byst cpe200 - w2003: lti system 79 cpe200 signals and systems chapter 2: linear time-invariant...

BYSTBYSTCPE200 - W2003: LTI SystemCPE200 - W2003: LTI System 1

CPE200 Signals CPE200 Signals and Systemsand Systems

Chapter 2: Linear Time-Invariant Syst

ems


Introduction2.1

In this chapter, we will consider a linear timlinear time-invariante-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]impulse response, h(t) or h[n] as a consequence of linear and time-invariance properties. The behavior of an LTI system


The impulse response is an output of the LTI system when the input is an impulse (unit sample) signal (t) or [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 sumconvolution sum” for d-t systems and the “convolution integralconvolution integral” for c-t systems.

can also be described by a linear constant- linear constant- coefficient differentialcoefficient differential or difference difference equationequation. Differential equations are used to represent c-t systems, while difference equations represent d-t systems.

In this chapter, we will define the impulse response and derives the convolution operation. Then properties of liner time- invariant systems will be discussed. Finally, we will briefly review a method for


solving differential and difference equations and we will discuss how to represent LTI systems using block diagram.

D-T LTI Systems: The2.2Convolution Sum

2.2.1 The Representation of Discrete-

Time Signals in Terms of Impulses

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.


i.e.

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,

As briefly mentioned in Ch. 1, the d-t unit impulse can be used to construct any d-t signal (See Eq. 1.47).

Any D-T signal is the sum of scaled and shifted unit impulses.

k

x[n] x[k] [n k]

(1.47)


0n,0

0n],0[x]n[]0[x

1n,0

1n],1[x]1n[]1[x

2n,0

2n],2[x]2n[]2[x

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][n+1]+x[0][n]+x[1][n-1]

+x[2][n-2] (2.1)

1n,0

1n],1[x]1n[]1[x


x[-1][n+1]

2

-1 + +

+

2

x[1][n-1]

1

x[2][n-2]

2

-2

x[0][n-0]

3

0

Figure 2.1 Decomposition of a discrete-time signal into aweighted sum of shifted impulses.

Eq. 1.47 is called the sifting propertysifting property of the d-t unit impulse since only the value of x[k] corresponding to k=n is preserved.

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.

=2

2

3

-1 0 1

-2

X[n]

2


2.2.2 The D-T Unit Impulse Response

Unit ImpulseResponse

The response of a linearlinear system when the input (excitation) signal is the impulimpulsese signal.

=

Since, in the case of d-t systems, the impulse signal is normally called the “unit sampleunit sample” signal, the unit impulse response for a linear d-t system is widely called the “unit sample unit sample responseresponse”.

We can derive the mathematical representation of the unit sample response by starting with an arbitrary linear d-t system defined as follows:


Since

Linear d-tSystem,

x[n] y[n]

k

]kn[]k[x]n[x

]}n[x{]n[y and

}]kn[]k[x{]n[yk

Because the system is linear, we can applied the operation to the shifted unit sample signal [n-k] before performing the summation operation. Hence,

.


}]kn[{]k[x]n[yk

(2.2)

Let {[n-k]} = h [n,k]. Hence hk[n] is the response of the linear system when the input is equal to [n-k].

i.e.

Linear d-tSystem,

[n-k] h [n,k]

hh [n,k][n,k] is known as the “unit impulse unit impulse

responseresponse” of a linear d-t system.


Therefore, once hh [n,k][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.:

k

]k,n[h]k[x]n[y (2.3)

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 nn and the time kk which is a time when the unit sample [n] is applied to the system. However, if the linear system is also time invatime invariantriant, then the time-shifted k is not an issuean issue. Thus, for an LTI d-t system,


h[n,k] = h[n-k] (2.4)

That is, the response of the LTI system when the input is [n] is defined as h[n]h[n] which is called the “unit sample responseunit sample response”. Then for an LTI system, Eq. 2.3 becomes

k

]kn[h]k[x]n[y (2.5)

This result is referred to as the convolution convolution sumsum or the superposition sum. The operation on the right-hand side of Eq. 2.5 is known as the convolutionconvolution of the sequence x[n] and h[n] which can be denoted as:

The output is the sum of scaled and shifted unit sample response.


y[n] = x[n]*h[n] (2.6)

The convolution process defined by Eq. 2.6 involves these following steps:

1. FLIPFLIP h[k] about k=0 which is h[-k]

2. SHIFTSHIFT h[-k] to the right by n which is

3. MULTIPLYMULTIPLY x[k] by h[n-k] which is the

h[n-k]

flipped and shifted versionof h[k].

4. ADDADD across all values of k to obtain thevalue of the output at one value ofn

5. RepeatRepeat step 2-4 for all possible value of n


Note: Useful Summation Formulas

Finite Summation Formulas

1aa1

a1a

n

0k

1nk

n

0k2

1nnk

)a1(

]naa)1n(1[aka

n

0k3

2n21n2n2k2

)a1(

]ana)1n2n2(a)1n()a1[(aak

n

0k 2

)1n(nk

n

0k

2

6

)1n2)(1n(nk

n

0k

223

4

)1n(nk


Infinite Summation Formulas

1|a|a1

1a

0k

k

1|a|)a1(

aka

0k2

k

1|a|3)a1(

aaak

0k

2k2

C-T LTI Systems: The2.3Convolution 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


superpositionsuperposition is evaluated by an integrationintegration instead of a summationsummation 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:

d)t()(x)t(x (2.7)

Here the scaled x() d is calculated from the value of x(t) at the time at which each impulse occurs, . 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)} as the output of the system in response to an


impulse input. Thus the response of the linear c-t system to any arbitrary input can be evaluated as:

If the linear system is time invariant, h(t,) in Eq. 2.8 will become h(t-). Hence, for an LTI c-t system, the response of the system to x(t) is defined as:

d),t(h)(x)t(y (2.8)

d)t(h)(x)t(y (2.9)


This result is referred to as the convolution iconvolution integralntegral or the superposition integral. As before, this operation is denoted by the symbol

“*”; that is

y(t) = x(t)*h(t) (2.10)

Properties of Linear Time-2.4Invariant Systems

The Commutative PropertiesThe Commutative Properties

x(t)*h(t) = h(t)*x(t)

x[n]*h[n] = h[n]*x[n]

(2.11)


The Distributive PropertiesThe 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)

The Associative PropertiesThe Associative Properties

x[n]*{h1[n]*h2[n]} = {x[n]*h1[n]} *

(2.13)

x(t)*{h1(t)*h2(t)} = {x(t)*h1(t)}*{x(t)*h2(t)}

Parallel Connection of SystemsParallel Connection of Systems

{x[n]*h2[n]}

Cascade Connection of SystemsCascade Connection of Systems


The Shifting PropertiesThe Shifting Properties

If y[n] = x[n]*h[n], then

(2.14)

Convolution with the unit impulseConvolution with the unit impulse

x[n]*[n] = x[n] (2.15)

y[n-k] = x[n-k]*h[n] = x[n]*h[n-k]

If h[n] = [n], then

x[n]*[n-k] = x[n-k] (2.16)

and

Invertibility of LTI SystemInvertibility of LTI System

If a system is invertible, there existsexists an inveinverse systemrse system such that when cascaded with the original system, yields an output


equal to the original input (see Sec. 1.6.2).

LTISystem

h(t)

InverseSystem

h-1(t)

x(t) y(t)

Figure 2.2 Cascade of an LTI system with impulse response

w(t) = x(t)

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) (2.17)


Compare Eq. 2.17 with Eq. 2.15, it implies that

{h(t)*h-1(t)} = (t) (2.18)

Causal LTI SystemsCausal LTI Systems

An LTI system is said to be causalcausal if and only if its impulse response is zero for negative values of n (or t).

Let consider the convolution sum which is:

0k

1

k

]kn[h]k[x]kn[h]k[x

(2.19)

kk

]kn[x]k[h]kn[h]k[x]n[y

Future inputsFuture inputs Pass and presentPass and presentinputsinputs


The first term in Eq. 2.19 is associated with indices k < 0k < 0 and can be expressed as:

(2.20)= …+h[-2]x[n+2]+h[-1]x[n+1]

The second term in Eq. 2.19 is associated with indices k ≥ 0k ≥ 0 and can be expressed as:

(2.21)= h[0]+h[1]x[n-1]+h[2]x[n-2]+...

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 causcausalal system, h[k] = 0 for k<0h[k] = 0 for k<0, and the convolution sum is reduced to

0k

]kn[x]k[h]n[yA causal LTI A causal LTI d-t systemd-t system

(2.22)


Similarly, a causalcausal c-t system has impulse response that satisfies h(h() = 0 for ) = 0 for <0<0. Thus, the output is expressed as the convolution integral

0

d)t(x)(h)t(yA causal LTI A causal LTI c-t systemc-t system

Stable LTI SystemsStable 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.

(2.22)


Since

then

k

]kn[x]k[h]n[y

k

|]kn[x||]k[h||]n[y|

Because all the input values are bounded, say by Mx, therefore,

k

x |]k[h|M|]n[y| (2.23)

From Eq. 2.23, if

k

|]k[h| is absolutely

summable, the output |y[n]| is bounded. Thus, for a stable LTI system, the impulse response must satisfies the following condition:


(2.24)

k

|]k[h|

Similarly, a c-t LTI system is BIBO stable if and only if the impulse response is absolutely integrable, that is,

A stable LTI d-t systemA stable LTI d-t system

(2.25)

d)(h| A stable LTI c-t systemA stable LTI c-t system

Unit Step Response of LTI2.5Systems

Unit StepUnit Step

Sudden ChangeSudden Change


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 responsethe step response as s[n]s[n]. Thus, the step response s[n] can be determined by the following equation:

]n[u]n[h]n[h]n[u]n[s

k

]kn[u]k[h (2.26)

Since u[n-k] = 0u[n-k] = 0 for k > nk > n and u[n-k] = 1u[n-k] = 1 for k ≤ nk ≤ n, hence

(2.27)

n

k

]k[h]n[s


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

d)(h)t(s (2.29)

From Eq. 2.29, the impulse response will be the first derivative of the unit step response, or


)t('sdt

)t(ds)t(h (2.30)

Causal LTI Systems2.6Described by Differentialand Difference Equations

An extremely important characteristic of d-t (or c-t) systems is that for which the input and output are related through a linear conlinear constant-coefficient difference (or differential) estant-coefficient difference (or differential) equationquation. That is, linear constant-coefficient difference and differential equations provide another representation for the input-output characteristics of LTI systems.


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:

M

0kk

N

0kk ]kn[xb]kn[ya (2.31)

where

y[n] = the output

x[n] = the input

and

ak and bk = the constant coefficients

N and M = the highest delayed orders


)t(xdt

db)t(y

dt

da

k

kM

0kk

N

0kk

k

k

(2.32)

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:

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


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)

The term yc[n] (or yc(t)) is known as the coco

mplementary solutionmplementary solution, whereas yp[n] (or yp(t)) is called the particular solutionparticular solution.

y(t) = yc(t) + yp(t)

Generally, the complementary solution will describe the response of a system when the ithe input is zeronput is zero. Such response is usually called the “natural responsenatural response” of a system.


The complementary solution is usually of the form:

ni

N

1ii

)n( C]n[y

tsN

1ii

)n( ieC)t(y

,for a difference equation, and

(2.34)

,for a differential equation.

(2.35)

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 inputthe given input.


Such response is usually called the “forced rforced responseesponse” 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 D-T

Input Particular Sol. Input Particular Sol.

1 C 1 C

e-st Ce-st n Cn

cos(t + ) C1cos(t)

+C2sin(t)

cos(t + ) C1cos(t)

+C2sin(t)


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,

0]kn[yaN

0kk

(2.36)

In other words, the complementary solution will describe the response of a system when the input is zero.


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:

0aN

0k

knk

(2.38)

or

n-N(N+a1N-1 +…+aN-1+aN) = 0

The polynomial “N+a1N-1 +…+aN-1+aN” is called the characteristic polynomialcharacteristic polynomial of the system. The roots of Eq. 2.38 can be real or complex valued but the coefficients “aakk”, in practice, are usually real.


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,

nNN

n22

n11 C...CC]n[y (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]

When x[n] = 0 and we substitute yc[n] = n in Eq. 2.40, we obtain

(2.40)


n+a1n-1 = 0

n-1(+a1) = 0

= -a1

Therefore, the solution to the homogeneous difference equation is

yc[n] = Cn = C(-a1)n

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]

(2.41)

(2.42)

(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.


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 11 is a root of multiplicity mmultiplicity m, then Eq. 2.39 will be expressed as:

...nCnCC]n[y n1

23

n12

n11

(2.39)

...CnC n1m1m

n1

1mm

nNC

2.6.2 The Particular Solution of the

Difference Equation

The particular solution yp[n] is required to satisfy the difference equation for the


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

yp[n] = C2n n ≥ 0

Substitute yp[n] into the difference equation, we obtain


C2nu[n] = (5/6)C2n-1u[n-1]-(1/6)C2n-2u[n-2]

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

Solving the above equation, we get C = 8/5. Therefore, the particular solution is

yp[n] = (8/5)2n n ≥ 0

+2nu[n]

4C = (5/6)2C - (1/6)C + 4

Ans.

byst cpe200 - w2003: lti system 79 cpe200 signals and systems chapter 2: linear time-invariant...

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