higher-order derivatives than dV /dt. In most equations representing real circuits,
the right-hand side is not equal to zero. A typical circuit is driven by one circuit,
and drives another: The output signal of one circuit is the input signal of the
next. The response of the circuit depends on what the driving term is. The most
important defining characteristic of any linear system is its natural response—
the way the circuit responds when it is out of equilibrium and is returning to
equilibrium with time. The natural response is a property of the circuit itself, if
you just leave that circuit alone; it does not have anything to do with what is
driving the circuit. Equation 8.4 is the natural response of Figure 8.1.
HIGHER-ORDER EQUATIONS
Of course, not all linear differential equations are first-order, as Equation 8.3
is. Circuits with second-order equations are common; we will study one in Chap-
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ter 11. A simple form of such a second-order equation is
d2 V
+ βV = 0
dt2
(8.5)
We can substitute Equation 8.4 into Equation 8.5, to obtain
1
+β =0
τ2
independent of the sign of the exponent. If β is negative, the solution is, indeed, an exponential. For positive β, we recall the derivatives of trigonometric
functions:
d
(sin x) = cos x
dx
d
(cos x) = − sin x
dx
(8.6)
(8.7)
Substituting Equation 8.6 into Equation 8.7, and comparing with Equation 8.5,
we obtain
V = cos β t
√
Similarly, sin β t also is a solution.
The solutions of Equation 8.5 thus will be sines and cosines
√ when β is positive, and exponentials when β is negative. In both cases, 1/ β has the units of
time. The annoying property of higher-order equations is that we get a solution
of a completely different kind when we change one parameter of the system by
a small amount. It would be nice to have a unified representation in which any
solution to a linear system could be expressed. The key to such a representation
is to use exponentials with complex arguments.
We will first review the nature of complex numbers. Then we will develop the
use of complex exponentials as a universal language for describing and analyzing
time-varying signals.
COMPLEX NUMBERS
A complex number N = x+jy has a real part Re(N ), x, and an imaginary
part Im(N ), y, where both x and y are real numbers. In all √complex-number
notation, we will use the electrical-engineering convention j = −1. Most other
fields use i for the unit imaginary number, but such usage prevents i from being
used as the current variable in electrical circuits. Such a loss is too great a price
for electrical engineers to pay.
Any complex number can be visualized as a point in a two-dimensional
complex plane, as shown in Figure 8.2. Mathematically, we have only redefined
the set of points in the familiar x, y plane, and have packaged them so that
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Im(N )
N
jy
FIGURE 8.2 Graphical representation of
a complex number N. Any number can be
represented as a point in the plane. The x
coordinate of the point is the real part of the
number; the y coordinate is the imaginary
part. Alternately, we can think of the number
as a vector from the origin to the point. The
complex conjugate N ∗ has the same real
part, but its imaginary part is of the opposite
sign.
R
φ
−φ
Re(N )
x
R
−jy
N∗
any point can be dealt with as a single entity. This repackaging underlies a
beautiful representation of signals, as we will see shortly—that representation is
dependent on several operations on complex numbers. The first operation of note
is addition; the sum of two complex numbers A = x1 + jy1 and B = x2 + jy2
is just the familiar form for the addition of ordinary vectors:
A + B = (x1 + x2 ) + j(y1 + y2 )
In other words, the real and imaginary parts of the numbers are simply added
separately.
Multiplication of two complex numbers also is defined by the rules of ordinary
algebra, where j 2 has the value −1:
A × B = x1 x2 + jx1 y2 + jx2 y1 + j 2 y1 y2
= x1 x2 − y1 y2 + j(x1 y2 + x2 y1 )
The product of complex numbers is much more simply expressed when the numbers are represented in polar form.
Polar Form
A natural alternate but completely equivalent form for specifying a point
in the complex plane is to use the polar form. In the polar form, we give the
point’s radius R from the origin, and the angle φ between the horizontal axis
and a line from the origin to the point. In the engineering community, R is called
the magnitude of the complex number, and φ is the phase angle, or simply
the phase. The key observation is that an expression for each unique point N in
the plane can be written as
N = x + jy = Rejφ
(8.8)
Because ejφ1 ×ejφ2 is equal to ej(φ1 +φ2 ) , multiplication by an exponential with an
imaginary argument is equivalent to a rotation in the complex plane. The polar
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form allows us to develop a natural definition of the multiplication operation
on complex numbers. The product of A = R1 ejφ1 and B = R2 ejφ2 is
A × B = (R1 × R2 )ej(φ1 +φ2 )
Complex Conjugate
We can verify these relations and develop insight into complex numbers in
the following way. The complex conjugate N ∗ of the complex number N has
the same magnitude as N does, but has the opposite phase angle:
N ∗ = x − jy = Re−jφ
(8.9)
∗
The product N × N will have an angle of zero, so it is a real number equal to
the square of the magnitude of N :
N × N ∗ = Rejφ × Re−jφ = R2 = (x + jy) × (x − jy) = x2 + y 2
The magnitude R of the number N (also called its length, modulus, or absolute value) is given by the square root of N × N ∗ :
R = x2 + y 2
From Figure 8.2, it is clear that the real part x is just the projection of the
vector onto the horizontal axis:
x = Re(N ) = R cos φ
(8.10)
Similarly, the imaginary part y is just the projection of the vector onto the
vertical axis:
y = Im(N ) = R sin φ
(8.11)
Substituting Equations 8.10 and 8.11 into Equations 8.8 and 8.9, we obtain definitions of the exponentials of imaginary arguments in terms of ordinary
trigonometric functions:
cos φ + j sin φ = ejφ
−jφ
cos φ − j sin φ = e
(8.12)
(8.13)
Addition and subtraction of Equations 8.12 and 8.13 lead to the well-known
identities that define ordinary trigonometric functions in terms of exponentials
of imaginary arguments:
sin φ =
ejφ − e−jφ
2j
ejφ + e−jφ
2
We have seen that there is a direct connection between exponentials with
imaginary arguments and the sine and cosine functions of ordinary trigonometry.
We might suspect that using exponentials with complex arguments would give
cos φ =
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us a way of representing functions with properties intermediate between the
oscillatory behavior of sine waves and the monotonically growing or decaying
behavior of exponentials. That suspicion will be confirmed.
COMPLEX EXPONENTIALS
Exponentials with complex arguments are unique in the world of mathematical functions. They form the vocabulary of a language that has evolved over
the years for representing signals that vary as a function of time. The complete
formalism is called the s-plane; it is a beautiful example of representation. Its
elegance stems in large measure from the fundamental mathematics that lies
behind it. The formal derivations are beyond the scope of our discussion; they
can be found in any text on complex variables, or in any text on linear system
theory [Siebert, 1986; Oppenheim et al., 1983]. The complex variable s = σ + jω
will be the new central player of our drama. As the plot unfolds, we will find this
complex variable multiplying the time t in every problem formulation involving
the time response of linear systems.
We can illustrate the complex-exponential approach by considering the general second-order equation
d2 V
dV
+α
+ βV = 0
dt2
dt
(8.14)
This equation can have solutions that are sine waves, or exponentials, or some
combination of the two. The observation was made a long time ago that, if we
use exponentials with a complex argument, we can always get a solution. We
therefore substitute est for V into Equation 8.14. Because
d st
(e ) = sest
dt
and
d2 st
(e ) = s2 est
dt2
Equation 8.14 becomes
s2 est + αsest + βest = 0
(8.15)
The complex exponential term est appears in every term of Equation 8.15, and
therefore can be canceled out, leaving the expression for s that must be satisfied
if est is to be a solution to the original differential equation:
s2 + αs + β = 0
(8.16)
The fact that derivatives of any order give back the original function est is
a special case of a much more general body of theory. If some operation on a
function gives a result that has the same form as the original function (differing
by only a constant factor), the function is said to be an eigenfunction of that
operation. Complex exponentials are eigenfunctions of any linear, time-invariant
operation, of which differentiation with respect to time is but one example.
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Equation 8.16 is a quadratic in s, with solutions
−α ± α2 − 4β
s=
2
(8.17)
Depending on the (real) values of α and β, the two values of s given by Equation 8.17 may be real, imaginary, or complex. If α2 − 4β is positive, both roots
will be real. If α2 − 4β is negative, the roots will be complex conjugates of each
other. That complex roots come in conjugate pairs turns out to be a general
result, although we have shown it for only the second-order system.
So what does it mean to have a complex waveform when we have to take
the measurement on a real oscilloscope? As we might guess, a real instrument
can measure only the real part of the signal. Real differential equations for real
physical systems must have real solutions. Simply looking at the real part of a
complex solution yields a solution also, because
Re(f (t)) =
f(t) + f ∗ (t)
2
and the roots come in conjugate pairs.
So, if our signal voltage is V = est , we will measure
Vmeas = Re (est ) = Re (e(σ+jω)t ) = eσt Re (ejωt ) = eσt cos (ωt)
For most signals, σ will be negative, and the response will be a damped
sinusoidal response. If σ is equal to zero, the signal is a pure sinusoid, neither
decaying or growing. If σ is greater than zero, the signal is an exponentially
growing sinusoid. If ω is equal to zero, the signal is a growing or decaying exponential, with no oscillating component. If we wish to represent a sinusoid with a
different phase φ—for example, a sine instead of a cosine—the signal multiplied
by ejφ still will be a solution:
Re(ejφ × est ) = Re(ejφ × e(σ+jω)t ) = eσt cos(ωt + φ)
The procedure we carried out on the second-order equation can be generalized to linear differential equations of higher order. No matter what the order of
the differential equation is, we can always substitute in est as a solution, and get
an equation in which the est appears in all the terms. Then we can go through
and cancel out the est from every term, which leaves us with an equation involving only the coefficients. For a differential equation of any order, we get a
polynomial in s with coefficients that are real numbers. They are real numbers
because they come from the values of real circuit elements, such as resistors and
capacitors.
We thus have a formal trick that we use with any time-invariant linear
differential equation. We convert the manipulation of a differential equation into
the manipulation of an algebraic equation in powers of the complex variable s.
We achieved this simplification by using a clever representation for signals.
If a term is differentiated once, we get an s. If a term contains a second
derivative, we get an s2 . (After a while, you will find that you can do this
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operation in your head.) What the procedure allows us to do is to treat s as an
operator, that operator meaning derivative with respect to time. If we apply the
s operator once, it means d/dt. If we take the derivative with respect to time
twice, that means we apply the operator s, and then apply it again, so we write
the composition of the operator with itself as s2 . In general,
sn ⇔
dn
dtn
Similarly, we can view 1/s as the operator signifying integration with respect to
time. This way of thinking about s as an operator was first noticed by a man by
the name of Oliver Heaviside [Josephs, 1950].
THE HEAVISIDE STORY
Oliver Heaviside was the person who discovered that the ionosphere would
reflect radio waves; there is a Heaviside layer [van der Pol, 1950] in the ionosphere that is named after him. He was thinking about the transient response
of electronic circuits, and he noticed that, if he was careful, he could treat the
variable s as the operator d/dt. He could perform the substitution and instantly
convert differential equations into algebraic equations. He derived a unified way
of treating circuits that involved only algebraic manipulations on these operators. It became known as the Heaviside operator calculus [van der Pol, 1950].
This was back in the old days, when the only thing most engineers worked with
was sine waves. Heaviside had the whole method worked out to the point that,
by multiplying and dividing by polynomials in s, he could actually predict the
response due to step functions, impulses, and other kinds of transient inputs—it
was beautiful stuff. It was a generalization of the behavior of linear systems in
response to signals other than sine waves.
Heaviside operator calculus is a bit like the Mendeleev periodic table. Mendeleev figured out that different elements were related systematically by simple
reasoning from special cases. Heaviside was smart enough to see that there had
to be a general method, and he was able to determine what it was, even though
he did not understand the deep mathematical basis for it. Ten years after he
developed his calculus and was able to predict the behavior of many circuits,
mathematicians started to notice that his method was successful and started
wondering what it really represented. Finally, somebody else said, “Ah! you mean
the Laplace transform.”
So, it turns out that these polynomials in s are Laplace transforms; they are
covered in all the first courses in complex variables. All the books about linear
systems are full of them now—they have become the standard lore. Notice, however, that the technique was invented by someone who did not work out the mathematical details; he just went ahead and started working real problems that were
very difficult to solve by any other approach. When he perfected the method and
made it useful, somebody noticed that Laplace had written down the mathemat-
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ics in abstract form 100 years before. So the Laplace transform should be called
the Heaviside–Laplace transform. But formalism often is viewed as standing on
its own, without regard for the applications insight that breathes life into it.
DRIVEN SYSTEMS
Systems that are not being driven are not very interesting—like your eye
when it is perfectly dark or your ear when it is perfectly quiet. Real systems do
not just decay away to nothing—they are driven by some input. The right-hand
side of a differential equation for a driven system is not equal to zero; rather, it
is some driving function that is a function of time. A differential equation for the
output voltage has the input voltage as its right-hand side. The input voltage
drives the system.
Now we need more than the homogeneous solution. Instead, we will examine
the driven solution to the equation. We apply some input waveform, and observe the output waveform. Most physical systems have the property that, if we
stop driving them, their output eventually dies away. Such systems are said to be
stable. Because the system is linear, no matter what shape of input waveform
we apply, the size of the output signal is proportional to the size of the input
signal. So the output voltage can be expressed in terms of the input voltage by
what electrical engineers call a transfer function, Vout /Vin . For a linear system,
Vout /Vin will be independent of the magnitude of Vin . If we double Vin , we double
Vout . That is why a transfer function has meaning—because it is independent of
the size of the input signal.
Consider the driven RC circuit of Figure 8.3. The output voltage satisfies
the circuit equation
C
dVout
1
= (Vin − Vout )
dt
R
(8.18)
In s-notation, Equation 8.18 can be written as
Vout (τ s + 1) = Vin
(8.19)
where, as before, τ is equal to RC. If the input voltage is zero, Equation 8.19
is identical to Equation 8.3. Rearranging Equation 8.19 into transfer function
form, we obtain
Vout
1
=
Vin
τs + 1
R
Vin
Vout
C
FIGURE 8.3 Simple RC circuit driven by a
voltage Vin . The homogeneous response of
this circuit is the same as that of Figure 8.1.
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The term 1/(τ s + 1) in Equation 8.20 is the s-plane transfer function H(s)
of the circuit of Figure 8.3. In general, for any time-invariant linear system,
Vout = H(s)Vin
As we noted, there is a form of input waveform Vin = est that has a special
place in the theory of linear systems, because it is an eigenfunction of any linear,
time-invariant operation. The response rs (t) of a system with transfer function
H(s) to the input est is
rs (t) = H(s)est
(8.21)
We will use the transfer function H(s) in two ways: to derive the circuit’s
frequency response when the circuit is driven by a sine-wave input, and to derive
the response of a circuit to any general input waveform. Equation 8.21 will allow
us to derive a fundamental relationship between the system’s impulse response
and its s-plane transfer function.
The reason we have devoted most of this chapter to developing s-notation
is that we wish to avoid re-deriving the response of each circuit for each kind of
input. Heaviside undoubtedly had the same motivation.
SINE-WAVE RESPONSE
Even in today’s world of digital electronics, people do apply sine waves to
circuits, and it is useful to understand what happens under these conditions. This
is what happens to your ear when you listen to an oboe. We use the transfer
function in s-notation to derive the response when a system is driven by a sine
wave.
Transfer Function
We saw how the complex exponential est can be used to represent both transient and sinusoidal waveforms. This approach allows us to use the s-notation
for derivatives, as described earlier. It also illustrates clearly how a linear circuit driven by a single-frequency sinusoidal input will have an output at that
frequency and at no other, as we will now show.
We can determine the system response to a cosine-wave input by evaluating
the system response when Vin is equal to ejωt . We notice that this form of input
is a special case of the more general form Vin = est used in Equation 8.21, with
s = jω. The real part of the response of a linear system to the input ejωt carries
exactly the same information as does the response to a cosine. Hence, by merely
setting s equal to jω in the transfer function, we can evaluate the amplitude and
phase of the response.
To understand what is going on, we will return to our derivation of the response of a linear differential equation, starting with Equation 8.21. Because the
equation is linear, the exponential ejωt will appear in every term. No matter how
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many derivatives we take, we cannot generate a signal with any other frequency.
The ejωt will cancel out of every term, leaving the output voltage equal to the
input voltage multiplied by some function of ω. That function is just the transfer
function with s = jω. For any given frequency, the transfer function, in general,
will be a complex number. It often is convenient to express the number in polar
form:
Vout (ω)
= R(ω)ejφ(ω)
Vin (ω)
R(ω) is the magnitude |Vout /Vin | of the transfer function, and φ is the phase.
RC-Circuit Example
We will, once again, use the RC circuit of Equation 8.19 as an example:
Vout
1
1
=
=
Vin
τs + 1
jωτ + 1
At very low frequencies, ωτ is much less than 1 and Vout is very nearly equal
to Vin . For very high frequencies, ωτ is much greater than 1, and
Vout
1
−j
≈
=
Vin
jωτ
ωτ
(8.22)
The −j indicates a phase lag of 90 degrees. The 1/ω dependence indicates that
in this range the circuit functions as an integrator.
The most convenient format for plotting the frequency response of a circuit
is shown in Figure 8.4. The horizontal axis is log frequency, the vertical axis is
log |Vout /Vin |. Logarithms are strange; zero is −∞ on a log scale. When we plot
log ω, zero frequency is way off the left edge of the world. By the time we ever
see it on a piece of graph paper, ω is not anywhere near zero anymore—it may
be three orders of magnitude below 1/τ . While we are crawling up out of the
1.0
| VVout |
in
0.1
101
102
103
Frequency
104
105
Hz
FIGURE 8.4 Response of the circuit of Figure 8.3 to a sine-wave input. The log of the
magnitude of Vout /Vin is plotted as a function of the log of the frequency. Because the
circuit is linear, the frequency of the output must be the same as that of the input.
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hole near zero, we are going through orders and orders of magnitude in ω, and
nothing is happening to the response of the system. As a result, the response on
a log plot is absolutely dead flat for decades, because −∞ is a long way away.
Eventually, we will get up to where ω is on the same scale as 1/τ is, so there
will be one order of magnitude at which the scale of frequency is the same as the
scale of 1/τ . Above that, ω will be very large compared with the scale of 1/τ .
When we get to the order of magnitude at which ωτ is near 1, the denominator
starts to increase and the response starts to decrease. At frequencies much higher
than 1/τ , the response decreases with a slope equal to −1, corresponding to the
limit given in Equation 8.22.
At low frequencies, the response is constant; at high frequencies, it varies as
1/ω; in between, it rounds off gracefully. At ω = 1/τ ,
Vout
1
=
Vin
j+1
(8.23)
2
2
Because the magnitude of a complex
√ number N is (Re(N )) + (Im(N )) , the
magnitude of √Equation 8.23 is 1/ 2. So, at that frequency, the amplitude is
decreased by 2.
The plot makes sense when we look at the circuit. The capacitor stores charge
when everything is DC (nothing is changing with respect to time); it charges up
to the input voltage and does not draw any more current. If there is no current,
there is no voltage drop across the resistor, so everything remains static, and
Vout is equal to Vin . If we suddenly change Vin , the output will exponentially
approach the new value with time constant τ .
As the input frequency increases, the imaginary part of the response becomes
more negative. This condition is called phase lag; the output waveform lags
behind the input waveform. The lag occurs because the capacitor takes time to
catch up! Another way of looking at it is that the capacitor is integrating the
input, and the integral of a cosine is a sine, which lags behind the cosine by
90 degrees. If the input starts up instantly, it still takes a while for the output
to catch up—it is lagging. By convention, we refer to the lagging phase angle φ
as negative.
RESPONSE TO ARBITRARY INPUTS
We can derive the response of any stable, linear, time-invariant system to
an arbitrary input waveform in the following manner. Let us suppose that we
apply an input f (t) to our system, starting at t = 0, as shown in Figure 8.5(a).
We can think of making up that waveform out of narrow pulses, like the one
shown in Figure 8.5(b); this pulse, which we will call p (t), has width and unit
amplitude. In other words, p (t) is equal to 1 in the narrow interval between
t = 0 and t = , and is zero for all other t. We can build up an approximation
to the arbitrary waveform f(t) starting at t = 0 by placing a pulse of height
f (0) at t = 0, followed by a pulse of height f() at t = , followed by a pulse of
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CHAPTER 8 TIME-VARYING SIGNALS
1
f(t )
p(t )
t
(a)
(b)
q(t )
t
(c)
(d)
∆
t
t
(e)
(f)
FIGURE 8.5 This sequence of graphs illustrates how an arbitrary waveform can be built
up from pulses, and how the response of a system to the waveform can be built up from the
responses to the individual pulses. The horizontal axis of all graphs is time; the vertical axis is
amplitude.
The arbitrary waveform (a) is the input to a linear, time-invariant system. We can approximate the input waveform as a sum of narrow pulses, as shown in (c). In (b), the pulse p(t )
starts at time t, and has unit amplitude, and width . The input waveform is approximated by a
sum of these pulses, each weighted by the value of the function at the time t, as shown in (c).
The response of the linear system to the pulse p(t ) is shown in (d). To illustrate the summation
process, we choose three individual pulses, as shown in (e). The individual responses of the
system to these three pulses are shown in (f). At any measurement time t , the response of the
system is just the sum of the individual values where the curves cross the time t . This picture
is a literal interpretation of Equation 8.25.
A more elegant formulation of the result is suggested by the response curve in (e), which
runs backward from the measurement time. The value of this curve at any time ∆ before the
measurement is taken can be used as a weighting function, determining the contribution to
the result of the pulse starting at that time. This interpretation is given in mathematical form
in Equation 8.26. In the limit of small , the shape of the system response q (t ) becomes
independent of , and its amplitude becomes directly proportional to . That limit allows us
to define the impulse response h(t ) in Equation 8.27. A continuous version of the sum shown
graphically in (e) can then be carried out; it is called the convolution of the waveform with the
impulse response, as given in Equation 8.28.
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PART III DYNAMIC FUNCTIONS
height f (2) at t = 2, and so on, as shown in Figure 8.5(c). Because the function
f (t) does not change appreciably over the time , we can express the pulse with
amplitude f(n) occurring at time t = n as f(n) p (n). Because the pulses do
not overlap, we can express the discrete approximation of Figure 8.5(c) as
f(t) ≈
∞
f(n)p (n)
(8.24)
n=0
The response of our system to the pulse p (t) will be some function q (t);
a typical response is shown in Figure 8.5(d). Because the system is linear, its
response to a pulse A p (t) will be A q (t); this property is, in fact, the definition
of a linear system—the form of the response does not depend on the amplitude
of the input, and the amplitude of the response is directly proportional to the
amplitude of the input. Because the system is time invariant, its response to a
pulse identical to p (t) but starting at a later time t + n is simply q (t + n). By
the principle of linear superposition mentioned in Chapter 2, we can approximate
the response of the system to the input f(t) as the sum of the responses to each
of the individual input pulses of Equation 8.24. Each pulse initiates a response
starting at its initiation time t. The measurement time t occurs a period t − t
later, and the response at t is therefore q (t − t). For the pulse train shown in
Figure 8.5(c), the response r(t ) of the system at measurement time t = N to
a pulse occurring at time t = n is therefore q ((N − n)).
Real physical systems, linear or nonlinear, are causal—inputs arriving after
the measurement time cannot affect the output at that time. Causality is expressed for our linear system by the fact that q (t) is zero for t less than zero.
For this reason, terms in Equation 8.25 with values of n greater than N will
make no contribution to the sum. We can therefore approximate the response of
our system to the input f (t) as a sum up to n = N :
N
r(N ) ≈
f(n) q ((N − n))
(8.25)
n=0
We can develop an intuition for the sum in Equation 8.25 by plotting a few of
the individual terms, as shown in Figure 8.5(f).
The time-invariant property of our system allows us to interpret this result
in a much more elegant way than is expressed by Equation 8.25. We notice that
the argument of q represents the interval between the time f is evaluated and the
time the measurement is taken. We can thus view q as a weighting function that
determines the contribution of f to the response as a function of time into the
past. By making a change of variables, we can express the result of Equation 8.25
in terms of the time-before-measurement ∆ = (N − n):
r(t ) ≈
N
f(t − ∆) q (∆)
∆=0
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CHAPTER 8 TIME-VARYING SIGNALS
This interpretation of the result is shown pictorially in Figure 8.5(e). Notice that
our original time scale has disappeared from the problem; it has been replaced
by the relevant variables associated with the time when the output is measured,
and with the time before the measurement was taken.
We can derive a continuous version of Equation 8.26 by noticing that no
physical system can follow an arbitrarily rapid change in input—a mechanical
system has mass, and an electronic system has capacitance. A finite current
flowing into a finite capacitor for an infinitesimal time can make only an infinitesimal change in the voltage across that capacitor. For that reason, there is
a lower limit to the width of our pulses, below which the form of the response
does not change. As the width of each pulse becomes smaller, the pulse amplitude stays the same in order to match the curve ever more closely, and the
number of pulses N increases to fill the duration of the observation.
We can reason about the behavior in that limit by considering the response
to two pulses of equal amplitude that are adjacent in time. Because is very
small compared to the time course of the response q (t), the response due to
each of the pulses will be nearly indistinguishable. The response when both
pulses are applied therefore will be nearly identical in form, and will be twice
the amplitude of the response to either pulse individually. In other words, in the
limit of very narrow pulses, the shape of the response becomes independent of
pulse width, and the amplitude of the response becomes directly proportional to
the pulse width. This limit allows us to define the impulse response h(t) of
the system:
h(t) = lim
→0
1
q (t)
(8.27)
The impulse response h(t) of any stable, linear, time-invariant analog system
must approach zero for infinite time. This property of finite impulse response
allows us to write Equation 8.26 as an integral in the continuous variables t and
∆, including the effect of inputs into the indefinite past. Because the original
time scale does not appear in the formulation, we use the variable t for the time
of measurement; we use ∆ for the time before measurement.
∞
r(t) =
f (t − ∆) h(∆) d∆
(8.28)
0
Notice that d∆ is the limit of , canceling the 1/ behavior of h(∆) given by
Equation 8.27.
Equation 8.28 is called the convolution of f with h, and it often is written
r = f ∗ h. It is the most fundamental result in the theory of linear systems.
In practical terms, it means that we can fully characterize a circuit operating in
its linear range by measuring the circuit’s impulse response in the manner we
have described. We then can compute the response of the system to any other
input waveform, as long as the circuit stays in its linear range. We will have
many occasions to apply this principle in later chapters.
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PART III DYNAMIC FUNCTIONS
Relationship of H(s) to the Impulse Response
We are now in a position to derive one of the truly deep results in linear
system theory by applying an input est to our system. From Equation 8.28, the
response rs (t) of the system to this input is
∞
rs (t) =
es(t−∆) h(∆) d∆
(8.29)
0
In Equation 8.21, we derived the response to this same input in terms of H(s):
rs (t) = H(s)est
(8.21)
But these two expressions represent the same response. Equating Equations 8.29
and 8.21, we find that
∞
es(t−∆) h(∆) d∆
H(s)est =
0
−st
Or, multiplying both sides by e
, we conclude that
∞
H(s) =
e−s∆ h(∆) d∆
(8.30)
0
Because ∆ is just a dummy variable of integration, it could be any other variable
as well. The impulse response is specified with respect to the time t, so we will
write Equation 8.30 using t as our variable of integration:
∞
H(s) =
e−st h(t) dt
0
The s-plane transfer function is the product of e−st and the impulse response h(t),
integrated over time. This integral is known as the Laplace transform of the
transfer function. The use of such transforms to determine the properties of linear
systems is a beautiful topic of study, and is highly recommended to those readers
who are not already familiar with the material [Seibert, 1986].
Impulse Response
In general, the transfer function of any circuit formed of discrete components
will be a quotient of two polynomials in s:
Vout
N (s)
=
Vin
D(s)
We can always rewrite such a transfer function as
Vout D(s) = Vin N (s)
The homogeneous response of the system is the solution when Vin is equal to 0:
Vout D(s) = 0
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CHAPTER 8 TIME-VARYING SIGNALS
In other words, the homogeneous response of the system is made up of signals corresponding to the roots of the denominator of the transfer function.
These roots are called the poles of the transfer function. Each root, or complexconjugate pair of roots, will represent one way in which the system can respond.
The response function always is est , with the value of s corresponding to the
particular root. Each such function is called a natural mode of the system.
We can illustrate the procedure with the simple RC circuit of Figure 8.3.
The denominator D(s) of the transfer function of Equation 8.19 was τ s + 1. The
solution of Equation 8.31 is s = −1/τ . We obtain the corresponding mode by
substituting this value of s into V = est ; the result is
V = V0 e−t/τ
This is the same result as the one we found by direct solution of the differential
equation in Equation 8.4.
The homogeneous response itself is sufficient to answer the most important
questions about a system—in particular, whether it is stable, and by what margin. The poles of a stable system must have a negative real part—those of an
unstable system have a positive real part. Questions of stability therefore are
related not to the form of the input waveform, but rather to only the nature of
the natural modes.
The total homogeneous response of the system always is the sum of the individual modes, each of which has a certain amplitude determined by the initial
conditions—the voltages stored on capacitors at t = 0. If we know the initial
values of all voltages and currents in the network, we have enough information
to determine uniquely the amplitude of each of the natural modes that sum together to make up the response of the system. An impulse and a step function
are examples of two common input waveforms that do not change after t = 0,
so the response to either of these inputs will be special cases of the natural response of the system. The amplitudes of the natural modes can be determined
from the values of the voltages and currents just after t = 0. These initial values can, in turn, be determined from the values of the voltages and currents
just before t = 0, and from the amplitude of the input waveform. Using this
approach, we can derive the impulse response of a circuit, and we can use that
response in Equation 8.28 to predict the behavior of the circuit to any input
waveform.
Although the application of these techniques gives us a powerful and useful method for deriving system properties, we must avoid the temptation to
apply this approach blindly to a new situation. All physical circuits are inherently nonlinear. Over a suitably limited range of signal excursions, it usually
is possible to make a linear approximation and to apply the results of linear
system theory. Normal operation of a circuit, however, may extend well beyond
the range of linear approximation. We must always check the conditions under
which a linear approximation is valid before we apply that approximation to
a specific circuit.
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SUMMARY
We have seen by the example of a simple RC circuit how the complex exponential notation can give us information about the response of the circuit to
both transient and sinusoidal inputs. We will use the s-notation in all discussions
of the time response of our systems to small signals. Once the systems are driven
into their nonlinear range, the results of linear system theory no longer apply.
Nonetheless, these results still provide a useful guide to help us determine the
large-signal response.
REFERENCES
Josephs, H.J. Heaviside’s Electric Circuit Theory. London: Methuen, 1950.
Oppenheim, A.V., Willsky, A.S., and Young, I.T. Signals and Systems.
Englewood Cliffs, NJ: Prentice-Hall, 1983.
Siebert, W.M. Circuits, Signals and Systems. Cambridge, MA: MIT Press 1986.
van der Pol, B. Heaviside’s operational calculus. The Heaviside Centenary
Volume. London: The Institution of Electrical Engineers, 1950, p. 70.
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