# Connexions

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# DIGITAL CORRELATION

Module by: Nguyen Huu Phuong. E-mail the author

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

(1)

or equivalently

(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

(3)

or equivalently

(4)

Thus

(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.

Final result :

### Example 2

Solution

The cross-correlation is

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 nm u(nm) ] = n= a n b nm u(n)u(nm) R xv (m)= n= [ a n u(n) ][ b nm u(nm) ] = n= a n b nm u(n)u(nm) 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

• For m0 m0 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 1ab ,| ab |<0 R xv + (m)= k=0 a k+m b k = a m k=0 (ab) k = 1 1ab ,| 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 :

(6)

or equivalently

(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 :

(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.

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=0ak+mak=amk=0a2k=am1a2Rxx+(m)=k=0ak+mak=amk=0a2k=am1a2 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} } } } } } } {}, a2<1a2<1 size 12{ lline a rline rSup { size 8{2} } <1} {}

Above result is for m0 m0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaad2gacqGHLjYScaaIWaaaaa@394F@ . Now for all m we just write mm size 12{ lline m rline } {} for m because of the even symmetry of the auto-correlation. So

Rxx(m)=am1a2Rxx(m)=am1a2 size 12{R rSub { size 8{ ital "xx"} } $$m$$ = { {a rSup { size 8{ lline m rline } } } over {1 - a rSup { size 8{2} } } } } {}

## Correlation and data communication

Consider a digital signal x(n) transmitted to the far end of the communication channel. It reaches the receiver n0n0 size 12{n rSub { size 8{0} } } {} samples later, becoming x(n - n 00 size 12{ {} rSub { size 8{0} } } {}), and it is also added with random noise z(n). Thus the total signal at the receiver is

Now let’s look at the cross-correlation betwwen y(n) and x(n) :

(9)

The result shows that the cross-correlation consists of two compoments : The auto-correlation Rxx(mm0)Rxx(mm0) size 12{R rSub { size 8{ ital "xx"} } $$m - m rSub { size 8{0} }$$ } {}of the transmitted signal but shifted in time, and the cross-correlation Rxz(m)Rxz(m) size 12{R"" lSub { size 8{ ital "xz"} } $$m$$ } {} between the transmitted signal x(n) and corrupting noise z(n). The meaning is that Rxx(mm0)Rxx(mm0) size 12{R rSub { size 8{ ital "xx"} } $$m - m rSub { size 8{0} }$$ } {} is usually larger than Rxz(m)Rxz(m) size 12{R"" lSub { size 8{ ital "xz"} } $$m$$ } {} and has peak at m = n 00 size 12{ {} rSub { size 8{0} } } {}, whereas Rxz(m)Rxz(m) size 12{R"" lSub { size 8{ ital "xz"} } $$m$$ } {} is usually smaller due to the random nature of noise and the independence of the signal and noise. Thus by examining Ryx(m)Ryx(m) size 12{R rSub { size 8{ ital "yx"} } $$m$$ } {}we know the delay n0n0 size 12{n rSub { size 8{0} } } {}of receiving signal.

### Example 4

Consider the transmitted signal and corrupting noise as follows x(n)=[ 4,3,1,2,7 ] x(n)=[ 4,3,1,2,7 ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadIhacaGGOaGaamOBaiaacMcacqGH9aqpdaWadaqaaiaaisdacaGGSaGaaGjbVlaaiodacaGGSaGaaGjbVlaaigdacaGGSaGaaGjbVlaaikdacaGGSaGaaGjbVlaaiEdaaiaawUfacaGLDbaaaaa@48C5@ x(n)=[ 0.7,0.5,0,0.8,0.6,0.4 ] x(n)=[ 0.7,0.5,0,0.8,0.6,0.4 ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadIhacaGGOaGaamOBaiaacMcacqGH9aqpdaWadaqaaiaaicdacaGGUaGaaG4naiaacYcacaaMe8UaeyOeI0IaaGimaiaac6cacaaI1aGaaiilaiaaysW7caaIWaGaaiilaiaaysW7cqGHsislcaaIWaGaaiOlaiaaiIdacaGGSaGaaGjbVlabgkHiTiaaicdacaGGUaGaaGOnaiaacYcacaaMe8UaeyOeI0IaaGimaiaac6cacaaI0aaacaGLBbGaayzxaaaaaa@5699@ The noise, generated by a random noise generator programme, has uniform destribution with amplitudes in the interval (-1, 1). The signal received at receiver is y(n)=x(n1)+z(n)n=0,1,2,... y(n)=x(n1)+z(n)n=0,1,2,... MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadMhacaGGOaGaamOBaiaacMcacqGH9aqpcaWG4bGaaiikaiaad6gacqGHsislcaaIXaGaaiykaiabgUcaRiaadQhacaGGOaGaamOBaiaacMcacaaMf8UaaGzbVlaaywW7caWGUbGaeyypa0JaaGimaiaacYcacaaMe8UaaGymaiaacYcacaaMe8UaaGOmaiaacYcacaGGUaGaaiOlaiaac6caaaa@535F@ Find the cross-correlation Ryx(m)Ryx(m) size 12{R rSub { size 8{ ital "yx"} } $$m$$ } {}.

Solution

Without going details of evalution, only the results are mentioned :

• Cross-correlation beween x(n) and z(n) : R zx (m)=[ 16,1.2,1.8,2.6,2.8,1.7,0.8,0.1,2.1 ] R zx (m)=[ 16,1.2,1.8,2.6,2.8,1.7,0.8,0.1,2.1 ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaqaaqaaaOqaaiaadkfadaWgaaWcbaGaamOEaiaadIhaaeqaaOGaaiikaiaad2gacaGGPaGaeyypa0ZaamWaaeaacqGHsislcaaIXaGaaGOnaiaacYcacaaMe8UaaGymaiaac6cacaaIYaGaaiilaiaaysW7cqGHsislcaaIXaGaaiOlaiaaiIdacaGGSaGaaGjbVlabgkHiTiaaikdacaGGUaGaaGOnaiaacYcacaaMe8UaeyOeI0IaaGOmaiaac6cacaaI4aGaaiilaiaaysW7caaIXaGaaiOlaiaaiEdacaGGSaGaaGjbVlabgkHiTiaaicdacaGGUaGaaGioaiaacYcacaaMe8UaeyOeI0IaaGimaiaac6cacaaIXaGaaiilaiaaysW7caaIYaGaaiOlaiaaigdaaiaawUfacaGLDbaaaaa@687C@

The highest value 38.2 of RyyRyy size 12{R rSub { size 8{ ital "yy"} } } {} oceurs at index m = 1 as expected.

## Correlation of periodic signals

For two period signals x(n) and v(n) having the same period of N indices (samples), the cross-correlation and auto-correlation are defined as

(10)

(11)

The two correlations also have a period of N samples.

Now let’s look at an application. The signal y(n) arrving at the receiver consists of the transmitted signal x(n) and adding noise z(n) :

The auto-correlation of the received signal for a duration of M samples, M is much greater than N, is

On replacing the expression of y(n) into above auto-correlation, we obtain

(12)

Because the signal x(n) is periodic with period N, the auto-correlation RxxRxx size 12{R rSub { size 8{ ital "xx"} } } {} is also periodic with peaks at m = 0, N, 2N ... The cross-correlation RxzRxz size 12{R rSub { size 8{ ital "xz"} } } {}(m) are Rzx(m) of the signal and noise are rather small because the signal and noise are uncorrelated. The last term Rzz(m) is the auto-correlation of noise, it has peak at m = 0 and decays fast to zero due to its random nature. Thus it remains RxxRxx size 12{R rSub { size 8{ ital "xx"} } } {} the largest. This feature allows us to detect the periodic signal x(n) even if the adding noise has amplitude comparable to that of the signal or even much higher. This method of correlation has been used to determine the pitch (fundamental frequency) of voice and music buried in noise.

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