Values of
x
p
[
n
] and
h
p
[
n
] outside the range (0,
N
 1)
are generated by periodic extension.
One way to visualize the process is to line up
x
[
k
]
clockwise around a circle &
h
[
k
]
counterclockwise
(folded).
51
Example 2.7
Find the periodic convolution of
and
with the period of
N
= 3 using the
cyclic method
3}
2,
,
1
{
n
x
p
2}
0,
,
1
{
n
h
p
52
Solution
Rotate outer
sequence
(folded
h
)
clockwise
Rotate outer
sequence
(folded
h
)
clockwise
5}
8,
,
5
{
n
y
53
Hardware implementation of convolution
Need memory, adder and multiplier
Perform operation of addition, multiplication and shifting
54
55
Discrete Correlation
Correlation is a measure of similarity between two
signals and is found using a process similar to
convolution.
Correlation is the convolution of one signal with a
folded version of the other.
The discrete crosscorrelation (denoted
) of
x
[
n
] and
h
[
n
] is defined by:
k
h
n
k
x
n
k
h
k
x
n
h
n
x
n
r
k
k
xh
k
x
n
k
h
n
k
x
k
h
n
x
n
h
n
r
k
k
hx
56
To find
r
xh
[
n
], the last element of
h
[
n
] is lined up with
the first element of
x
[
n
] & start shifting
h
[
n
] past
x
[
n
].
The pointwise product of the overlapping values are
summed up to generate the correlation.
This is equivalent to performing the convolution of
x
[
n
]
& the
folded
signal
h
[
n
]
.
The starting index of the correlation equals the sum of
the starting indices of
x
[
n
] and
h
[
n
]
.
Similarly,
r
hx
[
n
] equals the convolution of
x
[
n
] &
h
[
n
],
& its starting index equals the sum of the starting
indices of
x
[
n
] &
h
[
n
]
.
57
However,
r
xh
[
n
] does not equal to
r
hx
[
n
]
.
The two are folded versions of each other & related by
r
xh
[
n
] =
r
hx
[
n
]
.
Some equations need to be remembered:
Correlation length
:
N
x
+
N
h
 1
Correlation sum
:
n
h
n
x
n
h
n
x
n
r
xh
n
x
n
h
n
x
n
h
n
r
hx
n
h
n
x
n
r
58
Autocorrelation
The correlation
r
xx
[
n
] of a signal
x
[
n
] with itself
is called the
autocorrelation.
It is an even symmetric function
(
r
xx
[
n
] =
r
xx
[
n
])
with a maximum at
n
= 0 and satisfies the
inequality .
Correlation is an effective method of detecting
signals buried in noise.
Noise is essentially uncorrelated with the signal
59
It means that if we correlate a noisy signal with
itself, the correlation will be due only to the
signal (if present) and will exhibit a sharp peak
at
n
= 0.
The autocorrelation is always even symmetric
with a maximum at the origin.
Some equations need to be remembered:
n
x
n
x
n
x
n
x
n
r
xx
n
r
n
r
xx
xx
0
xx
xx
r
n
r
60
Question
61
Answer
62
Answer
63
Example 2.8
Given
x
[
n
] =
a
n
u
[
n
], 
a
 < 1. Find
r
xx
[
n
] for
n
≥ 0
64
Solution
Since
x
[
k

n
] =
a
kn
u
[
k

n
] starts at
k
=
n
, then
Since autocorrelation is an even symmetric function,
we have
n
k
m
m
n
m
n
m
n
m
n
k
k
k
xx
a
a
a
a
a
a
a
a
n
k
x
k
x
n
r
0
0
2
2
1
1
2
1
a
a
n
r
n
xx
65
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No man an ever reached to excellence in any
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 Spring '19
 Digital Signal Processing, Convolution Properties, y n