Transcript CPE200 Signals and Systems

CPE200 Signals and
Systems
Chapter 2:
Linear Time-Invariant
Systems
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2. Introduction
1
In this chapter, we will consider a linear
time-invariant (LTI) system which is a
system satisfying both linearity and the
time-invariance properties. Such systems
play a fundamental role in signal and
system analysis since highly useful tools and
concepts associated with LTI system
analysis offer the most insight into system
behavior. Although, only a small amount of
systems in the world are truly LTI,
nonlinear systems can still be approximated
as being linear within a small enough input
range.
An LTI system can be characterized in
terms of its impulse response, h(t) or h[n] as
a consequence of linear and time-invariance
properties. The behavior of an LTI system
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can also be described by a linear constantcoefficient differential or difference
equation. Differential equations are used to
represent c-t systems, while difference
equations represent d-t systems.
The impulse response is an output of the
LTI system when the input is an impulse
(unit sample) signal d(t) or d[n]. Knowing
the impulse response, we can determine the
output of the system to any arbitrary input
by a weighted sum of time-shifted impulse
responses. This operation is called the
“convolution sum” for d-t systems and the
“convolution integral” for c-t systems.
In this chapter, we will define the impulse
response and derives the convolution
operation. Then properties of liner timeinvariant systems will be discussed. Finally,
we will briefly review a method for
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solving differential and difference equations
and we will discuss how to represent LTI
systems using block diagram.
2. D-T LTI Systems: The
2 Convolution Sum
As mentioned in the previous section, the
output of an LTI system to any arbitrary
input can be determined by the convolution
process. We will discuss the convolution
process for d-t systems in this section first,
since it is much easier to understand than
one for c-t systems.
2.2.1 The Representation of DiscreteTime Signals in Terms of
Impulses
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As briefly mentioned in Ch. 1, the d-t unit
impulse can be used to construct any d-t
signal (See Eq. 1.47).
i.e.
Any D-T signal is the sum of scaled and shifted unit impulses.
x[n] 


x[k]d[n  k]
(1.47)
k 
This idea is fairly obvious to understand by
visualizing the graphical representation of
d-t signal x[n] as depicted in Fig. 2.1. From
Fig. 2.1, the d-t signal x[n] is decomposed
into four time-shifted, scaled unit impulse
signals where the scaling on each impulse
equals the value of x[n] at the particular
instant the unit sample occurs. For
example,
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x[1], n  1
x[1]d[n  1]  
n  1
0,
x[0], n  0
x[0]d[n ]  
n0
0,
x[1], n  1
x[1]d[n  1]  
n 1
0,
x[2], n  2
x[2]d[n  2]  
n2
0,
Hence, the sum of the four signals in Fig.
2.1 equals x[n] for -1 ≤ n ≤ 2 and we can
represent x[n] as follows:
x[n] = x[-1]d[n+1]+x[0]d[n]+x[1]d[n-1]
+x[2]d[n-2]
(2.1)
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At any time n, only one of the terms on the
right-hand side of Eq. 2.1 is nonzero.
Similarly, for any d-t signals, we can
represent them by Eq. 1.47.
3
3
2
2
x[0]d[n-0]
2
x[-1]d[n+1]
+
-1
2
-1
0
1
=
0
+
2
x[1]d[n-1]
X[n]
-2
1
+
2
x[2]d[n-2]
-2
Figure 2.1 Decomposition of a discrete-time signal into a
weighted sum of shifted impulses.
Eq. 1.47 is called the sifting property of the
d-t unit impulse since only the value of x[k]
corresponding to k=n is preserved.
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2.2.2 The D-T Unit Impulse Response
The response of a linear
Unit Impulse
Response = system when the input
(excitation) signal is the
impulse signal.
Since, in the case of d-t systems, the impulse
signal is normally called the “unit sample”
signal, the unit impulse response for a
linear d-t system is widely called the “unit
sample response”.
We can derive the mathematical
representation of the unit sample response
by starting with an arbitrary linear d-t
system defined as follows:
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x[n]
y[n]
Linear d-t
System, t
Since
y[n ]  t{x[n ]}
and
x[n ] 


 x[k]d[n  k] .
k  

y[n ]  t{  x[k ]d[n  k ]}
k  
Because the system is linear, we can applied
the operation t to the shifted unit sample
signal d[n-k] before performing the
summation operation. Hence,
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
y[n ] 

 x[k]t{d[n  k]}
(2.2)
k  
Let t{d[n-k]} = h [n,k]. Hence hk[n] is the
response of the linear system when the
input is equal to d[n-k].
i.e.
d[n-k]
h [n,k]
Linear d-t
System, t
h [n,k] is known as the “unit impulse
response” of a linear d-t system.
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Therefore, once h [n,k] of the linear d-t
system is determined, y[n] of the system for
any arbitrary x[n] can be evaluated by this
following Eq.:

y[n ] 

 x[k]h[n, k]
(2.3)
k  
Eq. 2.3 indicates that the response of a
linear d-t system to the input x[n] is a linear
combination of the responses to the
individual scaled and shifted impulses.
In general, the response h[n,k] is a function
of n and the time k which is a time when the
unit sample d[n] is applied to the system.
However, if the linear system is also time
invariant, then the time-shifted k is not an
issue. Thus, for an LTI d-t system,
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h[n,k] = h[n-k]
(2.4)
That is, the response of the LTI system
when the input is d[n] is defined as h[n]
which is called the “unit sample response”.
Then for an LTI system, Eq. 2.3 becomes
y[n ] 

 x[k]h[n  k]
(2.5)
k  
The output is the sum of scaled and shifted unit sample response.
This result is referred to as the convolution
sum or the superposition sum. The
operation on the right-hand side of Eq. 2.5
is known as the convolution of the sequence
x[n] and h[n] which can be denoted as:
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y[n] = x[n]*h[n]
(2.6)
The convolution process defined by Eq. 2.6
involves these following steps:
1. FLIP h[k] about k=0 which is h[-k]
2. SHIFT h[-k] to the right by n which is
h[n-k]
3. MULTIPLY x[k] by h[n-k] which is the
flipped and shifted version
of h[k].
4. ADD across all values of k to obtain the
value of the output at one value of
n
5. Repeat step 2-4 for all possible value of
n
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Note: Useful Summation Formulas
Finite Summation Formulas
n 1
1

a
k
a
  1 a
k 0
n
n
 ka

k
a[1  (n  1)a  na
k 0
n
k
a 
2 k
a 1
n
(1  a )
n 1
]
2
a[(1  a )  (n  1) 2 a n  (2n 2  2n  1)a n 1  n 2 a n  2 ]
(1  a ) 3
k 0
n (n  1)
k  2
k 0
n
n (n  1)(2n  1)
k 
6
k 0
n
2
2
2
n
(
n

1
)
3
k
  4
k 0
n
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Infinite Summation Formulas

1
a  1 a
k 0
k

 ka
k 0
k

| a | 1
a
(1  a )
2
| a | 1

2
a
a
2 k
 k a  (1  a )3 | a | 1
k 0
2. C-T LTI Systems: The
3 Convolution Integral
The output of a c-t LTI system can be
determined from knowledge of the input
and the impulse response of the system.
The approach and result are analogous to
the d-t case. For c-t systems, the
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superposition is evaluated by an integration
instead of a summation because of the
continuous nature of the input.
Similarly, any c-t signal x(t) may express as
the superposition of scaled and shifted
impulses:

x(t) 
 x(t)d(t  t)dt
(2.7)

Here the scaled x(t) dt is calculated from
the value of x(t) at the time at which each
impulse occurs, t. Eq. 2.7 is also called the
sifting property of the c-t impulses.
Now, for any linear c-t system, let define the
impulse response h(t) = t{d(t)} as the output
of the system in response to an
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impulse input. Thus the response of the
linear c-t system to any arbitrary input can
be evaluated as:


y(t ) 
 x(t)h(t, t)dt
(2.8)

If the linear system is time invariant, h(t,t)
in Eq. 2.8 will become h(t-t). Hence, for an
LTI c-t system, the response of the system
to x(t) is defined as:

y( t ) 
x
(
t
)
h
(
t

t
)
d
t

(2.9)

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This result is referred to as the convolution
integral or the superposition integral. As
before, this operation is denoted by the
symbol “*”; that is
y(t) = x(t)*h(t)
(2.10)
2. Properties of Linear Time4 Invariant Systems
The Commutative Properties
x[n]*h[n] = h[n]*x[n]
(2.11)
x(t)*h(t) = h(t)*x(t)
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The Distributive Properties
x[n]*{h1[n]+h2[n]} = x[n]*h1[n] + x[n]*h2[n]
(2.12)
x(t)*{h1(t)+h2(t)} = x(t)*h1(t) + x(t)*h2(t)
Parallel Connection of Systems
The Associative Properties
x[n]*{h1[n]*h2[n]} = {x[n]*h1[n]} *
{x[n]*h2[n]}
(2.13)
x(t)*{h1(t)*h2(t)} = {x(t)*h1(t)}*{x(t)*h2(t)}
Cascade Connection of Systems
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The Shifting Properties
y[n] = x[n]*h[n], then
If
y[n-k] = x[n-k]*h[n] = x[n]*h[n-k]
(2.14)
Convolution with the unit impulse
If
h[n] = d[n], then
x[n]*d[n] = x[n]
(2.15)
x[n]*d[n-k] = x[n-k]
(2.16)
and
Invertibility of LTI System
If a system is invertible, there exists an
inverse system such that when cascaded
with the original system, yields an output
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equal to the original input (see Sec. 1.6.2).
x(t)
LTI
System
h(t)
y(t)
Inverse
System
h-1(t)
w(t) = x(t)
Figure 2.2 Cascade of an LTI system with impulse response
h(t) and the inverse system with impulse response
h-1(t).
The relationship between the impulse
response of a system, h(t), and the
corresponding inverse system, h-1(t), is
easily derived. From Fig. 2.2, the impulse
response of the cascade connection is the
convolution of h(t) and h-1(t). Hence,
x(t)*{h(t)*h-1(t)} = x(t)
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(2.17)
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Compare Eq. 2.17 with Eq. 2.15, it implies
that
{h(t)*h-1(t)} = d(t)
(2.18)
Causal LTI Systems
An LTI system is said to be causal if and
only if its impulse response is zero for
negative values of n (or t).
Let consider the convolution sum which is:
y[n ] 



k  
1
k  

k  
k 0
 x[k]h[n  k]   h[k]x[n  k]
 x[k]h[n  k]   x[k]h[n  k]
Future inputs
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(2.19)
Pass and present
inputs
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The first term in Eq. 2.19 is associated with
indices k < 0 and can be expressed as:
= …+h[-2]x[n+2]+h[-1]x[n+1]
(2.20)
The second term in Eq. 2.19 is associated
with indices k ≥ 0 and can be expressed as:
= h[0]+h[1]x[n-1]+h[2]x[n-2]+... (2.21)
From Eq. 2.20 and 2.21, we can noticed that
future values of the input are associated
with indices k < 0 while present and past
values of the input are associated with
indices k ≥ 0 in the convolution sum.
Hence, for a causal system, h[k] = 0 for k<0,
and the convolution sum is reduced to

y[n ]   h[k ]x[n  k ]
k 0
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A causal LTI
d-t system
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Similarly, a causal c-t system has impulse
response that satisfies h(t) = 0 for t<0.
Thus, the output is expressed as the
convolution integral

y( t )   h (t) x ( t  t)dt
A causal LTI
c-t system
0
(2.22)
Stable LTI Systems
Recall from Ch. 1 that a system is bounded
input-bounded output (BIBO) stable if the
output is guaranteed to be bounded for
every bounded input.
I.e. , for a stable d-t system, if
|x[n]| ≤ Mx < ∞ for all n,
then the output must satisfy
|y[n]| ≤ My < ∞ for all n.
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Since
y[n ] 

 h[k]x[n  k]
k  
then
| y[n ] | 

 | h[k] || x[n  k] |
k  
Because all the input values are bounded,
say by Mx, therefore,
| y[n ] |  M x

From Eq. 2.23, if

 | h[k] |
(2.23)
k  
 | h[k] | is absolutely
k  
summable, the output |y[n]| is bounded.
Thus, for a stable LTI system, the impulse
response must satisfies the following
condition:
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
 | h[k] |  
A stable LTI d-t system
(2.24)
k  
Similarly, a c-t LTI system is BIBO stable if
and only if the impulse response is
absolutely integrable, that is,

 | h(t)dt  
A stable LTI c-t system
(2.25)

2. Unit Step Response of LTI
5 Systems
Unit Step
Sudden Change
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The unit step response of an LTI system
describes how the system responds to
sudden changes in the input. Let consider a
d-t LTI system having the impulse response
h[n] and denote the step response as s[n].
Thus, the step response s[n] can be
determined by the following equation:
s[n ]  u[n ]  h[n ]  h[n ]  u[n ]


 h[k]u[n  k]
(2.26)
k  
Since u[n-k] = 0 for k > n and u[n-k] = 1 for
k ≤ n, hence
s[n ] 
n
 h[k]
(2.27)
k  
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Eq. 2.27 indicates that the step response is
the running sum of the impulse response
and h[n] can be recovered from s[n] using
the relation
h[n] = s[n] - s[n-1]
(2.28)
Similarly, in c-t system, the step response of
an LTI system with impulse response h(t) is
the running integral of h(t), or
t
s( t ) 
h
(
t
)
d
t

(2.29)

From Eq. 2.29, the impulse response will be
the first derivative of the unit step response,
or
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ds(t )
h(t ) 
 s' ( t )
dt
(2.30)
2. Causal LTI Systems
6 Described by Differential
and Difference Equations
An extremely important characteristic of dt (or c-t) systems is that for which the input
and output are related through a linear
constant-coefficient difference (or
differential) equation. That is, linear
constant-coefficient difference and
differential equations provide another
representation for the input-output
characteristics of LTI systems.
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Difference equations are used to represent
d-t systems, while differential equations
represent c-t system. The general form of a
linear constant-coefficient difference
equation is:
N
M
k 0
k 0
 a k y[n  k]   b k x[n  k]
(2.31)
where
y[n] = the output
x[n] = the input
N and M = the highest delayed orders
and
ak and bk = the constant coefficients
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A linear constant-coefficient differential
equation has a similar form, with the
delayed values replaced by the derivative
values of the input x(t) and output y(t), as
shown in the following equation:
N
ak
k 0
d
k
dt
M
y( t )   b k
k
k 0
d
k
dt
k
x(t)
(2.32)
We can notice that Eq. 2.31 and 2.32
provide an implicit specification of the
system. That is, they describe a
relationship between the input and the
output, rather than an explicit expression
for the system output as a function of the
input. To determine an explicit expression,
we must solve the difference or differential
equation. In general, to solve Eq. 2.31 or
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2.32, we must specify a set of initial
conditions.
Generally, the solution of both Eq. 2.31 and
2.32 can be divided into two types of
solutions as shown below:
y[n] = yc[n] + yp[n]
(2.33)
y(t) = yc(t) + yp(t)
The term yc[n] (or yc(t)) is known as the
complementary solution, whereas yp[n] (or
yp(t)) is called the particular solution.
Generally, the complementary solution will
describe the response of a system when the
input is zero. Such response is usually
called the “natural response” of a system.
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The complementary solution is usually of
the form:
N
y ( n ) [n ]   Ci ni
(2.34)
i 1
,for a difference equation, and
N
y
(n )
(t )   Ci e
si t
(2.35)
i 1
,for a differential equation.
Where C, s, and  are constants to be
determined.
The particular solution, on the other hand,
represents any solution to the differential or
difference equation for the given input.
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Such response is usually called the “forced
response” of a system.
The particular solution is usually obtained
by assuming the system output has the
same general form as the input. Table 2.1
provides the general form of the particular
solution for common input signals.
Table 2.1 Form of a particular solution corresponding to
several types of common inputs.
C-T
Input
D-T
Particular Sol.
Input
Particular Sol.
1
C
1
C
e-st
Ce-st
n
Cn
cos(Wt + f)
C1cos(Wt)
+C2sin(Wt)
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cos(wt + f) C1cos(wt)
+C2sin(wt)
BYST
For a convenience, we will discussed only
how to solve a difference equation.
However, solving a differential equation can
be perform in the same manner.
2.6.1 The Complementary Solution of
the Difference Equation
To find the complementary solution, we
begin with writing the homogeneous
equation which is Eq. 2.31 with the left side
set equal to zero, that is,
N
 a k y[n  k]  0
(2.36)
k 0
In other words, the complementary solution
will describe the response of a system when
the input is zero.
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Basically, we assume that the solution of the
homogeneous equation is of the form:
yc[n] = n
(2.37)
If we substitute Eq. 2.37 into Eq. 2.36, we
obtain the polynomial equation:
N
 a k
n k
0
k 0
or
n-N(N+a1N-1 +…+aN-1+aN) = 0
(2.38)
The polynomial “N+a1N-1 +…+aN-1+aN”
is called the characteristic polynomial of the
system. The roots of Eq. 2.38 can be real or
complex valued but the coefficients “ak”, in
practice, are usually real.
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If we assume that the roots are distinct,
then, the most general solution to the
homogeneous difference equation is in the
form described by Eq. 2.34, that is,
y[n]  C1n1  C2n2  ...  C N nN
(2.39)
where C1, C2, …, CN are weighting
coefficients.
These coefficients are determined from the
initial conditions specified for the system.
Example 2.1 Determine the homogeneous
solution of the system described by the
first-order difference equation
y[n] + a1y[n-1] = x[n]
(2.40)
When x[n] = 0 and we substitute yc[n] = n
in Eq. 2.40, we obtain
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n+a1n-1 = 0
n-1(+a1) = 0
 = -a1
(2.41)
Therefore, the solution to the homogeneous
difference equation is
yc[n] = Cn = C(-a1)n
(2.42)
To determine the value of C, some of initial
conditions must be provided. From Eq.
2.41, when x[n] = 0 and at n = 0, we obtain
y [0] = -a1y[-1]
(2.43)
From Eq. 2.42, we have
yc [0] = C
Thus, the homogeneous solution of this
system is
yc [n] = (-a)n+1y[-1] n ≥ 0
Ans.
CPE200 - W2003: LTI System
116
BYST
Previously, we assumed that the
characteristic equation contains distinct
root. On the other hand, if the
characteristic equation contains multiple
roots, the form of the solution given in Eq.
2.39 must be modified. Let assume 1 is a
root of multiplicity m, then Eq. 2.39 will be
expressed as:
y[n]  C1n1  C2 nn1  C3n 2n1  ...
 Cm n m1n1  Cm1nm1  ...
 C N  n
(2.39)
2.6.2 The Particular Solution of the
Difference Equation
The particular solution yp[n] is required to
satisfy the difference equation for the
CPE200 - W2003: LTI System
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BYST
specific input signal x[n], n ≥ 0. It is usually
obtained by assuming the system output
has the same general form as the input.
That is, if x[n] is an exponential, we would
assume that the particular solution is also
an exponential.
Example 2.2 Determine the particular
solution of the difference equation
y[n]-(5/6)y[n-1]+(1/6)y[n-2] = x[n]
when the forcing function x[n] = 2n, n ≥ 0
and zero elsewhere.
To solve this problem, we begin with
assuming the particular solution is
n≥0
yp[n] = C2n
Substitute yp[n] into the difference
equation, we obtain
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BYST
C2nu[n] = (5/6)C2n-1u[n-1]-(1/6)C2n-2u[n-2]
+2nu[n]
To determine the value of K, we can
evaluate the above equation for any n ≥ 2,
where none of the terms vanish. Thus we
obtain
4C = (5/6)2C - (1/6)C + 4
Solving the above equation, we get C = 8/5.
Therefore, the particular solution is
yp[n] = (8/5)2n
n≥0
Ans.
CPE200 - W2003: LTI System
119
BYST