8
Linear System Theory
8.1 INTRODUCTION
In this chapter, the fundamental relationships between the input and output of
a linear time invariant system, as illustrated in Figure 8.1, are detailed.
Specifically, the relationships between the input and output time signals,
Fourier transforms and power spectral densities, are established. Such relation-
ships are fundamental to many aspects of system theory, including analysis of
noise in linear systems, and low noise amplifier design.
The relationships between the parameters defined in Figure 8.1, and proved
in this chapter are,
y(t) :

R

x()h(t 9 ) d (8.1)
Y (T, f ) H(T, f )X(T, f ) (8.2)
where X and Y are the respective Fourier transforms, evaluated on the interval
[0, T ], of the signals x and y. However, as will be shown in this chapter, the
relationship defined in Eq. (8.2) is an approximation. If both x, h + L , then the
relative error in this approximation can be made arbitrarily small by making
T sufficiently large. However, stationary random signals are not Lebesgue
integrable on the interval (0, -) and hence, this convergence is not guaranteed.
However, it is shown, for a broad class of signals and random processes,
including periodic signals and stationary random processes, that the corre-
sponding relationship between the input and output power spectral densities,
namely,
G
7
(T, f ) "H(T, f )"G
6

Y
, respectively, are the power spectral densities of the input and output
random processes.
∆
t
∞
∫
–∞
δ
∆
(t)dt = 1
δ
∆
(t)
1
∆
−
Figure 8.2 Definition of the function 

.
8.2 IMPULSE RESPONSE
Fundamental to defining the impulse function of a time invariant linear system,
is the function 

defined by the graph shown in Figure 8.2. The response of a
linear time invariant system to the input signal 

is denoted h

.

β
i − 1
i + 1
i +
i
1 + 3β ⁄ 2
1
Figure 8.3 Illustration of a function that is Lebesgue integrable on [0, -], but is not square
Lebesgue integrable on the same interval.
The following are two examples where, as  ; 0, the integrated error between
h and h

is finite. First, the ‘‘identity’’ system where h

(t) : 

(t) and second,
the system where
h

(t) :


0
t+ [,  ; 1/]
elsewhere
(8.6)
For both systems, and for t + (0, -), it follows that
h(t) : lim



(t)"-h

for t+ [0, -].
This second condition excludes a signal such as 1/(t
, which is integrable
on [0, -], but has infinite energy on all intervals of the form [0, t
M
]. It
also excludes signals such as the one shown in Figure 8.3, whose integral
equals 

G
(1/i>@), which from the comparison test (Knopp, 1956
pp. 56f ), is finite for 90, but whose energy is given by 

G
(1/i\@) and is
infinite when 90.
IMPULSE RESPONSE
231
8.3 INPUT ‒OUTPUT RELATIONSHIP
Consider the causal linear time invariant system illustrated in Figure 8.1. The
well-known relationship between the input and output signals is specified in
the following theorem.
T 8.1. I—O R   L S If the
input signal x to, and the system impulse response h of, a linear time invariant
system are both causal, are locally integrable, and have bounded variation on all
finite intervals, then the output signal, y, is given by
y(t) :

WNH2
N>H2
x()h(p)e\HLDN>H d dp
232
LINEAR SYSTEM THEORY
Figure 8.4 Illustration of area of integration for Y and I.
and the integration regions for both Y and I are as shown in Figure 8.4.
W ith
X(T, s) :

2

x(t)e\QR dt (8.11)
and similarly for other L aplace transformed variables, it is the case that
Y (T, s) :

WNH2
N>HW2
x()h(p)e\QN>H d dp
(8.12)
: X(T, s)H(T, s) 9 I(T, s)
where
I(T, s) :

WNH2
N>H2
x()h(p)e\QN>H d dp (8.13)
Proof. The proof of this theorem is given in Appendix 2.
For the Fourier transform case Y


increase, then Re[s] 9 0 is a sufficient condition for
lim
2
Y (T, s) : lim
2
X(T, s)H(T, s) (8.16)
Proof. The proof is given in Appendix 3.
8.4.1 Windowed Input and Nonwindowed Output
For completeness, the response of a linear time invariant system for the case
where the input and impulse response are windowed, but the output is not
windowed, as illustrated in Figure 8.5, is stated in the following theorem.
T 8.4 T  O S:N C If both
x, h + L [0, T ], and have bounded variation on [0, T ], then the Fourier and
L aplace transforms Y

of the output signal y, which is not windowed, are given by
Y

(2T, f ) : X(T, f )H(T, f ) (8.17)
Y

(2T, s) : X(T, s)H(T, s)(8.18)
Proof. The proof of this result is given in Appendix 4.
This result has application, when the output signal y is to be derived for the
interval [0, T ]. The procedure is as follows for the Fourier transform case.
First, evaluate X(T, f ) and H(T, f ), second, evaluate Y

(2T, f ):X(T, f )H(T, f ),
and third, evaluate y by taking the inverse Fourier transform of Y



(T, f ) " lim
2
Y (T, f ) almost everywhere. The following example
illustrates this point.
8.4.2.1 Example Consider a linear system with an impulse response and
input signal, respectively, defined according to
h(t) :
h
M
e\RO

t 9 0, 90
x(t) :(2 
V
sin(2f
V
t) t 9 0
(8.19)
For the case where 
V
: 1, h
M
: 1,  : 0.1, T : 1, and f
V
: 4, the output signal
y is plotted in Figure 8.6. For these parameters, the magnitude of the true, Y,
and approximate, Y

, Fourier transforms, as well as the magnitude of the error,

d : (2 
V
h
M


19e\2O 9
Te\2O


(8.20)
When T is sufficiently large, such that Te\2O/ 1, it follows that "I(T, f )"-
(2 
V
h
M
, which is independent of the interval length T, and only depends on
FOURIER AND LAPLACE TRANSFORM OF OUTPUT
235
0.5 1 5 10 50
0.005
0.01
0.05
0.1
0.5
True
Frequency (Hz)
Magnitude Approximation
Error
Figure 8.7 Magnitude of the true and approximate Fourier transform of the output signal as

1
2
Frequency (Hz)
Magnitude
error
Figure 8.9 Magnitude of the approximate Fourier transform Y

of the output signal, for the
cases where T : 2 (lower peak) and T : 4 (higher peak). The magnitude of the error between
the true and approximate Fourier transform is identical for T : 2 and T : 4, and is the smooth
curve.
magnitude of the true Fourier transform Y, is plotted for cases T : 2 and
T : 4. In Figure 8.9, the magnitude of approximate Fourier transform Y

,as
well as the error "I", are graphed for cases T : 2 and T : 4. As T increases,
the lobe at the frequency of the input (4Hz) increases in height, and decreases
in width. Away from the lobe, the envelope of the magnitude of both Y and Y

remains constant as T increases and, consistent with this, I does not change
with T. Clearly, for this example the approximate Fourier transform Y

, does
not converge to the true Fourier transform Y, defined in Eq. (8.9).
8.4.2.2 Explanation An explanation of the nonconvergence of Y

(T, f )to
Y (T, f )asT ; -, for signals with constant average power, can be found by
noting that I can be approximated by an integral over the region defined in
Figure 8.10, where t

λ
T
T
p
to I
p = T −λ
t
h
Region where there is
significant contribution
T – t
h
Figure 8.10 Region of integration where there is a significant contribution to the integral I. The
time t
h
is the time when the impulse response has negligible magnitude as defined in the text.
as I(T, f ) is finite. However, for the infinite interval, it follows, as I(T, f ) does
not increase with T, that
lim
2
G
'
(T, f ) : lim
2
"I(T, f )"
T
: 0 (8.22)
A consequence of this result is that
lim
2

3 R uncountable case

(8.24)
where P[x(, t)] : P[] : p
A
for the countable case, and P[x(, t)"
AZA
M
A
M
>BA
] :
P[+ [
M
, 
M
; d]] : f

(
M
) d for the uncountable case. Here, f

is the prob-
ability density function associated with the index random variable , whose
sample space is S

.
238
LINEAR SYSTEM THEORY