ch2-3

Copyright © 2001, S. K. Mitra
1
Stability Condition of an LTI
Discrete-Time System
• BIBO Stability Condition - A discrete-
time is BIBO stable if the output sequence
{y[n]} remains bounded for all bounded
input sequence {x[n]}
• An LTI discrete-time system is BIBO stable
if and only if its impulse response sequence
{h[n]} is absolutely summable, i.e.
∞
<
= ∑
∞
−∞
=
n
n
h ]
[
S

2
• Proof: Assume h[n] is a real sequence
• Since the input sequence x[n] is bounded we
have
• Therefore
∞
<
≤ x
B
n
x ]
[
]
[
]
[
]
[
]
[
]
[ k
n
x
k
h
k
n
x
k
h
n
y
k
k
−
≤
−
= ∑
∑
∞
−∞
=
∞
−∞
=
x
k
x B
k
h
B =
≤ ∑
∞
−∞
=
]
[ S

3
• Thus, S < implies
indicating that y[n] is also bounded
• To prove the converse, assume y[n] is
bounded, i.e.,
• Consider the input given by
∞ ∞
<
≤ y
B
n
y ]
[
y
B
n
y ≤
]
[



=
−
≠
−
−
=
0
0
]
[
if
,
]
[
if
]),
[
sgn(
]
[
n
h
K
n
h
n
h
n
x

4
where sgn(c) = +1 if c > 0 and sgn(c) =
if c < 0 and
• Note: Since , {x[n]} is obviously
bounded
• For this input, y[n] at n = 0 is
• Therefore, implies S <
1
−
1
≤
K
y
B
n
y ≤
]
[ ∞
1
≤
]
[n
x
∑
∞
−∞
=
=
=
k
k
h
k
h
y ]
[
])
[
sgn(
]
[0 S ∞
<
≤ y
B

5
• Example - Consider a causal LTI discrete-
time system with an impulse response
• For this system
• Therefore S < if for which the
system is BIBO stable
• If , the system is not BIBO stable
∞ 1
<
α
1
=
α
]
[
)
(
]
[ n
n
h n
µ
α
=
α
α
µ
α
−
=
=
= ∑
∑
∞
=
∞
−∞
= 1
1
0
n
n
n
n
n]
[
S 1
<
α
if

6
Causality Condition of an LTI
• Let and be two input sequences
with
• The corresponding output samples at
of an LTI system with an impulse response
{h[n]} are then given by
]
[n
x1 ]
[n
x2
]
[
]
[ n
x
n
x 2
1 = o
n
n ≤
for
o
n
n =

7
∑
∑
∞
=
∞
−∞
=
−
=
−
=
0
2
2
2
k
o
k
o
o k
n
x
k
h
k
n
x
k
h
n
y ]
[
]
[
]
[
]
[
]
[
∑
−
−∞
=
−
+
1
2
k
o k
n
x
k
h ]
[
]
[
∑
∑
∞
=
∞
−∞
=
−
=
−
=
0
1
1
1
k
o
k
o
o k
n
x
k
h
k
n
x
k
h
n
y ]
[
]
[
]
[
]
[
]
[
∑
−
−∞
=
−
+
1
1
k
o k
n
x
k
h ]
[
]
[

8
• If the LTI system is also causal, then
• As
• This implies
]
[
]
[ n
x
n
x 2
1 = o
n
n ≤
for
]
[
]
[ o
o n
y
n
y 2
1 =
∑
∑
∞
=
∞
=
−
=
−
0
2
0
1
k
o
k
o k
n
x
k
h
k
n
x
k
h ]
[
]
[
]
[
]
[
∑
∑
−
−∞
=
−
−∞
=
−
=
−
1
2
1
1
k
o
k
o k
n
x
k
h
k
n
x
k
h ]
[
]
[
]
[
]
[

9
• As for the only way
the condition
will hold if both sums are equal to zero,
which is satisfied if
]
[
]
[ n
x
n
x 2
1 ≠ o
n
n >
∑
∑
−
−∞
=
−
−∞
=
−
=
−
1
2
1
1
k
o
k
o k
n
x
k
h
k
n
x
k
h ]
[
]
[
]
[
]
[
0
=
]
[k
h for k < 0

10
• An LTI discrete-time system is causal
if and only if its impulse response {h[n]} is a
causal sequence
• Example - The discrete-time system defined
by
is a causal system as it has a causal impulse
response
]
3
[
]
2
[
]
1
[
]
[
]
[ 4
3
2
1 −
α
+
−
α
+
−
α
+
α
= n
x
n
x
n
x
n
x
n
y
}
{
]}
[
{ 4
3
2
1 α
α
α
α
=
n
h
↑

11
• Example - The discrete-time accumulator
defined by
is a causal system as it has a causal impulse
response given by
]
[
]
[
]
[ n
n
y
n
µ
=
δ
= ∑
−∞
=
l
l
]
[
]
[
]
[ n
n
h
n
µ
=
δ
= ∑
−∞
=
l
l

12
• Example - The factor-of-2 interpolator
defined by
is noncausal as it has a noncausal impulse
response given by
( )
]
1
[
]
1
[
]
[
]
[
2
1 +
+
−
+
= n
x
n
x
n
x
n
y u
u
u
}
5
.
0
1
5
.
0
{
]}
[
{ =
n
h
↑

13
• Note: A noncausal LTI discrete-time system
with a finite-length impulse response can
often be realized as a causal system by
inserting an appropriate amount of delay
• For example, a causal version of the factor-
of-2 interpolator is obtained by delaying the
input by one sample period:
( )
]
[
]
2
[
]
1
[
]
[
2
1 n
x
n
x
n
x
n
y u
u
u +
−
+
−
=

14
Finite-Dimensional LTI
Discrete-Time Systems
• An important subclass of LTI discrete-time
systems is characterized by a linear constant
coefficient difference equation of the form
• x[n] and y[n] are, respectively, the input and
the output of the system
• and are constants characterizing
the system
}
{ k
d }
{ k
p
∑
∑
=
=
−
=
−
M
k
k
N
k
k k
n
x
p
k
n
y
d
0
0
]
[
]
[

15
• The order of the system is given by
max(N,M), which is the order of the difference
equation
• It is possible to implement an LTI system
characterized by a constant coefficient
difference equation as here the computation
involves two finite sums of products

16
• If we assume the system to be causal, then
the output y[n] can be recursively computed
using
provided
• y[n] can be computed for all ,
knowing x[n] and the initial conditions
0
0 ≠
d
o
n
n ≥
]
[
,
.
.
.
],
2
[
],
1
[ N
n
y
n
y
n
y o
o
o −
−
−
]
[
]
[
]
[
1 0
1 0
k
n
x
d
p
k
n
y
d
d
n
y
M
k
k
N
k
k −
+
−
−
= ∑
∑
=
=

17
Classification of LTI Discrete-
Time Systems
Time Systems
Based on Impulse Response Length -
• If the impulse response h[n] is of finite
length, i.e.,
then it is known as a finite impulse
response (FIR) discrete-time system
• The convolution sum description here is
2
1
2
1 ,
and
for
0
]
[ N
N
N
n
N
n
n
h <
>
<
=
∑
=
−
=
2
1
]
[
]
[
]
[
N
N
k
k
n
x
k
h
n
y

18
Time Systems
Time Systems
• The output y[n] of an FIR LTI discrete-time
system can be computed directly from the
convolution sum as it is a finite sum of
products
• Examples of FIR LTI discrete-time systems
are the moving-average system and the
linear interpolators

19
Time Systems
Time Systems
• If the impulse response is of infinite length,
then it is known as an infinite impulse
response (IIR) discrete-time system
• The class of IIR systems we are concerned
with in this course are characterized by
linear constant coefficient difference
equations

20
Time Systems
Time Systems
• Example - The discrete-time accumulator
defined by
is seen to be an IIR system
]
[
]
1
[
]
[ n
x
n
y
n
y +
−
=

21
Time Systems
Time Systems
• Example - The familiar numerical
integration formulas that are used to
numerically solve integrals of the form
can be shown to be characterized by linear
constant coefficient difference equations,
and hence, are examples of IIR systems
∫ τ
τ
=
t
d
x
t
y
0
)
(
)
(

22
Time Systems
Time Systems
• If we divide the interval of integration into n
equal parts of length T, then the previous
integral can be rewritten as
where we have set t = nT and used the
notation
∫
−
τ
τ
+
−
=
nT
T
n
d
x
T
n
y
nT
y
)
1
(
)
(
)
)
1
((
)
(
∫ τ
τ
=
nT
d
x
nT
y
0
)
(
)
(

23
Time Systems
Time Systems
• Using the trapezoidal method we can write
• Hence, a numerical representation of the
definite integral is given by
)}
(
)
)
1
((
{
)
(
2
)
1
(
nT
x
T
n
x
d
x T
nT
T
n
+
−
=
τ
τ
∫
−
)}
(
)
)
1
((
{
)
)
1
((
)
(
2
nT
x
T
n
x
T
n
y
nT
y T
+
−
+
−
=

24
Time Systems
Time Systems
• Let y[n] = y(nT) and x[n] = x(nT)
• Then
reduces to
which is recognized as the difference
equation representation of a first-order IIR
discrete-time system
)}
(
)
)
1
((
{
)
)
1
((
)
(
2
nT
x
T
n
x
T
n
y
nT
y T
+
−
+
−
=
]}
1
[
]
[
{
]
1
[
]
[
2
−
+
+
−
= n
x
n
x
n
y
n
y T

25
Time Systems
Time Systems
Based on the Output Calculation Process
• Nonrecursive System - Here the output can
be calculated sequentially, knowing only
the present and past input samples
• Recursive System - Here the output
computation involves past output samples in
addition to the present and past input
samples

26
Time Systems
Time Systems
Based on the Coefficients -
• Real Discrete-Time System - The impulse
response samples are real valued
• Complex Discrete-Time System - The
impulse response samples are complex
valued

27
Correlation of Signals
• There are applications where it is necessary
to compare one reference signal with one or
more signals to determine the similarity
between the pair and to determine additional
information based on the similarity

28
• For example, in digital communications, a
set of data symbols are represented by a set
of unique discrete-time sequences
• If one of these sequences has been
transmitted, the receiver has to determine
which particular sequence has been received
by comparing the received signal with every
member of possible sequences from the set

29
• Similarly, in radar and sonar applications,
the received signal reflected from the target
is a delayed version of the transmitted
signal and by measuring the delay, one can
determine the location of the target
• The detection problem gets more
complicated in practice, as often the
received signal is corrupted by additive
ransom noise

30
Definitions
• A measure of similarity between a pair of
energy signals, x[n] and y[n], is given by the
cross-correlation sequence defined by
• The parameter called lag, indicates the
time-shift between the pair of signals
l
]
[l
xy
r
...
,
,
,
],
[
]
[
]
[ 2
1
0 ±
±
=
−
= ∑
∞
−∞
=
l
l
l
n
xy n
y
n
x
r

31
• y[n] is said to be shifted by samples to the
right with respect to the reference sequence
x[n] for positive values of , and shifted by
samples to the left for negative values of
• The ordering of the subscripts xy in the
definition of specifies that x[n] is the
reference sequence which remains fixed in
time while y[n] is being shifted with respect
to x[n]
]
[l
xy
r
l
l
l

32
• If y[n] is made the reference signal and shift
x[n] with respect to y[n], then the
corresponding cross-correlation sequence is
given by
• Thus, is obtained by time-reversing
∑∞
−∞
= −
= n
yx n
x
n
y
r ]
[
]
[
]
[ l
l
]
[
]
[
]
[ l
l −
=
+
= ∑∞
−∞
= xy
m r
m
x
m
y
]
[l
yx
r
]
[l
xy
r

33
• The autocorrelation sequence of x[n] is
given by
obtained by setting y[n] = x[n] in the
definition of the cross-correlation sequence
• Note: , the energy
of the signal x[n]
∑∞
−∞
= −
= n
xx n
x
n
x
r ]
[
]
[
]
[ l
l
]
[l
xy
r
x
n
xx n
x
r E
∑∞
−∞
= =
= ]
[
]
[ 2
0

34
• From the relation it follows
that implying that is
an even function for real x[n]
• An examination of
reveals that the expression for the cross-
correlation looks quite similar to that of the
linear convolution
]
[
]
[ l
l −
= xy
yx r
r
]
[
]
[ l
l −
= xx
xx r
r ]
[l
xx
r
∑∞
−∞
= −
= n
xy n
y
n
x
r ]
[
]
[
]
[ l
l

35
• This similarity is much clearer if we rewrite
the expression for the cross-correlation as
• The cross-correlation of y[n] with the
reference signal x[n] can be computed by
processing x[n] with an LTI discrete-time
system of impulse response ]
[ n
y −
]
[
]
[
)]
(
[
]
[
]
[ l
l
l
l −
=
−
−
= ∑∞
−∞
= y
x
n
y
n
x
r n
xy *
]
[ n
y −
]
[n
x ]
[n
rxy

36
• Likewise, the autocorrelation of x[n] can be
computed by processing x[n] with an LTI
discrete-time system of impulse response
]
[ n
x −
]
[n
x ]
[n
rxx
]
[ n
x −

37
Properties of
Properties of Autocorrelation
Autocorrelation and
and
Cross-correlation Sequences
• Consider two finite-energy sequences x[n]
and y[n]
• The energy of the combined sequence
is also finite and
nonnegative, i.e.,
]
[
]
[ l
−
+ n
y
n
x
a
∑
∑ ∞
−∞
=
∞
−∞
= =
−
+ n
n n
x
a
n
y
n
x
a ]
[
])
[
]
[
( 2
2
2
l
0
2 2
≥
−
+
−
+ ∑
∑ ∞
−∞
=
∞
−∞
= n
n n
y
n
y
n
x
a ]
[
]
[
]
[ l
l

38
Properties of
Autocorrelation and
and
• Thus
where and
• We can rewrite the equation on the previous
slide as
for any finite value of a
0
0
2
0
2
≥
+
+ ]
[
]
[
]
[ yy
xy
xx r
r
a
r
a l
0
0 >
= x
xx
r E
]
[ 0
0 >
= y
yy
r E
]
[
[ ] 0
1
0
0
1 ≥











 a
r
r
r
r
a
yy
xy
xy
xx
]
[
]
[
]
[
]
[
l
l

39
Properties of
Autocorrelation and
and
• Or, in other words, the matrix
is positive semidefinite
•
or, equivalently,






]
[
]
[
]
[
]
[
0
0
yy
xy
xy
xx
r
r
r
r
l
l
0
0
0 2
≥
− ]
[
]
[
]
[ l
xy
yy
xx r
r
r
y
x
yy
xx
xy r
r
r E
E
=
≤ ]
[
]
[
|
]
[
| 0
0
l

40
Properties of
Autocorrelation and
and
• The last inequality on the previous slide
provides an upper bound for the cross-
correlation samples
• If we set y[n] = x[n], then the inequality
reduces to
x
xx
xy r
r E
=
≤ ]
[
|
]
[
| 0
l

41
Properties of
Autocorrelation and
and
• Thus, at zero lag ( ), the sample value
of the autocorrelation sequence has its
maximum value
• Now consider the case
where N is an integer and b > 0 is an
arbitrary number
• In this case
0
=
l
x
y b E
E 2
=
]
[
]
[ N
n
x
b
n
y −
±
=

42
Properties of
Autocorrelation and
and
• Therefore
• Using the above result in
we get
x
x
y
x b
b E
E
E
E =
= 2
2
y
x
yy
xx
xy r
r
r E
E
=
≤ ]
[
]
[
|
]
[
| 0
0
l
]
[
]
[
]
[ 0
0 xx
xy
xx r
b
r
r
b ≤
≤
− l

43
Correlation Computation
Using MATLAB
Using MATLAB
• The cross-correlation and autocorrelation
sequences can easily be computed using
MATLAB
• Example - Consider the two finite-length
sequences
[ ]
2
4
4
1
2
1
2
3
1 −
−
=
]
[n
x
[ ]
3
2
1
4
1
2 −
−
=
]
[n
y

44
Using MATLAB
Using MATLAB
• The cross-correlation sequence
computed using Program 2_7 of text is
plotted below
]
[n
rxy
-4 -2 0 2 4 6 8
-10
0
10
20
30
Lag index
Amplitude

45
Using MATLAB
Using MATLAB
• The autocorrelation sequence
computed using Program 2_7 is shown below
• Note: At zero lag, is the maximum
]
[l
xx
r
]
[0
xx
r
-5 0 5
-20
0
20
40
60
Lag index
Amplitude

46
Using MATLAB
Using MATLAB
• The plot below shows the cross-correlation
of x[n] and for N = 4
• Note: The peak of the cross-correlation is
precisely the value of the delay N
]
[
]
[ N
n
x
n
y −
=
-10 -5 0 5
-20
0
20
40
60
Lag index
Amplitude

47
Using MATLAB
Using MATLAB
• The plot below shows the autocorrelation of
x[n] corrupted with an additive random
noise generated using the function randn
• Note: The autocorrelation still exhibits a
peak at zero lag
-5 0 5
0
20
40
60
80
Lag index
Amplitude

48
Using MATLAB
Using MATLAB
• The autocorrelation and the cross-
correlation can also be computed using the
function xcorr
• However, the correlation sequences
generated using this function are the time-
reversed version of those generated using
Programs 2_7 and 2_8

49
Normalized Forms of
Normalized Forms of
Correlation
Correlation
• Normalized forms of autocorrelation and
cross-correlation are given by
• They are often used for convenience in
comparing and displaying
• Note: and
independent of the range of values of x[n]
and y[n]
]
[
]
[
]
[
]
[
,
]
[
]
[
]
[
0
0
0 yy
xx
xy
xy
xx
xx
xx
r
r
r
r
r l
l
l
l =
= ρ
ρ
1
≤
|
]
[
| l
xx
ρ 1
≤
|
]
[
| l
xy
ρ

50
Correlation Computation for
Power Signals
Power Signals
• The cross-correlation sequence for a pair of
power signals, x[n] and y[n], is defined as
• The autocorrelation sequence of a power
signal x[n] is given by
∑
−
=
∞
→
−
+
=
K
K
n
K
xy n
y
n
x
K
r ]
[
]
[
lim
]
[ l
l
1
2
1
∑
−
=
∞
→
−
+
=
K
K
n
K
xx n
x
n
x
K
r ]
[
]
[
lim
]
[ l
l
1
2
1

51
Periodic Signals
Periodic Signals
• The cross-correlation sequence for a pair of
periodic signals of period N, and ,
is defined as
• The autocorrelation sequence of a periodic
signal of period N is given by
]
[n
x
~
]
[n
y
~
]
[n
x
~
∑ −
= −
= 1
0
1 N
n
N
xy n
y
n
x
r ]
[
]
[
]
[ l
l ~ ~
~~
∑ −
= −
= 1
0
1 N
n
N
xx n
x
n
x
r ]
[
]
[
]
[ l
l ~ ~
~~

52
Periodic Signals
Periodic Signals
• Note: Both and are also
periodic signals with a period N
• The periodicity property of the
autocorrelation sequence can be exploited to
determine the period of a periodic signal
that may have been corrupted by an additive
random disturbance
]
[l
xy
r~~ ]
[l
xx
r~~

53
Periodic Signals
Periodic Signals
• Let be a periodic signal corrupted by
the random noise d[n] resulting in the signal
which is observed for where
]
[n
x
~
]
[
]
[
]
[ n
d
n
x
n
w +
= ~
1
0 −
≤
≤ M
n
N
M >
>

54
Periodic Signals
Periodic Signals
• The autocorrelation of w[n] is given by
∑ −
= −
= 1
0
1
]
[
]
[
]
[ M
n
M
ww n
w
n
w
r l
l
∑ −
= −
+
−
+
= 1
0
1
])
[
]
[
])(
[
]
[
(
M
n
M
n
d
n
x
n
d
n
x l
l
~
~
∑
∑ −
=
−
= −
+
−
= 1
0
1
1
0
1
]
[
]
[
]
[
]
[ M
n
M
M
n
M
n
d
n
d
n
x
n
x l
l
∑
∑ −
=
−
= −
+
−
+ 1
0
1
1
0
1
]
[
]
[
]
[
]
[ M
n
M
M
n
M
n
x
n
d
n
d
n
x l
l
~ ~
~ ~
]
[
]
[
]
[
]
[ l
l
l
l dx
xd
dd
xx r
r
r
r +
+
+
= ~
~
~
~

55
Periodic Signals
Periodic Signals
• In the last equation on the previous slide,
is a periodic sequence with a period N and
hence will have peaks at
with the same amplitudes as approaches M
• As and d[n] are not correlated, samples
of cross-correlation sequences and
are likely to be very small relative to the
amplitudes of
.
.
.
,
2
,
,
0 N
N
=
l
l
]
[l
xx
r ~
~
]
[n
x
~
]
[l
xx
r ~
~
]
[l
xd
r~ ~ ]
[l
dx
r

56
Periodic Signals
Periodic Signals
• The autocorrelation of d[n] will show
a peak at = 0 with other samples having
rapidly decreasing amplitudes with
increasing values of
• Hence, peaks of for > 0 are
essentially due to the peaks of and can
be used to determine whether is a
periodic sequence and also its period N if
the peaks occur at periodic intervals
l
l
|
|l
]
[l
ww
r
]
[l
xx
r ~
~
]
[n
x
~
]
[l
dd
r

57
Correlation Computation of a
Periodic Signal Using MATLAB
• Example - We determine the period of the
sinusoidal sequence ,
corrupted by an additive
uniformly distributed random noise of
amplitude in the range
• Using Program 2_8 of text we arrive at the
plot of shown on the next slide
)
25
.
0
cos(
]
[ n
n
x =
95
0 ≤
≤ n
]
5
.
0
,
5
.
0
[−
]
[l
ww
r

58
• As can be seen from the plot given above,
there is a strong peak at zero lag
• However, there are distinct peaks at lags that
are multiples of 8 indicating the period of the
sinusoidal sequence to be 8 as expected
-20 -10 0 10 20
-60
-40
-20
0
20
40
60
Lag index
Amplitude

59
• Figure below shows the plot of
• As can be seen shows a very strong
peak at only zero lag
-20 -10 0 10 20
-2
0
2
4
6
8
Lag index
Amplitude
]
[l
dd
r
]
[l
dd
r

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