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Linear-time invariant systems Impulse 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) 3Signals and systems analysis د. عامر الخيري
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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)
3Signals and systems analysis الخيري. عامر د
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
4Signals and systems analysis الخيري. عامر د
)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).
5Signals and systems analysis الخيري. عامر د
)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.
6Signals and systems analysis الخيري. عامر د
)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).
7Signals and systems analysis الخيري. عامر د
Linear-time invariant systemsConvolution (continued):• Properties of the Convolution Integral:• Commutative:
• Associative:
• Distributive:
8Signals and systems analysis الخيري. عامر د
)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
9Signals and systems analysis الخيري. عامر د
)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:
10Signals and systems analysis الخيري. عامر د
)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
11Signals and systems analysis الخيري. عامر د
)()( 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
12Signals and systems analysis الخيري. عامر د
)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
13Signals and systems analysis الخيري. عامر د
)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.
14Signals and systems analysis الخيري. عامر د
)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).
15Signals and systems analysis الخيري. عامر د
Discrete LTI systemsProperties of Convolution Sum:• The following properties of the convolution sum are
analogous to the convolution integral properties:• Commutative:
• Associative:
• Distributive:
16Signals and systems analysis الخيري. عامر د
)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)
17Signals and systems analysis الخيري. عامر د
)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
18Signals and systems analysis الخيري. عامر د
)22(][][ nxKny
)23(][][ nKnh
Properties of Discrete-time LTI Systems
• 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
19Signals and systems analysis الخيري. عامر د
)24(00][ nnh
)25(][][][0
k
knxkhny
)26(][][][
n
k
knhkxny
Properties of Discrete-time LTI Systems
• 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
20Signals and systems analysis الخيري. عامر د
)26(00][ nnx
)28(][][][][][0 0
n
k
n
k
knhkxknxkhny
)27(00][ nnx
Properties of Discrete-time LTI Systems
• Stability:It can be shown that a discrete-time LTI system is
BIB0 stable if its impulse response is absolutely summable, that is,
21Signals and systems analysis الخيري. عامر د
)29(][
k
kh
Differential and DifferenceLTI Systems
• Causal CT LTI Systems Described by Differential Equations whereas Causal DT LTI Systems Described by Difference Equations.
22Signals and systems analysis الخيري. عامر د
Differential and DifferenceLTI Systems
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).
23Signals and systems analysis الخيري. عامر د
)30()()(0 0
N
k
M
kk
k
kk
k
k dttxdb
dttyda
Differential and DifferenceLTI Systems
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.
24Signals and systems analysis الخيري. عامر د
)31()()()(
)()()(
01
01
txbdttdxb
dttxdb
tyadttdya
dttyda
M
M
M
N
N
N
Differential and DifferenceLTI Systems
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)
25Signals and systems analysis الخيري. عامر د
Differential and DifferenceLTI Systems
• 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)
26Signals and systems analysis الخيري. عامر د
)32()()()( tytyty hp
)33(0)(0
N
kk
k
k dttyda
Differential and DifferenceLTI Systems
• 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).
27Signals and systems analysis الخيري. عامر د
1
1 )(,,)(),(
N
N
dttyd
dttdyty
Differential and DifferenceLTI Systems
• 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
28Signals and systems analysis الخيري. عامر د
0
)()(
0)()()(
0
10
10
0
ttk
k
k
k
N
N
dttyd
dttyddt
tyddttdyty
Differential and DifferenceLTI Systems
• 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:
29Signals and systems analysis الخيري. عامر د
)34()()(300)(1000 txtydttdy
)()()( tytyty hp
Differential and DifferenceLTI Systems
• 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.
30Signals and systems analysis الخيري. عامر د
ttt eCeCe 222 50003002000
Differential and DifferenceLTI Systems
• 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
31Signals and systems analysis الخيري. عامر د
0)(300)(1000 tydttdy
hh
ststst esAAeAse )3001000(30010000
Differential and DifferenceLTI Systems
• 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.
32Signals and systems analysis الخيري. عامر د
.0,941.2)()()( 23.0 teAetytyty tthp
Differential and DifferenceLTI Systems
• 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
33Signals and systems analysis الخيري. عامر د
0)0()0()0( 1
1
N
N
dtyd
dtdyy
0941.2)0( Ay
Differential and DifferenceLTI Systems
• 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:
34Signals and systems analysis الخيري. عامر د
.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
35Signals and systems analysis الخيري. عامر د
)35(][][0 0
N
k
M
kkk knxbknya
Causal LTI systems described by Difference equations.
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.
36Signals and systems analysis الخيري. عامر د
)36(][]1[][][]1[][
01
01
nxbnxbMnxbnyanyaNnya
M
N
MiiNii ba 11
0Na
Causal LTI systems described by Difference equations.
• 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.
37Signals and systems analysis الخيري. عامر د
Causal LTI systems described by Difference equations.
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.
38Signals and systems analysis الخيري. عامر د
)37(][][][ nynyny hp
N
kk knya
0
0][
Causal LTI systems described by Difference equations.
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
39Signals and systems analysis الخيري. عامر د
00 ,0][,0][ nnnynnnx
)38(][)8.0(]1[5.0][ nunyny n
)39(0]1[5.0][ nyny hh
Causal LTI systems described by Difference equations.
• 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
40Signals and systems analysis الخيري. عامر د
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][
Causal LTI systems described by Difference equations.
• 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.
41Signals and systems analysis الخيري. عامر د
5.005.01
05.01
1
zz
BzBz nn
nh Bny )5.0(][
nh Bzny ][
Causal LTI systems described by Difference equations.
• 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.
42Signals and systems analysis الخيري. عامر د
nnph Bnynyny )8.0(
38)5.0(][][][
Causal LTI systems described by Difference equations.
• 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.
43Signals and systems analysis الخيري. عامر د
110)8.0(]1[5.0]0[:0
][)8.0(]1[5.0][0
yyn
nunyny n
Causal LTI systems described by Difference equations.
• In our example, the coefficient is computed as follows:
• Therefore, the complete solution is
44Signals and systems analysis الخيري. عامر د
35
38)8.0(
38)5.0(1]0[ 00
B
BBy
].[)8.0(38][)5.0(
35][ nununy nn
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