signals and systems analysis net 351 instructor: dr. amer el-khairy د. عامر الخيري

Signals and Systems Analysis

NET 351Instructor: Dr. Amer El-Khairy

الخيري. عامر د

Linear-time invariant systemsTwo most important attributes of systems are linearity

and time-invariance.• Develop the fundamental input-output relationship for

systems having these attributes. • show that the input-output relationship for LTI systems is

described in terms of a convolution operation.• The importance of the convolution operation in LTI• Systems stems from the fact that knowledge of the

response of an LTI system to the unit impulse input allows us to find its output to any input signals.

• Specify the input-output relationships for LTI systems by differential and difference equations.

2Signals and systems analysis الخيري. عامر د

Linear-time invariant systemsImpulse Response:• The impulse response h(t) of a continuous-time LTI

system (represented by T) is defined to be the response of the system when the input is (t), that is,

h(t) = T {(t)} (1)


Linear-time invariant systemsResponse to an arbitrary input:The input x(t) can be expressed

Since the system is linear, the response y(t) of the system to an arbitrary input x( t ) can be expressed as


)2()()()(

dtxtx

)3()()(

)()()}({)(

dtTx

dtxTtxTty

Linear-time invariant systemsResponse to an arbitrary input (continued):Since the system is time-invariant, we have

Substituting in equation (3) we get

Equation (5) indicates that a continuous-time LTI system is completely characterized by its impulse response h(t).


)4()}({)( tTth

)5()()()(

dthxty

Linear-time invariant systemsConvolution integral:Equation (5) defines the convolution of two

continuous-time signals x(t) and h(t) denoted by

Equation (6) is commonly called the convolution integral. Thus, we have the fundamental result that the output of any continuous-time LTI system is the convolution of the input x(t) with the impulse response h(t) of the system.


)6()()()()()(

dthxthtxty

Linear-time invariant systemsConvolution integral (continued):The figure below illustrates the definition of the

impulse response h(t) and the relationship of Equation (6).


Linear-time invariant systemsConvolution (continued):• Properties of the Convolution Integral:• Commutative:

• Associative:

• Distributive:


)7()()()()( txththtx

)8()()()()()()( 2121 ththtxththtx

)9()()()()()()()( 2121 thtxthtxththtx

Linear-time invariant systemsConvolution with unit impulse:• Convolving with Unit Impulse

Proof:For this proof, we will let (t) be the unit impulse

located at the origin. Using the definition of convolution we start with the convolution integral


)10()()()( txttx

)11()()()()(

dtxttx

Linear-time invariant systemsConvolution with unit impulse:Proof (continued):From the definition of the unit impulse, we know that

() = 0 whenever ≠ 0.We use this fact to reduce the above equation to the

following:


)12()()()(

)()()()(

txdtx

dtxttx

Linear-time invariant systemsConvolution:Width:• If Duration x1(t) = T1 and Duration x2(t) = T2, then

duration of is (T1 + T2)

Causality:• If both x1(t) and x2(t) are causal, then

is also causal.• when the input x(t) is causal, the output y(t) of a

causal continuous-time LTI system is given by


)()( 21 txtx

)13()()()(0 t

dthxty

)()( 21 txtx

Discrete LTI systemsImpulse Response:The impulse response (or unit sample response) h[n]

of a discrete-time LTI system (represented by T) is defined to be the response of the system when the input is [n], that is,

Response to an Arbitrary Input:As we know, an input x[n] can be expressed as


)14(]}[{][ nTnh

)15(][][][

k

knkxnx

Discrete LTI systemsResponse to an Arbitrary Input (continued):• Since the system is linear, the response y[n] of the

system to an arbitrary input x[n] can be expressed as


)15(][][

][][

][][][][

k

k

k

knhkx

knTkx

knkxTnxTny

Discrete LTI systemsConvolution Sum:• Equation (15) defines the convolution of two

sequences x[n] and h[n] denoted by

• Equation (16) is commonly called the convolution sum. Thus, again, we have the fundamental result that the output of any discrete-time LTI system is the convolution of the input x[n] with the impulse response h[n] of the system.


)16(][][][][][

k

knhkxnhnxny

Discrete LTI systemsConvolution Sum:• The Figure below illustrates the definition of the

impulse response h[n] and the relationship of Eq. (16).


Discrete LTI systemsProperties of Convolution Sum:• The following properties of the convolution sum are

analogous to the convolution integral properties:• Commutative:

• Associative:

• Distributive:


)17(][][][][ nxnhnhnx

)18(][][][][][][ 2121 nhnhnxnhnhnx

)19(][][][][][][][ 2121 nhnxnhnxnhnhnx

Discrete LTI systems• Step Response:The step response s[n] of a discrete-time LTI

system with the impulse response h[n] is readily obtained from Eq. (20)


)20(][][][][][][

k

n

k

khknukhnunhns

)21(]1[][][ nsnsnh

Properties of Discrete-time LTI Systems

• Systems with or without Memory:Since the output y[n] of a memoryless system

depends on only the present input x[n], then, if the system is also linear and time-invariant, this relationship can only be of the form

where K is a (gain) constant. Thus, the corresponding impulse response is simply


)22(][][ nxKny

)23(][][ nKnh


• Causality:Similar to the continuous-time case, the causality

condition for a discrete-time LTI system is

Applying the causality condition Eq.(24) to Eq.(16), the output of a causal discrete-time LTI system is expressed as

or


)24(00][ nnh

)25(][][][0

k

knxkhny

)26(][][][

n

k

knhkxny


• Causality:As in the continuous-time case, we say that any

sequence x[n] is called causal if

and is called anticausal if

Then, when the input x[n] is causal, the output y[n] of a causal discrete-time LTI system is given by


)26(00][ nnx

)28(][][][][][0 0

n

k

n

k

knhkxknxkhny

)27(00][ nnx


• Stability:It can be shown that a discrete-time LTI system is

BIB0 stable if its impulse response is absolutely summable, that is,


)29(][

k

kh

Differential and DifferenceLTI Systems

• Causal CT LTI Systems Described by Differential Equations whereas Causal DT LTI Systems Described by Difference Equations.



Differential equations play a central role in describing the input-output relationships of a wide variety of electrical, mechanical, chemical, and biological system

A general Nth-order linear constant-coefficient differential equation is given by where coefficients a, and b, are real constants. The order N refers to the highest derivative of y(t) in Eq. (30).


)30()()(0 0

N

k

M

kk

k

kk

k

k dttxdb

dttyda


Equation (30) can be expanded to

• To find a solution to a differential equation of this form, we need more information than the equation provides. We need N initial conditions (or auxiliary conditions) on the output variable y(t) and its derivatives to be able to calculate a solution.


)31()()()(

)()()(

01

01

txbdttdxb

dttxdb

tyadttdya

dttyda

M

M

M

N

N

N


The complete solution to Eq. (30) is given by the sum of the homogeneous solution of the differential equation (a solution with the input signal set to zero) and of a particular solution (an output signal that satisfies the differential equation), also called the forced response of the system.

The usual terminology is as follows:• Forced response of the system = particular solution

(usually has the same form as the input signal).• Natural response of the system = homogeneous solution

(depends on initial conditions and forced response)



• Therefore, the solution is composed of a homogeneous response (natural response) and a particular solution (forced response) of the system:

where yp(t) is a solution of Eq. (30) and yh(t)


)32()()()( tytyty hp

)33(0)(0

N

kk

k

k dttyda


• The exact form of yh(t) is determined by N auxiliary conditions. In general, a set of auxiliary conditions are the values of

at some point in time.• In order for the linear system described by Eq. (30)

to be causal we must assume the condition of initial rest (or an initially relaxed condition).


1

1 )(,,)(),(

N

N

dttyd

dttdyty


• That is, if x(t) = 0 for t ≤ to, then assume y(t) = 0 for t ≤ to. Thus, the response for t > to can be calculated from Eq. (30) with the initial conditions

where


0

)()(

0)()()(

0

10

10

0

ttk

k

k

k

N

N

dttyd

dttyddt

tyddttdyty


• Example #1:Consider the LTI system described by the causal

linear constant coefficient differential equation

Calculate the output of this system y(t) in response to the input signal x(t) = 5000e-2tu(t).

• As stated above, the solution is composed of a homogeneous response and a particular solution of the system:


)34()()(300)(1000 txtydttdy

)()()( tytyty hp


• Solution/step #1:For the particular solution for t > 0, we consider a

signal yp(t) of the same form as x(t) for t > 0: yp(t)=Ce-2t , where coefficient C is to be determined. Substituting the exponentials for x(t) and yp(t) in Equation (34), we get

which simplifies to -2000C + 300C = 5000and yields C = -2.941Thus, we have yp(t) = -2.941e-2t , t > 0.


ttt eCeCe 222 50003002000


• Solution/step #2:Now we want to determine yh(t), the natural response

of the system. We assume a solution of the form of an exponential: yh(t) = Aest , where A ≠ 0.

In accordance to Eq. (33) we have

Substituting the value of yh(t), we get


0)(300)(1000 tydttdy

hh

ststst esAAeAse )3001000(30010000


• Solution/step #2:which simplifies to s + 0.3 = 0. This equation holds for

s = -0.3. Also, with this value for s, Ae-0.3t is a solution to the homogeneous response for any value of A.

Combining both responses yh(t) and yp(t), we get

Now, because we have not yet specified an initial condition on y(t), this response is not completely determined, as the value of A is still unknown.


.0,941.2)()()( 23.0 teAetytyty tthp


• Solution/step #3:Strictly speaking, for causal LTI systems defined by

linear constant-coefficient differential equations, the initial conditions must be

which is called initial rest.In our Example, initial rest implies that y(0) = 0, so that


0)0()0()0( 1

1

N

N

dtyd

dtdyy

0941.2)0( Ay


• Solution/step #3:and we get A = 2.941. Thus, the response of the

system is given by

What about the negative times t < 0? The condition of initial rest and the causality of the system imply that y(t) = 0, t < 0 since x(t) = 0, t < 0. Therefore, we can write the output as follows:


.0,)(941.2)( 23.0 teety tt

)()(941.2)( 23.0 tueety tt

Causal LTI systems described by Difference equations.

• In a causal LTI difference system, the discrete-time input and output signals are related implicitly through a linear constant-coefficient difference equation.

• In general, an Nth-order linear constant coefficient difference equation has the form


)35(][][0 0

N

k

M

kkk knxbknya


which can be expanded to

• The constant coefficients and are assumed to be real, and although some of them may be equal to zero, it is assumed that without loss of generality.

• The order of the difference equation is defined as the longest time delay of the output present In the equation.


)36(][]1[][][]1[][

01

01

nxbnxbMnxbnyanyaNnya

M

N

MiiNii ba 11

0Na


• To find a solution to the difference equation, we need more information than what the equation provides. We need N initial conditions (or auxiliary conditions) on the output variable (its N past values) to be able to compute a specific solution.



General Solution:• A general solution to Equation (35) can be

expressed as the sum of a homogeneous solution

(natural response) to

and a particular solution (forced response), in a manner analogous to the continuous-time case.


)37(][][][ nynyny hp

N

kk knya

0

0][


General Solution:• The concept of initial rest of the LTI causal system

described by the difference equation here means that implies .

• Example #2:Consider the first-order difference equation initially at

rest:

the homogeneous solution satisfies


00 ,0][,0][ nnnynnnx

)38(][)8.0(]1[5.0][ nunyny n

)39(0]1[5.0][ nyny hh


• For the particular solution for n ≥ 0, we look for a signal yp[n] of the same form as x[n]:

Then, we get

which yields


3/8

1)8.0(5.01

)8.0()8.0(5.0)8.0(1

1

AA

AA nnn

np Any )8.0(][

np ny )8.0(

38][


• Now let us determine yh[n], the natural response of the system. We hypothesize a solution of the form of an exponential signal: .

Substituting this exponential in Equation (39), we get

• With this value for z, is a solution to the homogeneous equation for any choice of B.


5.005.01

05.01

1

zz

BzBz nn

nh Bny )5.0(][

nh Bzny ][


• Combining the natural response and the forced response, we find the solution to the difference Equation (38) for n ≥ 0:

• The assumption of initial rest implies y[–1] = 0, but we need to use an initial condition at a time where the forced response exists ( for n ≥ 0), that is, y[0], which can be obtained by a simple recursion.


nnph Bnynyny )8.0(

38)5.0(][][][


• Note that this remark also holds for higher-order systems. For instance, the response of a second-order system initially at rest satisfies the conditions y[–2] = y[–1] = 0, but y[0], y[1] must be computed recursively and used as new initial conditions in order to obtain the correct coefficients in the homogeneous response.


110)8.0(]1[5.0]0[:0

][)8.0(]1[5.0][0

yyn

nunyny n


• In our example, the coefficient is computed as follows:

• Therefore, the complete solution is


35

38)8.0(

38)5.0(1]0[ 00

B

BBy

].[)8.0(38][)5.0(

35][ nununy nn

signals and systems analysis net 351 instructor: dr. amer el-khairy د. عامر الخيري

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