Copyright ? 2001,S,K,Mitra
1
Discrete-Time Systems
? A discrete-time system processes a given
input sequence x[n] to generates an output
sequence y[n] with more desirable
properties
? In most applications,the discrete-time
system is a single-input,single-output
system,
Sy s te m tim eD is c r e te ?x[n] y[n]
Input sequence Output sequence
Copyright ? 2001,S,K,Mitra
2
Discrete-Time Systems,
Examples
? 2-input,1-output discrete-time systems -
Modulator,adder
? 1-input,1-output discrete-time systems -
Multiplier,unit delay,unit advance
Copyright ? 2001,S,K,Mitra
3
Discrete-Time Systems,Examples
? Accumulator -
? The output y[n] at time instant n is the sum
of the input sample x[n] at time instant n
and the previous output at time
instant which is the sum of all
previous input sample values from to
? The system cumulatively adds,i.e.,it
accumulates all input sample values
??
???
n xny
?
? ][][
][]1[][][1 nxnynxxn ?????? ?
????
?
]1[ ?ny
,1?n
1?n??
Copyright ? 2001,S,K,Mitra
4
Discrete-Time Systems:Examples
? Accumulator - Input-output relation can
also be written in the form
? The second form is used for a causal input
sequence,in which case is called
the initial condition
????
?
?
???
n xxny
0
1 ][][][
??
??
,][]1[
0
????
?
n xy
?
?
]1[?y
0?n
Copyright ? 2001,S,K,Mitra
5
Discrete-Time Systems:Examples
? M-point moving-average system -
? Used in smoothing random variations in
data
? An application in denoising,Consider
x[n] = s[n] + d[n],
where s[n] is the signal corrupted by a noise
d[n]
? ??
?
?
1
0
][1][ M
k
knxMny
Copyright ? 2001,S,K,Mitra
6
d[n] - random signal ],)9.0([2][ nnns ?
Discrete-Time Systems:Examples
0 10 20 30 40 50
0
1
2
3
4
5
6
7
T i m e i nde x n
A
m
pl
i
t
ude
s [ n]
y[ n]
Copyright ? 2001,S,K,Mitra
7
Discrete-Time Systems:Examples
? Linear interpolation - Employed to estimate
sample values between pairs of adjacent
sample values of a discrete-time sequence
? Factor-of-4 interpolation
Copyright ? 2001,S,K,Mitra
8
Discrete-Time Systems,
Examples
? Factor-of-2 interpolator -
? Factor-of-3 interpolator -
? ?]1[]1[21][][ ????? nxnxnxny uuu
? ?]2[]1[31][][ ????? nxnxnxny uuu
? ?]1[]2[32 ???? nxnx uu
Copyright ? 2001,S,K,Mitra
9
Discrete-Time Systems,
Classification
? Linear Systems
? Shift-Invariant Systems
? Causal Systems
? Stable Systems
? Passive and Lossless Systems
Copyright ? 2001,S,K,Mitra
10
Linear Discrete-Time Systems
? Definition - If is the output due to an
input and is the output due to an
input then for an input
the output is given by
? Above property must hold for any arbitrary
constants and and for all possible
inputs and
][1 ny
][1 nx
][2 nx
][2 ny
][][][ 21 nxnxnx ?? ??
][][][ 21 nynyny ?? ??
?,?
][1 nx ][2 nx
Copyright ? 2001,S,K,Mitra
11
Linear Discrete-Time Systems
? Accumulator -
? For an input
the output is
? Hence,the above system is linear
????
??????
nn xnyxny
??
?? ][][,][][ 2211
][][][ 21 nxnxnx ?? ??
? ?? ??
???
n xxny
?
?? ][][][ 21 ??
][][][][ 2121 nynyxx nn ???? ??????
?????? ??
??
Copyright ? 2001,S,K,Mitra
12
Nonlinear Discrete-Time
System
? Consider
? Outputs and for inputs
and are given by
][][][][ 112 ???? nxnxnxny
][ny1
][][][][ 11 11211 ???? nxnxnxny
][ny2
][][][][ 11 22222 ???? nxnxnxny
][nx2
][nx1
Copyright ? 2001,S,K,Mitra
13
Nonlinear Discrete-Time
System
? Output y[n] due to an input
is given by
][][ nxnx 21 ?? ?
212[ ] { [ ] [ ] }y n x n x n????
]}[][] } {[][{ 1111 2121 ??????? nxnxnxnx ????
]}[][][{ 11 11212 ???? nxnxnx?
]}[][][{ 11 22222 ???? nxnxnx?
]}[][][][][][{ 11112 212121 ??????? nxnxnxnxnxnx??
Copyright ? 2001,S,K,Mitra
14
Nonlinear Discrete-Time
System
? On the other hand
? Hence,the system is nonlinear
][][ nyny 21 ?? ?
]}[][][{ 11 1121 ???? nxnxnx?
]}[][][{ 11 2222 ???? nxnxnx?
][ny?
Copyright ? 2001,S,K,Mitra
15
Shift-Invariant System
? For a shift-invariant system,if is the
response to an input,then the response
to an input
is simply
where is any positive or negative integer
? The above relation must hold for any
arbitrary input and its corresponding output
][ny1
][nx1
][][ onnxnx ?? 1
][][ onnyny ?? 1
on
Copyright ? 2001,S,K,Mitra
16
Shift-Invariant System
? In the case of sequences and systems with
indices n related to discrete instants of time,
the above property is called time-invariance
property
? Time-invariance property ensures that for a
specified input,the output is independent of
the time the input is being applied
Copyright ? 2001,S,K,Mitra
17
Shift-Invariant System
? Example - Consider the up-sampler with an
input-output relation given by
? For an input the output
is given by
??
? ????
o th e r w ise,
.,,,,,,,],/[][
0
20 LLnLnxnx
u
][][ onnxnx ??1 ][,nx u1
??
? ????
o th e r w is e,
.....,,,],/[][
,0
201
1
LLnLnxnx
u
??
? ?????
o th e r w is e,
.,,,,,,,],/)[(
0
20 LLnLLnnx o
Copyright ? 2001,S,K,Mitra
18
Shift-Invariant System
? However from the definition of the up-sampler
? Hence,the up-sampler is a time-varying system
][ ou nnx ?
??
? ?????
o th e r w is e,
.,,,,,,,],/)[(
0
2 LnLnnnLnnx oooo
][,nx u1?
Copyright ? 2001,S,K,Mitra
19
Linear Time-Invariant System
? Linear Time-Invariant (LTI) System -
A system satisfying both the linearity and
the time-invariance property
? LTI systems are mathematically easy to
analyze and characterize,
? Therefore,easy to design
Copyright ? 2001,S,K,Mitra
20
Causal System
? In a causal system,the -th output sample
depends only on input samples x[n]
for and does not depend on input
samples for
? Let and be the responses of a
causal discrete-time system to the inputs
and,respectively
on
onn ?
onn ?
][ ony
][ny1 ][ny2
][nx2
][nx1
Copyright ? 2001,S,K,Mitra
21
Causal System
? Then
for n < N
implies also that
for n < N
? For a causal system,changes in output
samples do not precede changes in the input
samples
][][ 21 nxnx ?
][][ 21 nyny ?
Copyright ? 2001,S,K,Mitra
22
Causal System
? Examples of causal systems,
? Examples of noncausal systems,
][][][][][ 321 4321 ??????? nxnxnxnxny ????
][][][][ 21 210 ????? nxbnxbnxbny
][][ 21 21 ???? nyanya
][][][ nxnyny ??? 1
])[][(][][ 1121 ????? nxnxnxny uuu
])[][(][][ 2131 ????? nxnxnxny uuu
])[][( 1232 ???? nxnx uu
Copyright ? 2001,S,K,Mitra
23
Causal System
? A noncausal system can be implemented as
a causal system by delaying the output by
an appropriate number of samples
? For example a causal implementation of the
factor-of-2 interpolator is given by
])[][(][][ nxnxnxny uuu ????? 21 21
Copyright ? 2001,S,K,Mitra
24
Stable System
? There are various definitions of stability
? We consider here the bounded-input,
bounded-output (BIBO) stability
? If y[n] is the response to an input x[n] and if
for all values of n
then
for all values of n
xBnx ?][
yBny ?][
Copyright ? 2001,S,K,Mitra
25
Stable System
? Example - The M-point moving average
filter is BIBO stable,
? For a bounded input we have
?
?
?
??
1
0
1 M
kM
knxny ][][
xBnx ?][
??
?
?
?
?
????
1
0
11
0
1 M
kM
M
kM
knxknxny ][][][
xxM BMB ?? )(1
Copyright ? 2001,S,K,Mitra
26
Passive and Lossless Systems
? A discrete-time system is defined to be
passive if,for every finite-energy input x[n],
the output y[n] has,at most,the same energy,
i.e,
? For a lossless system,the above inequality is
satisfied with an equal sign for every input
??? ??
?
???
?
??? nn
nxny 22 ][][
Copyright ? 2001,S,K,Mitra
27
Passive and Lossless Systems
? Example - Consider the discrete-time
system defined by with N
a positive integer
? Its output energy is given by
? Hence,it is a passive system if and is
a lossless system if
][][ Nnxny ?? ?
2 2 2 2 2[ ] [ ] [ ]
n n n
y n x n N x n??
? ? ?
? ? ? ? ? ? ? ? ?
? ? ?? ? ?
1??
1??
Copyright ? 2001,S,K,Mitra
28
Impulse and Step Responses
? The response of a discrete-time LTI system
to a unit sample sequence {d[n]} is called
the unit sample response or,simply,the
impulse response,and is denoted by {h[n]}
? The response of a discrete-time LTI system
to a unit step sequence {m[n]} is called the
unit step response or,simply,the step
response,and is denoted by {s[n]}
Copyright ? 2001,S,K,Mitra
29
Impulse and Step Responses
LTI [ ] [ ]x n nd? [ ] [ ]y n h n?
? impulse response
? step response
LTI [ ] [ ]x n nm? [ ] [ ]y n s n?
Copyright ? 2001,S,K,Mitra
30
Impulse Response
? Example - The impulse response of the
system
is obtained by setting x[n] = d[n] resulting in
? The impulse response is thus a finite-length
sequence of length 4 given by
][][][][][ 321 4321 ??????? nxnxnxnxny ????
][][][][][ 321 4321 ??????? nnnnnh d?d?d?d?
},,,{]}[{ 4321 ????
?
?nh
Copyright ? 2001,S,K,Mitra
31
Impulse Response
? Example - The impulse response of the
discrete-time accumulator
is obtained by setting x[n] = d[n] resulting
in
?
???
?
n
xny
?
? ][][
][][][ nnh
n
md ?? ?
????
?
Copyright ? 2001,S,K,Mitra
32
Impulse Response
? Example - The impulse response {h[n]} of
the factor-of-2 interpolator
is obtained by setting and is
given by
? The impulse response is thus a finite-length
sequence of length 3,
])[][(][][ 1121 ????? nxnxnxny uuu
])[][(][][ 1121 ????? nnnnh ddd
}.,.{]}[{ 50150
?
?nh
][][ nnx u d?
Copyright ? 2001,S,K,Mitra
33
Time-Domain Characterization
of Discrete-Time LTI Systems
? Input-Output Relationship – A consequence
of the linear,time-invariance property is that
a discrete-time LTI system is completely
characterized by its impulse response
? Therefore,knowing the impulse response,
one can compute the output of the system for
any arbitrary input
Copyright ? 2001,S,K,Mitra
34
? A discrete-time LTI system is denoted as
[]xn []yn
Time-Domain Characterization
of Discrete-Time LTI Systems
[]hn
Copyright ? 2001,S,K,Mitra
35
Time-Domain Characterization
of Discrete-Time LTI Systems
? Let h[n] denote the impulse response of a
discrete-time LTI system
? We compute its output y[n] for the input,
? As the system is linear,we can compute its
outputs for each member of the input
separately and add the individual outputs to
determine y[n]
]5[75.0]2[]1[5.1]2[5.0][ ?d??d??d??d? nnnnnx
Copyright ? 2001,S,K,Mitra
36
Time-Domain Characterization
of Discrete-Time LTI Systems
? Since the system is time-invariant
input output
]2[]2[ ???d nhn
]1[]1[ ???d nhn
]2[]2[ ???d nhn
]5[]5[ ???d nhn
Copyright ? 2001,S,K,Mitra
37
Time-Domain Characterization
of Discrete-Time LTI Systems
? Likewise,as the system is linear
? Hence,because of the linearity property we
get
]5[75.0]5[75.0 ???d nhn
input output
]2[5.0]2[5.0 ???d nhn
]2[]2[ ????d? nhn
]1[5.1]1[5.1 ???d nhn
][.][.][ 151250 ???? nhnhny
][.][ 57502 ???? nhnh
Copyright ? 2001,S,K,Mitra
38
Time-Domain Characterization
of Discrete-Time LTI Systems
? Now,any arbitrary input sequence x[n] can
be expressed as a linear combination of
delayed and advanced unit sample
sequences in the form
? The response of the LTI system to an input
will be
? ?d?
?
???k
knkxnx ][][][
][][ knkx ?d ][][ knhkx ?
Copyright ? 2001,S,K,Mitra
39
Time-Domain Characterization
of Discrete-Time LTI Systems
? Hence,the response y[n] to an input
will be
which can be alternately written as
? ?d?
?
???k
knkxnx ][][][
? ??
?
???k
knhkxny ][][][
?
?
???
??
k
khknxny ][][][
Copyright ? 2001,S,K,Mitra
40
Convolution Sum
? The summation
is called the convolution sum of the
sequences x[n] and h[n] and represented
compactly as
??
?
???
?
???
????
kk
nhknxknhkxny ][][][][][
y[n] = x[n] h[n] *
Copyright ? 2001,S,K,Mitra
41
Convolution Sum
? Properties
– Commutative property,
– Associative property,
– Distributive property,
x[n] h[n] = h[n] x[n] * *
(x[n] h[n]) y[n] = x[n] (h[n] y[n]) * * * *
x[n] (h[n] + y[n]) = x[n] h[n] + x[n] y[n] * * *
Copyright ? 2001,S,K,Mitra
42
Convolution Sum
? Interpretation,
1) Time-reverse h[k] to form
2) Shift to the right by n sampling
periods if n > 0 or shift to the left by n
sampling periods if n < 0 to form
3) Form the product
4) Sum all samples of v[k] to develop the
n-th sample of y[n] of the convolution sum
][ kh ?
][ kh ?
][ knh ?
][][][ knhkxkv ??
Copyright ? 2001,S,K,Mitra
43
Convolution Sum
? Schematic Representation,
? The computation of an output sample using
the convolution sum is simply a sum of
products
? Involves fairly simple operations such as
additions,multiplications,and delays
?nz ][ knh ?][ kh ?
][kx
][kv ][ny?
k
Copyright ? 2001,S,K,Mitra
44
Convolution Using MATLAB
? The M-file conv implements the convolution
sum of two finite-length sequences
? If
then conv(a,b) yields
]31102[a ???
]1-021[b ?
]31513142[ ???
Copyright ? 2001,S,K,Mitra
45
Simple Interconnection
Schemes
? Two simple interconnection schemes are,
– Cascade Connection
– Parallel Connection
Copyright ? 2001,S,K,Mitra
46
Cascade Connection
? Impulse response h[n] of the cascade of two
discrete-time LTI systems with impulse
responses and is given by
][nh1][nh2][nh1 ][nh2?
][][ nhnh 1? ][2][nh1 * ?
][nh1 ][nh2
][nh2 ][][ nnh 1? *
Copyright ? 2001,S,K,Mitra
47
Cascade Connection
? Note,The ordering of the systems in the
cascade has no effect on the overall impulse
response because of the commutative
property of convolution
? A cascade connection of two stable systems
is stable
? A cascade connection of two passive
(lossless) systems is passive (lossless)
Copyright ? 2001,S,K,Mitra
48
Cascade Connection
? An application is in the development of an
inverse system
? If the cascade connection satisfies the
relation
then the LTI system is said to be the
inverse of and vice-versa
][nh1
][nh2
][nh2][1 nh ][nd?*
Copyright ? 2001,S,K,Mitra
49
Cascade Connection
? An application of the inverse system
concept is in the recovery of a signal x[n]
from its distorted version appearing at
the output of a transmission channel
? If the impulse response of the channel is
known,then x[n] can be recovered by
designing an inverse system of the channel
][? nx
][nh2][nh1][nx ][nx
channel inverse system ][nx^
][nh2][nh1 ][nd?*
Copyright ? 2001,S,K,Mitra
50
Cascade Connection
? Example - Consider the discrete-time
accumulator with an impulse response m[n]
? Its inverse system satisfies the condition
? It follows from the above that for
n < 0 and
for
02 ?][nh
1]0[2 ?h
0
0
2 ??
?
n
h
?
?][ 1?n
][nh2][nm ][nd?*
Copyright ? 2001,S,K,Mitra
51
Cascade Connection
? Thus the impulse response of the inverse
system of the discrete-time accumulator is
given by
which is called a backward difference
system
]1[][][2 ?d?d? nnnh
Copyright ? 2001,S,K,Mitra
52
? Impulse response h[n] of the parallel
connection of two discrete-time LTI
systems with impulse responses and
is given by
][nh2
][nh1
? ][][ nhnh 1? ][2][nh1? ?
][nh1
][nh2
][][][ nhnhnh 21 ??
Parallel Connection
Copyright ? 2001,S,K,Mitra
53
Simple Interconnection Schemes
? Consider the discrete-time system where
][nh2
][nh1 ?
?
][nh4
][nh3
],1[5.0][][1 ?d?d? nnnh
],1[25.0][5.0][2 ?d?d? nnnh
],[2][3 nnh d?
][)5.0(2][4 nnh n m??
Copyright ? 2001,S,K,Mitra
54
Simple Interconnection Schemes
? Simplifying the block-diagram we obtain
][nh2
][nh1 ?
][][ 43 nhnh ?
][nh1 ?
])[][(][ 432 nhnhnh ?*
?
Copyright ? 2001,S,K,Mitra
55
Simple Interconnection Schemes
? Overall impulse response h[n] is given by
? Now,
][][][][][ nhnhnhnhnh 42321 ???
])[][(][][][ nhnhnhnhnh 4321 ???*
* *
][2])1[][(][][ 412132 nnnnhnh d?d?d?
]1[][ 21 ?d?d? nn
* *
Copyright ? 2001,S,K,Mitra
56
Simple Interconnection Schemes
? Therefore
]1[)(][)( 1212121 ?m?m?? ? nn nn
]1[)(][)( 2121 ?m?m?? nn nn
][][)( 21 nnn d??d??
][][]1[][]1[][][ 2121 nnnnnnnh d?d??d?d??d?d?
? ?][)(2])1[][(][][ 21412142 nnnnhnh n m??d?d?* *
1
Discrete-Time Systems
? A discrete-time system processes a given
input sequence x[n] to generates an output
sequence y[n] with more desirable
properties
? In most applications,the discrete-time
system is a single-input,single-output
system,
Sy s te m tim eD is c r e te ?x[n] y[n]
Input sequence Output sequence
Copyright ? 2001,S,K,Mitra
2
Discrete-Time Systems,
Examples
? 2-input,1-output discrete-time systems -
Modulator,adder
? 1-input,1-output discrete-time systems -
Multiplier,unit delay,unit advance
Copyright ? 2001,S,K,Mitra
3
Discrete-Time Systems,Examples
? Accumulator -
? The output y[n] at time instant n is the sum
of the input sample x[n] at time instant n
and the previous output at time
instant which is the sum of all
previous input sample values from to
? The system cumulatively adds,i.e.,it
accumulates all input sample values
??
???
n xny
?
? ][][
][]1[][][1 nxnynxxn ?????? ?
????
?
]1[ ?ny
,1?n
1?n??
Copyright ? 2001,S,K,Mitra
4
Discrete-Time Systems:Examples
? Accumulator - Input-output relation can
also be written in the form
? The second form is used for a causal input
sequence,in which case is called
the initial condition
????
?
?
???
n xxny
0
1 ][][][
??
??
,][]1[
0
????
?
n xy
?
?
]1[?y
0?n
Copyright ? 2001,S,K,Mitra
5
Discrete-Time Systems:Examples
? M-point moving-average system -
? Used in smoothing random variations in
data
? An application in denoising,Consider
x[n] = s[n] + d[n],
where s[n] is the signal corrupted by a noise
d[n]
? ??
?
?
1
0
][1][ M
k
knxMny
Copyright ? 2001,S,K,Mitra
6
d[n] - random signal ],)9.0([2][ nnns ?
Discrete-Time Systems:Examples
0 10 20 30 40 50
0
1
2
3
4
5
6
7
T i m e i nde x n
A
m
pl
i
t
ude
s [ n]
y[ n]
Copyright ? 2001,S,K,Mitra
7
Discrete-Time Systems:Examples
? Linear interpolation - Employed to estimate
sample values between pairs of adjacent
sample values of a discrete-time sequence
? Factor-of-4 interpolation
Copyright ? 2001,S,K,Mitra
8
Discrete-Time Systems,
Examples
? Factor-of-2 interpolator -
? Factor-of-3 interpolator -
? ?]1[]1[21][][ ????? nxnxnxny uuu
? ?]2[]1[31][][ ????? nxnxnxny uuu
? ?]1[]2[32 ???? nxnx uu
Copyright ? 2001,S,K,Mitra
9
Discrete-Time Systems,
Classification
? Linear Systems
? Shift-Invariant Systems
? Causal Systems
? Stable Systems
? Passive and Lossless Systems
Copyright ? 2001,S,K,Mitra
10
Linear Discrete-Time Systems
? Definition - If is the output due to an
input and is the output due to an
input then for an input
the output is given by
? Above property must hold for any arbitrary
constants and and for all possible
inputs and
][1 ny
][1 nx
][2 nx
][2 ny
][][][ 21 nxnxnx ?? ??
][][][ 21 nynyny ?? ??
?,?
][1 nx ][2 nx
Copyright ? 2001,S,K,Mitra
11
Linear Discrete-Time Systems
? Accumulator -
? For an input
the output is
? Hence,the above system is linear
????
??????
nn xnyxny
??
?? ][][,][][ 2211
][][][ 21 nxnxnx ?? ??
? ?? ??
???
n xxny
?
?? ][][][ 21 ??
][][][][ 2121 nynyxx nn ???? ??????
?????? ??
??
Copyright ? 2001,S,K,Mitra
12
Nonlinear Discrete-Time
System
? Consider
? Outputs and for inputs
and are given by
][][][][ 112 ???? nxnxnxny
][ny1
][][][][ 11 11211 ???? nxnxnxny
][ny2
][][][][ 11 22222 ???? nxnxnxny
][nx2
][nx1
Copyright ? 2001,S,K,Mitra
13
Nonlinear Discrete-Time
System
? Output y[n] due to an input
is given by
][][ nxnx 21 ?? ?
212[ ] { [ ] [ ] }y n x n x n????
]}[][] } {[][{ 1111 2121 ??????? nxnxnxnx ????
]}[][][{ 11 11212 ???? nxnxnx?
]}[][][{ 11 22222 ???? nxnxnx?
]}[][][][][][{ 11112 212121 ??????? nxnxnxnxnxnx??
Copyright ? 2001,S,K,Mitra
14
Nonlinear Discrete-Time
System
? On the other hand
? Hence,the system is nonlinear
][][ nyny 21 ?? ?
]}[][][{ 11 1121 ???? nxnxnx?
]}[][][{ 11 2222 ???? nxnxnx?
][ny?
Copyright ? 2001,S,K,Mitra
15
Shift-Invariant System
? For a shift-invariant system,if is the
response to an input,then the response
to an input
is simply
where is any positive or negative integer
? The above relation must hold for any
arbitrary input and its corresponding output
][ny1
][nx1
][][ onnxnx ?? 1
][][ onnyny ?? 1
on
Copyright ? 2001,S,K,Mitra
16
Shift-Invariant System
? In the case of sequences and systems with
indices n related to discrete instants of time,
the above property is called time-invariance
property
? Time-invariance property ensures that for a
specified input,the output is independent of
the time the input is being applied
Copyright ? 2001,S,K,Mitra
17
Shift-Invariant System
? Example - Consider the up-sampler with an
input-output relation given by
? For an input the output
is given by
??
? ????
o th e r w ise,
.,,,,,,,],/[][
0
20 LLnLnxnx
u
][][ onnxnx ??1 ][,nx u1
??
? ????
o th e r w is e,
.....,,,],/[][
,0
201
1
LLnLnxnx
u
??
? ?????
o th e r w is e,
.,,,,,,,],/)[(
0
20 LLnLLnnx o
Copyright ? 2001,S,K,Mitra
18
Shift-Invariant System
? However from the definition of the up-sampler
? Hence,the up-sampler is a time-varying system
][ ou nnx ?
??
? ?????
o th e r w is e,
.,,,,,,,],/)[(
0
2 LnLnnnLnnx oooo
][,nx u1?
Copyright ? 2001,S,K,Mitra
19
Linear Time-Invariant System
? Linear Time-Invariant (LTI) System -
A system satisfying both the linearity and
the time-invariance property
? LTI systems are mathematically easy to
analyze and characterize,
? Therefore,easy to design
Copyright ? 2001,S,K,Mitra
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Causal System
? In a causal system,the -th output sample
depends only on input samples x[n]
for and does not depend on input
samples for
? Let and be the responses of a
causal discrete-time system to the inputs
and,respectively
on
onn ?
onn ?
][ ony
][ny1 ][ny2
][nx2
][nx1
Copyright ? 2001,S,K,Mitra
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Causal System
? Then
for n < N
implies also that
for n < N
? For a causal system,changes in output
samples do not precede changes in the input
samples
][][ 21 nxnx ?
][][ 21 nyny ?
Copyright ? 2001,S,K,Mitra
22
Causal System
? Examples of causal systems,
? Examples of noncausal systems,
][][][][][ 321 4321 ??????? nxnxnxnxny ????
][][][][ 21 210 ????? nxbnxbnxbny
][][ 21 21 ???? nyanya
][][][ nxnyny ??? 1
])[][(][][ 1121 ????? nxnxnxny uuu
])[][(][][ 2131 ????? nxnxnxny uuu
])[][( 1232 ???? nxnx uu
Copyright ? 2001,S,K,Mitra
23
Causal System
? A noncausal system can be implemented as
a causal system by delaying the output by
an appropriate number of samples
? For example a causal implementation of the
factor-of-2 interpolator is given by
])[][(][][ nxnxnxny uuu ????? 21 21
Copyright ? 2001,S,K,Mitra
24
Stable System
? There are various definitions of stability
? We consider here the bounded-input,
bounded-output (BIBO) stability
? If y[n] is the response to an input x[n] and if
for all values of n
then
for all values of n
xBnx ?][
yBny ?][
Copyright ? 2001,S,K,Mitra
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Stable System
? Example - The M-point moving average
filter is BIBO stable,
? For a bounded input we have
?
?
?
??
1
0
1 M
kM
knxny ][][
xBnx ?][
??
?
?
?
?
????
1
0
11
0
1 M
kM
M
kM
knxknxny ][][][
xxM BMB ?? )(1
Copyright ? 2001,S,K,Mitra
26
Passive and Lossless Systems
? A discrete-time system is defined to be
passive if,for every finite-energy input x[n],
the output y[n] has,at most,the same energy,
i.e,
? For a lossless system,the above inequality is
satisfied with an equal sign for every input
??? ??
?
???
?
??? nn
nxny 22 ][][
Copyright ? 2001,S,K,Mitra
27
Passive and Lossless Systems
? Example - Consider the discrete-time
system defined by with N
a positive integer
? Its output energy is given by
? Hence,it is a passive system if and is
a lossless system if
][][ Nnxny ?? ?
2 2 2 2 2[ ] [ ] [ ]
n n n
y n x n N x n??
? ? ?
? ? ? ? ? ? ? ? ?
? ? ?? ? ?
1??
1??
Copyright ? 2001,S,K,Mitra
28
Impulse and Step Responses
? The response of a discrete-time LTI system
to a unit sample sequence {d[n]} is called
the unit sample response or,simply,the
impulse response,and is denoted by {h[n]}
? The response of a discrete-time LTI system
to a unit step sequence {m[n]} is called the
unit step response or,simply,the step
response,and is denoted by {s[n]}
Copyright ? 2001,S,K,Mitra
29
Impulse and Step Responses
LTI [ ] [ ]x n nd? [ ] [ ]y n h n?
? impulse response
? step response
LTI [ ] [ ]x n nm? [ ] [ ]y n s n?
Copyright ? 2001,S,K,Mitra
30
Impulse Response
? Example - The impulse response of the
system
is obtained by setting x[n] = d[n] resulting in
? The impulse response is thus a finite-length
sequence of length 4 given by
][][][][][ 321 4321 ??????? nxnxnxnxny ????
][][][][][ 321 4321 ??????? nnnnnh d?d?d?d?
},,,{]}[{ 4321 ????
?
?nh
Copyright ? 2001,S,K,Mitra
31
Impulse Response
? Example - The impulse response of the
discrete-time accumulator
is obtained by setting x[n] = d[n] resulting
in
?
???
?
n
xny
?
? ][][
][][][ nnh
n
md ?? ?
????
?
Copyright ? 2001,S,K,Mitra
32
Impulse Response
? Example - The impulse response {h[n]} of
the factor-of-2 interpolator
is obtained by setting and is
given by
? The impulse response is thus a finite-length
sequence of length 3,
])[][(][][ 1121 ????? nxnxnxny uuu
])[][(][][ 1121 ????? nnnnh ddd
}.,.{]}[{ 50150
?
?nh
][][ nnx u d?
Copyright ? 2001,S,K,Mitra
33
Time-Domain Characterization
of Discrete-Time LTI Systems
? Input-Output Relationship – A consequence
of the linear,time-invariance property is that
a discrete-time LTI system is completely
characterized by its impulse response
? Therefore,knowing the impulse response,
one can compute the output of the system for
any arbitrary input
Copyright ? 2001,S,K,Mitra
34
? A discrete-time LTI system is denoted as
[]xn []yn
Time-Domain Characterization
of Discrete-Time LTI Systems
[]hn
Copyright ? 2001,S,K,Mitra
35
Time-Domain Characterization
of Discrete-Time LTI Systems
? Let h[n] denote the impulse response of a
discrete-time LTI system
? We compute its output y[n] for the input,
? As the system is linear,we can compute its
outputs for each member of the input
separately and add the individual outputs to
determine y[n]
]5[75.0]2[]1[5.1]2[5.0][ ?d??d??d??d? nnnnnx
Copyright ? 2001,S,K,Mitra
36
Time-Domain Characterization
of Discrete-Time LTI Systems
? Since the system is time-invariant
input output
]2[]2[ ???d nhn
]1[]1[ ???d nhn
]2[]2[ ???d nhn
]5[]5[ ???d nhn
Copyright ? 2001,S,K,Mitra
37
Time-Domain Characterization
of Discrete-Time LTI Systems
? Likewise,as the system is linear
? Hence,because of the linearity property we
get
]5[75.0]5[75.0 ???d nhn
input output
]2[5.0]2[5.0 ???d nhn
]2[]2[ ????d? nhn
]1[5.1]1[5.1 ???d nhn
][.][.][ 151250 ???? nhnhny
][.][ 57502 ???? nhnh
Copyright ? 2001,S,K,Mitra
38
Time-Domain Characterization
of Discrete-Time LTI Systems
? Now,any arbitrary input sequence x[n] can
be expressed as a linear combination of
delayed and advanced unit sample
sequences in the form
? The response of the LTI system to an input
will be
? ?d?
?
???k
knkxnx ][][][
][][ knkx ?d ][][ knhkx ?
Copyright ? 2001,S,K,Mitra
39
Time-Domain Characterization
of Discrete-Time LTI Systems
? Hence,the response y[n] to an input
will be
which can be alternately written as
? ?d?
?
???k
knkxnx ][][][
? ??
?
???k
knhkxny ][][][
?
?
???
??
k
khknxny ][][][
Copyright ? 2001,S,K,Mitra
40
Convolution Sum
? The summation
is called the convolution sum of the
sequences x[n] and h[n] and represented
compactly as
??
?
???
?
???
????
kk
nhknxknhkxny ][][][][][
y[n] = x[n] h[n] *
Copyright ? 2001,S,K,Mitra
41
Convolution Sum
? Properties
– Commutative property,
– Associative property,
– Distributive property,
x[n] h[n] = h[n] x[n] * *
(x[n] h[n]) y[n] = x[n] (h[n] y[n]) * * * *
x[n] (h[n] + y[n]) = x[n] h[n] + x[n] y[n] * * *
Copyright ? 2001,S,K,Mitra
42
Convolution Sum
? Interpretation,
1) Time-reverse h[k] to form
2) Shift to the right by n sampling
periods if n > 0 or shift to the left by n
sampling periods if n < 0 to form
3) Form the product
4) Sum all samples of v[k] to develop the
n-th sample of y[n] of the convolution sum
][ kh ?
][ kh ?
][ knh ?
][][][ knhkxkv ??
Copyright ? 2001,S,K,Mitra
43
Convolution Sum
? Schematic Representation,
? The computation of an output sample using
the convolution sum is simply a sum of
products
? Involves fairly simple operations such as
additions,multiplications,and delays
?nz ][ knh ?][ kh ?
][kx
][kv ][ny?
k
Copyright ? 2001,S,K,Mitra
44
Convolution Using MATLAB
? The M-file conv implements the convolution
sum of two finite-length sequences
? If
then conv(a,b) yields
]31102[a ???
]1-021[b ?
]31513142[ ???
Copyright ? 2001,S,K,Mitra
45
Simple Interconnection
Schemes
? Two simple interconnection schemes are,
– Cascade Connection
– Parallel Connection
Copyright ? 2001,S,K,Mitra
46
Cascade Connection
? Impulse response h[n] of the cascade of two
discrete-time LTI systems with impulse
responses and is given by
][nh1][nh2][nh1 ][nh2?
][][ nhnh 1? ][2][nh1 * ?
][nh1 ][nh2
][nh2 ][][ nnh 1? *
Copyright ? 2001,S,K,Mitra
47
Cascade Connection
? Note,The ordering of the systems in the
cascade has no effect on the overall impulse
response because of the commutative
property of convolution
? A cascade connection of two stable systems
is stable
? A cascade connection of two passive
(lossless) systems is passive (lossless)
Copyright ? 2001,S,K,Mitra
48
Cascade Connection
? An application is in the development of an
inverse system
? If the cascade connection satisfies the
relation
then the LTI system is said to be the
inverse of and vice-versa
][nh1
][nh2
][nh2][1 nh ][nd?*
Copyright ? 2001,S,K,Mitra
49
Cascade Connection
? An application of the inverse system
concept is in the recovery of a signal x[n]
from its distorted version appearing at
the output of a transmission channel
? If the impulse response of the channel is
known,then x[n] can be recovered by
designing an inverse system of the channel
][? nx
][nh2][nh1][nx ][nx
channel inverse system ][nx^
][nh2][nh1 ][nd?*
Copyright ? 2001,S,K,Mitra
50
Cascade Connection
? Example - Consider the discrete-time
accumulator with an impulse response m[n]
? Its inverse system satisfies the condition
? It follows from the above that for
n < 0 and
for
02 ?][nh
1]0[2 ?h
0
0
2 ??
?
n
h
?
?][ 1?n
][nh2][nm ][nd?*
Copyright ? 2001,S,K,Mitra
51
Cascade Connection
? Thus the impulse response of the inverse
system of the discrete-time accumulator is
given by
which is called a backward difference
system
]1[][][2 ?d?d? nnnh
Copyright ? 2001,S,K,Mitra
52
? Impulse response h[n] of the parallel
connection of two discrete-time LTI
systems with impulse responses and
is given by
][nh2
][nh1
? ][][ nhnh 1? ][2][nh1? ?
][nh1
][nh2
][][][ nhnhnh 21 ??
Parallel Connection
Copyright ? 2001,S,K,Mitra
53
Simple Interconnection Schemes
? Consider the discrete-time system where
][nh2
][nh1 ?
?
][nh4
][nh3
],1[5.0][][1 ?d?d? nnnh
],1[25.0][5.0][2 ?d?d? nnnh
],[2][3 nnh d?
][)5.0(2][4 nnh n m??
Copyright ? 2001,S,K,Mitra
54
Simple Interconnection Schemes
? Simplifying the block-diagram we obtain
][nh2
][nh1 ?
][][ 43 nhnh ?
][nh1 ?
])[][(][ 432 nhnhnh ?*
?
Copyright ? 2001,S,K,Mitra
55
Simple Interconnection Schemes
? Overall impulse response h[n] is given by
? Now,
][][][][][ nhnhnhnhnh 42321 ???
])[][(][][][ nhnhnhnhnh 4321 ???*
* *
][2])1[][(][][ 412132 nnnnhnh d?d?d?
]1[][ 21 ?d?d? nn
* *
Copyright ? 2001,S,K,Mitra
56
Simple Interconnection Schemes
? Therefore
]1[)(][)( 1212121 ?m?m?? ? nn nn
]1[)(][)( 2121 ?m?m?? nn nn
][][)( 21 nnn d??d??
][][]1[][]1[][][ 2121 nnnnnnnh d?d??d?d??d?d?
? ?][)(2])1[][(][][ 21412142 nnnnhnh n m??d?d?* *
