Convolution is very useful and powerful concept. It appears quite frequently in DSP discussion. It is begun with a rather twisted definition (folding before shifting), but it then becomes the representation of linear systems, and is linked to the Fourier transform and the z-transform.
As for convolution, correlation is defined for both analog and digital signals. Correlation of two signals measure the degree of their similarity. But correlation of a signal with itself also has meaning and application. The strength of convolution lies in the fact that if applies to signals as well as systems, whereas correlation only applies to signals. Correlation is used in many areas such as radar, geophysics, data communications, and, especially, random processes.
Cross-correlation and auto-correlation
Cross-correlation, or correlation for short, between two discrete-time signals x(n) and v(n), assumed real-valued, is defined as
R
xv
(m)=
∑
n=−∞
∞
x(n)v(n−m)
m=0, ±1, ±2,...
R
xv
(m)=
∑
n=−∞
∞
x(n)v(n−m)
m=0, ±1, ±2,...
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadkfadaWgaaWcbaGaamiEaiaadAhaaeqaaOGaaiikaiaad2gacaGGPaGaeyypa0ZaaabCaeaacaWG4bGaaiikaiaad6gacaGGPaGaamODaiaacIcacaWGUbGaeyOeI0IaamyBaiaacMcaaSqaaiaad6gacqGH9aqpcqGHsislcqGHEisPaeaacqGHEisPa0GaeyyeIuoakiaaywW7caaMf8UaamyBaiabg2da9iaaicdacaGGSaGaaGjbVlabgglaXkaaigdacaGGSaGaaGjbVlabgglaXkaaikdacaGGSaGaaiOlaiaac6cacaGGUaaaaa@5F17@
(1)
or equivalently
R
xv
(m)=
∑
n=−∞
∞
x(n+m)v(n)
m=0, ±1, ±2,...
R
xv
(m)=
∑
n=−∞
∞
x(n+m)v(n)
m=0, ±1, ±2,...
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadkfadaWgaaWcbaGaamiEaiaadAhaaeqaaOGaaiikaiaad2gacaGGPaGaeyypa0ZaaabCaeaacaWG4bGaaiikaiaad6gacqGHRaWkcaWGTbGaaiykaiaadAhacaGGOaGaamOBaiaacMcaaSqaaiaad6gacqGH9aqpcqGHsislcqGHEisPaeaacqGHEisPa0GaeyyeIuoakiaaywW7caaMf8UaamyBaiabg2da9iaaicdacaGGSaGaaGjbVlabgglaXkaaigdacaGGSaGaaGjbVlabgglaXkaaikdacaGGSaGaaiOlaiaac6cacaGGUaaaaa@5F0C@
(2)
Notice that correlation at index n is the summation of the product of one signal and other signal shifted.
When the signals x(n) and v(n) are interchanged, we get
R
vx
(m)=
∑
n=−∞
∞
v(n)x(n−m)
m=0, ±1, ±2,...
R
vx
(m)=
∑
n=−∞
∞
v(n)x(n−m)
m=0, ±1, ±2,...
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadkfadaWgaaWcbaGaamODaiaadIhaaeqaaOGaaiikaiaad2gacaGGPaGaeyypa0ZaaabCaeaacaWG2bGaaiikaiaad6gacaGGPaGaamiEaiaacIcacaWGUbGaeyOeI0IaamyBaiaacMcaaSqaaiaad6gacqGH9aqpcqGHsislcqGHEisPaeaacqGHEisPa0GaeyyeIuoakiaaywW7caaMf8UaamyBaiabg2da9iaaicdacaGGSaGaaGjbVlabgglaXkaaigdacaGGSaGaaGjbVlabgglaXkaaikdacaGGSaGaaiOlaiaac6cacaGGUaaaaa@5F17@
(3)
or equivalently
R
vx
(m)=
∑
n=−∞
∞
v(n+m)x(n)
m=0, ±1, ±2,...
R
vx
(m)=
∑
n=−∞
∞
v(n+m)x(n)
m=0, ±1, ±2,...
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadkfadaWgaaWcbaGaamODaiaadIhaaeqaaOGaaiikaiaad2gacaGGPaGaeyypa0ZaaabCaeaacaWG2bGaaiikaiaad6gacqGHRaWkcaWGTbGaaiykaiaadIhacaGGOaGaamOBaiaacMcaaSqaaiaad6gacqGH9aqpcqGHsislcqGHEisPaeaacqGHEisPa0GaeyyeIuoakiaaywW7caaMf8UaamyBaiabg2da9iaaicdacaGGSaGaaGjbVlabgglaXkaaigdacaGGSaGaaGjbVlabgglaXkaaikdacaGGSaGaaiOlaiaac6cacaGGUaaaaa@5F0C@
(4)
Thus
R
xv
(m)=
R
xv
(−m)
R
xv
(m)=
R
xv
(−m)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadkfadaWgaaWcbaGaamiEaiaadAhaaeqaaOGaaiikaiaad2gacaGGPaGaeyypa0JaamOuamaaBaaaleaacaWG4bGaamODaaqabaGccaGGOaGaeyOeI0IaamyBaiaacMcaaaa@4270@
(5)
This result shows that one correlation is the flipped version (mirror-imaged) of the other, but otherwise contains the same information.
The evalution of correlation is similar to that of convolution expect no signal flipping is need, hence the computing steps are
slide (shift) – multiply – add. The
method of sequence (vector), as for the convolution (
section), is one of the possible ways.
Example 1 Find the cross-correlation of the following signals
x(n)=[
2, 5, 2, 4
]
x(n)=[
2, 5, 2, 4
]
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadIhacaGGOaGaamOBaiaacMcacqGH9aqpdaWadaqaaiaaikdacaGGSaGaaGjbVlaaiwdacaGGSaGaaGjbVlaaikdacaGGSaGaaGjbVlaaisdaaiaawUfacaGLDbaaaaa@45CA@
v(n)=[
2, −3, 1
]
v(n)=[
2, −3, 1
]
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadAhacaGGOaGaamOBaiaacMcacqGH9aqpdaWadaqaaiaaikdacaGGSaGaaGjbVlabgkHiTiaaiodacaGGSaGaaGjbVlaaigdaaiaawUfacaGLDbaaaaa@43B7@
The figures in bold face are samples at origin.
Solution
First we choose the shorter sequence, in this case v(n), to be shifted, and the longer sequence, x(n), to stay stationary. Next the evaluate the correlation at m = 0 (no shifting yet), then the correlation at m = 1, 2, 3 … (shifting v(n) to the right) until v(n) has gone past x(n) completely. Next, we evaluate the correlation at = -1, -2, -3 … (shifting v(n) to the left) until v(n) has gone past x(n) completely. At each value of m, we do the multiplication and summing. The evaluation is arranged as follows. Remember to align the values of x(n) and v(n) at origin at be beginning.
x(n)=2, 5, 2, 4,
m=0: v(n)=0, 2, 3, −1 ⇒ R(0)=8
m=1: v(n−1)=0, 0, 2, −3 ⇒ R(1)=−8
m=2: v(n−2)=0, 0, 0, 2 ⇒ R(2)=8
m=−1: v(n+1)=2, −3, 1, 0 ⇒ R(−1)=−9
m=−2: v(n+2)=−3, 1, 0, 0 ⇒ R(−2)=−1
m=−3: v(n+3)=1, 0, 0, 0 ⇒ R(−3)=2
x(n)=2, 5, 2, 4,
m=0: v(n)=0, 2, 3, −1 ⇒ R(0)=8
m=1: v(n−1)=0, 0, 2, −3 ⇒ R(1)=−8
m=2: v(n−2)=0, 0, 0, 2 ⇒ R(2)=8
m=−1: v(n+1)=2, −3, 1, 0 ⇒ R(−1)=−9
m=−2: v(n+2)=−3, 1, 0, 0 ⇒ R(−2)=−1
m=−3: v(n+3)=1, 0, 0, 0 ⇒ R(−3)=2
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@10BC@
Final result :
R
xv
(m)=[
2, −1, −9, 8, −8, 8
]
R
xv
(m)=[
2, −1, −9, 8, −8, 8
]
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadkfadaWgaaWcbaGaamiEaiaadAhaaeqaaOGaaiikaiaad2gacaGGPaGaeyypa0ZaamWaaeaacaaIYaGaaiilaiaaysW7cqGHsislcaaIXaGaaiilaiaaysW7cqGHsislcaaI5aGaaiilaiaaysW7caaI4aGaaiilaiaaysW7cqGHsislcaaI4aGaaiilaiaaysW7caaI4aaacaGLBbGaayzxaaaaaa@509D@
Example 2 Given two signals
x(n)=
a
n
u(n)
v(n)=
b
n
u(n)
x(n)=
a
n
u(n)
v(n)=
b
n
u(n)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOabaeqabaGaamiEaiaacIcacaWGUbGaaiykaiabg2da9iaadggadaahaaWcbeqaaiaad6gaaaGccaWG1bGaaiikaiaad6gacaGGPaaabaaabaGaamODaiaacIcacaWGUbGaaiykaiabg2da9iaadkgadaahaaWcbeqaaiaad6gaaaGccaWG1bGaaiikaiaad6gacaGGPaaaaaa@492E@
Compute the cross-corelation.
Solution
The cross-correlation is
R
vx
(m)=
∑
n=−∞
∞
[
a
n
u(n)
]
[
b
n−m
u(n−m)
]
=
∑
n=−∞
∞
a
n
b
n−m
u(n)u(n−m)
R
vx
(m)=
∑
n=−∞
∞
[
a
n
u(n)
]
[
b
n−m
u(n−m)
]
=
∑
n=−∞
∞
a
n
b
n−m
u(n)u(n−m)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@72C3@
The summation is divided into two ranges of of m depending on the shifting direction of v(n) with respect to x(n).
- For m < 0, v(n) is shifted to the left of x(n), the summation lower limit is n = 0 :
R
xv
−
(m)=
∑
n=−∞
∞
[
a
n
u(n)
][
b
n−m
u(n−m)
]
=
∑
n=−∞
∞
a
n
b
n−m
u(n)u(n−m)
R
xv
−
(m)=
∑
n=−∞
∞
[
a
n
u(n)
][
b
n−m
u(n−m)
]
=
∑
n=−∞
∞
a
n
b
n−m
u(n)u(n−m)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@73B1@
Where the formula of infinite geometric serics (
Equation) has been used. Since m < 0, we can write
R
xv
−
(m)=
1
1−ab
b
−m
u(m−1)
R
xv
−
(m)=
1
1−ab
b
−m
u(m−1)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadkfadaqhaaWcbaGaamiEaiaadAhaaeaacqGHsislaaGccaGGOaGaamyBaiaacMcacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIXaGaeyOeI0IaamyyaiaadkgaaaGaamOyamaaCaaaleqabaGaeyOeI0IaamyBaaaakiaadwhacaGGOaGaamyBaiabgkHiTiaaigdacaGGPaaaaa@494B@
- For
m≥0
m≥0
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaad2gacqGHLjYScaaIWaaaaa@394F@
, v(n) is shifted to the right, the summation lower limit is n = m :
R
xv
+
(
m
)
=
∑
n
=
m
∞
a
n
b
n
−
m
R
xv
+
(
m
)
=
∑
n
=
m
∞
a
n
b
n
−
m
size 12{R rSub { size 8{ ital "xv"} } rSup { size 8{+{}} } \( m \) = Sum cSub { size 8{n=m} } cSup { size 8{ infinity } } {a rSup { size 8{n} } b rSup { size 8{n - m} } } } {}
Let’s make a change of variable k = n – m to get
R
xv
+
(m)=
∑
k=0
∞
a
k+m
b
k
=
a
m
∑
k=0
∞
(ab)
k
=
1
1−ab
, |
ab
|<0
R
xv
+
(m)=
∑
k=0
∞
a
k+m
b
k
=
a
m
∑
k=0
∞
(ab)
k
=
1
1−ab
, |
ab
|<0
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@734D@
Where the formula finite geometric serics (
Equation) has been used. Since m
size 12{ >= {}} {} 0, we can write
R
xv
+
(
m
)
=
1
1
−
ab
a
m
u
(
m
)
R
xv
+
(
m
)
=
1
1
−
ab
a
m
u
(
m
)
size 12{R rSub { size 8{ ital "xv"} } rSup { size 8{+{}} } \( m \) = { {1} over {1 - ital "ab"} } a rSup { size 8{m} } u \( m \) } {}
On combining the two parts, the overall cross-correlation results
R
xv
(
m
)
=
R
xv
−
(
m
)
+
R
xv
+
(
m
)
=
1
1
−
ab
[
b
−
m
u
(
m
−
1
)
+
a
m
u
(
m
)
]
R
xv
(
m
)
=
R
xv
−
(
m
)
+
R
xv
+
(
m
)
=
1
1
−
ab
[
b
−
m
u
(
m
−
1
)
+
a
m
u
(
m
)
]
size 12{R rSub { size 8{ ital "xv"} } \( m \) =R rSub { size 8{ ital "xv"} } rSup { size 8{ - {}} } \( m \) +R rSub { size 8{ ital "xv"} } rSup { size 8{+{}} } \( m \) = { {1} over {1 - ital "ab"} } \[ b rSup { size 8{ - m} } u \( m - 1 \) +a rSup { size 8{m} } u \( m \) \] } {}
Auto-correlation
Auto-correlation of a signal x(n) is the cross-correlation with itself :
R
xx
(m)=
∑
n=−∞
∞
x(n)x(n−m)
R
xx
(m)=
∑
n=−∞
∞
x(n)x(n−m)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadkfadaWgaaWcbaGaamiEaiaadIhaaeqaaOGaaiikaiaad2gacaGGPaGaeyypa0ZaaabCaeaacaWG4bGaaiikaiaad6gacaGGPaGaamiEaiaacIcacaWGUbGaeyOeI0IaamyBaiaacMcaaSqaaiaad6gacqGH9aqpcqGHsislcqGHEisPaeaacqGHEisPa0GaeyyeIuoaaaa@4CB0@
(6)
or equivalently
R
xx
(m)=
∑
n=−∞
∞
x(n+m)x(n)
R
xx
(m)=
∑
n=−∞
∞
x(n+m)x(n)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadkfadaWgaaWcbaGaamiEaiaadIhaaeqaaOGaaiikaiaad2gacaGGPaGaeyypa0ZaaabCaeaacaWG4bGaaiikaiaad6gacqGHRaWkcaWGTbGaaiykaiaadIhacaGGOaGaamOBaiaacMcaaSqaaiaad6gacqGH9aqpcqGHsislcqGHEisPaeaacqGHEisPa0GaeyyeIuoaaaa@4CA5@
(7)
At m = 0 (no shifting yet) the auto-correlation is maximum because the signal superimposes completely with itself. The correlation decreases as m increases in both directions.
The auto-correlation is an even symmetric function of m :
R
xx
(m)=
R
xx
(−m)
R
xx
(m)=
R
xx
(−m)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadkfadaWgaaWcbaGaamiEaiaadIhaaeqaaOGaaiikaiaad2gacaGGPaGaeyypa0JaamOuamaaBaaaleaacaWG4bGaamiEaaqabaGccaGGOaGaeyOeI0IaamyBaiaacMcaaaa@4274@
(8)
Example 3 Find the expression for the auto-correlation of the signal given in Example 2.8.2
x(n)=anu(n)x(n)=anu(n) size 12{x \( n \) =a rSup { size 8{n} } u \( n \) } {}
Solution
We have
R
xx
(
m
)
=
∑
n
=
−
∞
∞
x
(
n
)
x
(
n
−
m
)
=
∑
n
=
−
∞
∞
a
n
a
n
−
m
u
(
n
)
u
(
n
−
m
)
R
xx
(
m
)
=
∑
n
=
−
∞
∞
x
(
n
)
x
(
n
−
m
)
=
∑
n
=
−
∞
∞
a
n
a
n
−
m
u
(
n
)
u
(
n
−
m
)
size 12{R rSub { size 8{ ital "xx"} } \( m \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {x \( n \) x \( n - m \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {a rSup { size 8{n} } a rSup { size 8{n - m} } } } u \( n \) u \( n - m \) } {}
Since
Rxx(m)Rxx(m) size 12{R rSub { size 8{ ital "xx"} } \( m \) } {} iseven symmetric we need to compute only the
Rxx+(m)Rxx+(m) size 12{R rSub { size 8{ ital "xx"} } rSup { size 8{+{}} } \( m \) } {} for m
size 12{ >= {}} {} 0 then generalize the result for the correlation.
For
m≥0
m≥0
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaad2gacqGHLjYScaaIWaaaaa@394F@
R
xx
+
(
m
)
=
∑
n
=
m
∞
a
n
a
n
−
m
R
xx
+
(
m
)
=
∑
n
=
m
∞
a
n
a
n
−
m
size 12{R rSub { size 8{ ital "xx"} } rSup { size 8{+{}} } \( m \) = Sum cSub { size 8{n=m} } cSup { size 8{ infinity } } {a rSup { size 8{n} } a rSup { size 8{n - m} } } } {}
Make a change of varible k = n – m as in previous example :
Rxx+(m)=∑k=0∞ak+mak=am∑k=0∞a2k=am1−a2Rxx+(m)=∑k=0∞ak+mak=am∑k=0∞a2k=am1−a2 size 12{R rSub { size 8{ ital "xx"} } rSup { size 8{+{}} } \( m \) = Sum cSub { size 8{k=0} } cSup { size 8{ infinity } } {a rSup { size 8{k+m} } a rSup { size 8{k} } =a rSup { size 8{m} } Sum cSub { size 8{k=0} } cSup { size 8{ infinity } } {a rSup { size 8{2k} } = { {a rSup { size 8{m} } } over {1 - a rSup { size 8{2} } } } } } } {},
∣a∣2<1∣a∣2<1 size 12{ lline a rline rSup { size 8{2} } <1} {}
Above result is for
m≥0
m≥0
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaad2gacqGHLjYScaaIWaaaaa@394F@
. Now for all m we just write
∣m∣∣m∣ size 12{ lline m rline } {} for m because of the even symmetry of the auto-correlation. So
Rxx(m)=a∣m∣1−a2Rxx(m)=a∣m∣1−a2 size 12{R rSub { size 8{ ital "xx"} } \( m \) = { {a rSup { size 8{ lline m rline } } } over {1 - a rSup { size 8{2} } } } } {}
