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Asked: January 17, 20232023-01-17T21:57:33+05:30 2023-01-17T21:57:33+05:30In: 5G_PRACH

Correlation of 2 signals

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How do we calculate the correlation of 2 discrete signals (real or complex)?

What is the formula? Can you explain with an example (take discrete signals of length 10)?

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Sourav · Answer 1 · 2023-01-23T15:04:23+05:30

Sourav

2023-01-23T15:04:23+05:30Added an answer on January 23, 2023 at 3:04 pm

This answer was edited.

Correlation: Correlation is the measure of similarities between two signals.
If we compare between 2 same signals (main signal and it’s time delayed version), it is known as auto-correlation.
If we compare between 2 different types of signals, it is known as cross-correlation.

Auto-correlation

$R_{xx}(m)=\Sigma_{-\infty}^{\infty}x[n]x^*[n-m]$

Cross-correlation

$R_{xy}(m)=\Sigma_{-\infty}^{\infty}x[n]y^*[n-m]$

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VINODKUMAR · Answer 2 · 2023-01-23T17:08:01+05:30

VINODKUMAR

2023-01-23T17:08:01+05:30Added an answer on January 23, 2023 at 5:08 pm

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Correlation describes the mutual relationship which exists between the two or more things.The same definition which holds good in the case of signals.Correlation between signals indicates the measure up to which the given signal resembles the another signal.

Auto correlation of discrete time signal is expressed as

$Rxx(m)=\sum_{n=-\infty}^{\infty} x(n) x^*(n-m)$

and cross correlation of discrete time signals is expressed as

$Rxy(m)=\sum_{n=-\infty}^{\infty} x(n) y^*(n-m)$

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VINODKUMAR
2023-01-23T20:46:38+05:30Replied to answer on January 23, 2023 at 8:46 pm
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$X_n =[\, 1,1,0,2,1,0,1,1,1,2]\,$

$Y_n =[\,1,1,2,1,0,1,1,1,2,1]\,$

Cross correlation of two signals is

$R_x_y =[\,1,3,3,4,7,5,5,9,8,10,13,8,7,6,4,6,6,3,2]\,$
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parv_chandola · Answer 3 · 2023-01-23T17:17:04+05:30

parv_chandola

2023-01-23T17:17:04+05:30Added an answer on January 23, 2023 at 5:17 pm

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It is a measure of similarity between two signals or signals with itself.

Auto-correlation of discrete time signal is expressed as

$R_{xx}[k] = \sum_{m=-\infty}^{\infty} x[m]x^{*}[m-k]$

and cross-correlation of discrete time signals is expressed as

$R_{xy}[k] = \sum_{m=-\infty}^{\infty} x[m]y^{*}[m-k]$

Example: Given

$x_{1}[n] = [1,0,0,1,1,1,0,0,0,1]$

$x_{2}[n] = [1,0,1,1,1,1,1,0,0,1]$

The cross correlation yields the output.

$y_{1}[n] = [1,0,0,2,2,2,2,3,3,5,3,2,3,2,2,1,1,0,1]$

which can also be shown according to the plot generated given below.

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KIRAN · Answer 4 · 2023-01-23T19:34:48+05:30

KIRAN

2023-01-23T19:34:48+05:30Added an answer on January 23, 2023 at 7:34 pm

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Correlation:

correlation is the measure of similarity between a time shifted signal and its original signal.

We will calculate the correlation of two discrete signals using the formula :

$R_{XY}(n) = \sum_{n=-\infty}^{\infty}X(k)Y^*(n-k)$

Let us consider an example with 2 discrete sequences as :

$x(n) = [ 1,2,3,4,5,5,2,6,1,4 ]$

$y(n) = [2,5,5,2,6,1,2,1,4,5]$

after correlating the above sequence we get result as ;

$z(n) = [5,14,24,36,49,63,60,84,91,104,110,85,107,67,76,47,37,22,8]$

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nagajayanth · Answer 5 · 2023-01-23T19:40:41+05:30

Correlation: Correlation is defined as measure of similarity or relationship between two signals.

Autocorrelation: The correlation between the two same discrete sequences.

Auto correlation of discrete time signal is expressed as

$R_{xx}[m]=\sum_{n=-\infty}^{\infty} x[n] x^*[n-m]$

Cross correlation:The correlation between two different discrete sequences.

cross-correlation of discrete time signals is expressed as

$R_{xy}[k] = \sum_{m=-\infty}^{\infty} x[m]y^*[m-k]$

Example:

$x[n] = [1 \ 0 \ 0 \ 1 \ 0 \ 1 \ 0 \ 0 \ 1 \ 0 ]$

$y[n] = [ 0 \ 1 \ 1 \ 0 \ 1 \ 0 \ 1 \ 0 \ 0 \ 1]$

The cross correlation of the above sequences gives the output as follows:

$z[n] = [ 1 \ 0 \ 0 \ 2 \ 0 \ 2 \ 1 \ 1 \ 4 \ 0 \ 2 \ 2 \ 1 \ 2 \ 0 \ 1 \ 1 \ 0 \ 0 ]$

Vamsi · Answer 6 · 2023-01-23T19:46:16+05:30

Vamsi

2023-01-23T19:46:16+05:30Added an answer on January 23, 2023 at 7:46 pm

This answer was edited.

The correlation of two signals is the convolution between one signal with the functional inverse version of the other signal.

If the two signals are the same then it is autocorrelation otherwise it is cross-correlation.

$R_{xy}(l)=\sum_{n=-\infty}^{\infty}x(n)y^*(n-l)$

Example: let us take the two discrete sequences as

$x_1[n]={[1,2,2,1,1,2,2,1,2,0]}$

$x_2[n]={[1,2,0,1,2,0,1,2,0,1]}$

The output of the correlation of these two signals is

$y[n]={[1,2,4,6,7,8,10,12,14,12,16,12,7,10,8,4,5,2,0]}$

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Akhil · Answer 7 · 2023-01-23T20:21:18+05:30

Correlation is a mathematical tool that measures the amount of similarity present between two signals. In communications, it is widely used at the receiver side to detect the required signals and discard the unwanted signals.

If correlation is performed in between same signals, it is termed as “Autocorrelation”. It are calculated as follows:

$R_{xx}{k} = \sum x[m]x^{*}[m-k]$

If correlation is performed between two different sequences, it is termed as “Cross Correlation”. It are calculated as follows:

$R_{xy}{k} = \sum x[m]y^{*}[m-k]$

As an example, consider the two sequences each of length 10

$x_1(n) &= {{1,2,1,0,-1,1,2,3,3,2}}$

$x_2(n) &= {{0,-1,1,4,-1,1,0,3,0,1}}$

Then the autocorrelation of the first sequence with itself is

$R_xx(n) &= {{2,7,11,11,6,2,3,13,26,34,26,13,3,2,6,11,11,7,2}}$

Then the crosscorrelation of the two sequences is

$R_xy(n) &= {{1,2,4,6,3,2,2,14,13,12,5,10,10,12,11,18,-1,-2,0}}$

Pavan · Answer 8 · 2023-01-23T20:26:38+05:30

Correlation is the measure of similarity between two signals.

Correlation between two discrete signals x[n] and y[n] is denoted by Rxy[k]. It is calculated by using following formula.

$Rxy(k)=\sum_{n=-\infty}^{\infty} x(n) y^*(n-k)$

If the correlation is between same signal, it is called Auto-Correlation.

If correlation is between two different signals, it is called Cross-Correlation

Example:

x[n] = { 1,0,2,0,1,1,3,2,1,2}

y[n] = {2,0,1,2,3,4,3,2,1,2}

Cross correlation: {2 , 1, 6, 5, 10, 12, 19, 19, 21, 29, 27, 31, 23, 20, 12, 11, 6, 2, 4}

Aditya Nirmal · Answer 9 · 2023-01-23T20:30:02+05:30

When we have two distinct signals a[n] and b[n] , we calculate correlation of them using

$Out[k]=\sum_{n=-\infty}^{\infty}a[n]b[k-n]$

Correlation is of two types:

Auto Correlation
Cross Correlation

For real and discrete signals a[n] and b[n] :

Auto correlation is

$R_{aa}[n]=\Sigma_{-\infty}^{\infty}a[n]a[k-n]$

$R_{bb}[n]=\Sigma_{-\infty}^{\infty}b[n]b[k-n]$

Cross correlation is

$R_{ab}[n]=\Sigma_{-\infty}^{\infty}a[n]b[k-n]$

$R_{ba}[n]=\Sigma_{-\infty}^{\infty}b[n]a[k-n]$

For complex and discrete signals a[n] and b[n] :

Auto correlation is

$R_{aa}[n]=\Sigma_{-\infty}^{\infty}a[n]a^*[k-n]$

$R_{bb}[n]=\Sigma_{-\infty}^{\infty}b[n]b^*[k-n]$

Cross correlation is

$R_{ab}[n]=\Sigma_{-\infty}^{\infty}a[n]b^*[k-n]$

$R_{ba}[n]=\Sigma_{-\infty}^{\infty}b[n]a^*[k-n]$

Example :

$a[n] = [1,1,0,2,3,0,1,1,2,3]$

$b[n] = [0,0,1,1,2,2,3,3,0,0]$

The cross correlation of a[n] and b[n] is

$z[n] = [ 0 \ 0 \ 1 \ 2 \ 3 \ 6 \ 10\ 13\ 14 \ 14 \ 20 \ 18 \ 12 \ 16 \ 15 \ 15 \ 9 \ 0 \ 0 ]$

tarun_gupta · Answer 10 · 2023-01-24T00:46:48+05:30

Correlation refers to the similarity or relationship between two signals. It measures how similar two signals are in terms of shape or pattern.

Cross-correlation: This type of correlation measures the similarity between two signals as a function of the time lag between them. It is commonly used in signal processing to detect patterns or features in a signal or to filter the unwanted noise

$R_{xy}(k)=\Sigma_{-\infty}^{\infty}x(n)y^*(n-k)$

Where Rxy(k) is the cross-correlation at lag k, x(n) is the value of the first signal at time n, y(n) is the value of the second signal at time n, and k is the time lag.

Auto-correlation: This type of correlation measures the similarity between a signal and a time-shifted version of itself. It is used to detect periodic patterns in a signal or to find statistical properties of the signal

$R_{xx}(k)=\Sigma_{-\infty}^{\infty}x(n)x^*(n-k)$

Where Rxx(k) is the auto-correlation at lag k, x(n) is the value of the signal at time n, and k is the time lag.

Here i took two discrete signal each of length 10 element

$x[n] = [1,2,3,1,2,3,1,2,3,1]$

$y[n] = [1,0,0,1,1,1,0,0,1,1]$

The cross-correlation of x[n] and y[n] is

$z[n] = [ 1 \ 3 \ 5 \ 4 \ 4 \ 8 \ 10\ 9\ 11 \ 10 \ 8\ 9 \ 7 \ 7 \6 \ 1 \ 2 \ 3 \ 0 ]$

vasanthi123 · Answer 11 · 2023-01-24T15:00:29+05:30

correlation describe the relationship between two signals.it is measure of similarities and degree of similarities between two signals.

it has two type of correlation

1.auto correlation

2.cross correlation

auto correlation

the autocorrelation function is defined as the measure of similarity between a signal and its time delayed version.The auto correlation between two same signals or waveforms.

cross correlation

the cross correlation function is defined as the measure of similarity between a signal and its time delayed version.The cross correlation between two different signals or waveforms.

Example

In this example i have taken two sequence x1(n) and X2(n) with length of 10 elements.

x(n)=[1,1,1,1,1,2,2,2,2,2]

Y(n)=[3,3,3,3,3,4,4,4,4,4]

take x(n) sequence itself and invert y(n) sequence and perform convolution between two sequence it results correlation.

If the same sequence perform this operation it results auto correlation.

Rxx(n) =[2,4,6,8,10,13,16,19,22,25,22,19,16,13,10,8,6,4,2]

If the different sequence perform this operation it results cross correlation.

Rxy(n) = [4,8,12,16,20,27,34,41,48,55,50,45,40,35,30,24,18,12,6]

Swaraj · Answer 12 · 2023-01-24T20:03:55+05:30

Given two discrete-time real signals (sequences) x[n] and y[n] .

The autocorrelation and crosscorrelation functions are respectively defined by

Autocorrelation of x[n]

$begin{equation*} R_{text {xx}}[k]=sum_{m=-infty }^{infty } text x[m].text x[m-k] end{equation*}$

Autocorrelation of y[n]

$begin{equation*} R_{text {yy}}[k]=sum_{m=-infty }^{infty } text y[m].text y[m-k] end{equation*}$

Crosscorrelation of x[n] and y[n]

$begin{equation*} R_{text {xy}}[k]=sum_{m=-infty }^{infty } text x[m].text y[m-k] end{equation*}$

Crosscorrelation of y[n] and x[n]

$begin{equation*} R_{text {yx}}[k]=sum_{m=-infty }^{infty } text y[m].text x[m-k] end{equation*}$

Let us understand using an example;

Example:

Find the Correlation of the given below given two sequences

x[n]={1,2,1,5,4,3,1,2,6,8}

p[n]={2,2,4,5,3,6,1,1,2,2}

We can use the above formula or we can also write MATLAB code for correlation.

MATLAB code:

x=[1,2,1,5,4,3,1,2,6,8];

subplot(311);

stem(x)

title(“x[n]”);

p=[2,2,4,5,3,6,1,1,2,2];

subplot(312);

stem(p);

title(“p[n]”);

subplot(313);

y=xcorr(x,p);

stem(y);

title(“y[n]”);

y[n]={2,6,7,15,27,35,34,60,74,96,85,72,85,94,70,70,48,28,16}

The cross correlation plot of x[n] and p[n] is attached below

M. Akhil · Answer 13 · 2023-01-25T01:23:21+05:30

Correlation :A signal operation similar to signal convolution, but with completely different physical meaning, is signal correlation. The signal correlation operation can be performed either with one signal (autocorrelation) or between two different signals (cross correlation). Physically, signal autocorrelation indicates how the signal energy (power) is distributed within the signal, and as such is used to measure the signal power. Applications of signal autocorrelation are in radar, sonar, satellite, and wireless communications systems. There are also many applications of signal cross correlation in signal processing systems, especially when the signal is corrupted by another undesirable signal (noise) so that the signal estimation (detection) from a noisy signal has to be performed. Signal can be also considered as a measure of similarity of two signals.

Formula :

Given two discrete-time real signals (sequences) and . The autocorrelation and cross correlation functions are respectively defined by

$begin{equation*} R_{text {xx}}[k]=sum_{m=-infty }^{infty } text x[m].text x[m-k] end{equation*}$

is the auto correlation of x[n]

$begin{equation*} R_{text {yy}}[k]=sum_{m=-infty }^{infty } text y[m].text y[m-k] end{equation*}$

is the auto correlation of y[n]

$begin{equation*} R_{text {xy}}[k]=sum_{m=-infty }^{infty } text x[m].text y[m-k] end{equation*}$

is the cross correlation of x[n] and y[n]

$begin{equation*} R_{text {yx}}[k]=sum_{m=-infty }^{infty } text y[m].text x[m-k] end{equation*}$

is the cross correlation of y[n] and x[n]

Example :

let x[n] and y[n] are two discrete time signals

x=[1,2,1,1,2,2,1,2,1,1]

y=[2,1,2,1,2,1,1,1,2,2]

R is the correlation output when both the signals are correlated and is given by

R=[2,6,7,7,10,14,15,17,19,20,19,15,15,13,12,7,7,3,2]

Rajita · Answer 14 · 2023-01-25T21:38:16+05:30

Correlation: Correlation is a simple mathematical operation to compare two functions or signals or waveforms. It is defined as the measure of similarity between those signals. There are two types of correlations −

Cross-correlation
Autocorrelation

Cross-correlation: It is the measure of similarity or coherence between one signal and the time-delayed version of another signal.

Autocorrelation: It is the measure of similarity or coherence between a signal and its time delayed version.

Both cross-correlation and autocorrelation is also defined separately for energy (or aperiodic) signals and power (periodic) signals.

autocorrelation:

$R_{xx}{k} = \sum x[m]x^{*}[m-k]$

cross-correlation:

$R_{xy}{k} = \sum x[m]y^{*}[m-k]$

Ex: let X[n] and Y[n] are two discrete time signals

X[n]=[1,2,3,4,0,4,3,2,1,0]

Y[n]=[5,4,3,2,1,0,1,2,3,4]

The cross-correlation of X[n] and Y[n] is

R[n]=[4,11,20,30,20,28,32,35,37,45,42,45,40,30,40,26,14,5,0].

Shivalika Tripathi · Answer 15 · 2023-01-25T21:57:19+05:30

Correlation:

It is a measure of similarity between signals and is found using a process similar to convolution.
Correlation is used to compare two signals.
It is used in radar & sonar systems to find the location of a target , it’s other applications are in image processing ,control engineering etc.
The correlation is of two types:

Cross Correlation
Auto Correlation

Cross Correlation: This is a kind of correlation, in which the signal in-hand is correlated with another signal so as to know how much resemblance exists between them.

Cross correlation for real and discrete signals x[n] and y[n] :

$R_{xy}[n]=\Sigma_{-\infty}^{\infty}x[n]y[k-n]$

$R_{yx}[n]=\Sigma_{-\infty}^{\infty}y[n]x[k-n]$

Cross correlation for complex and discrete signals x[n] and y[n] :

$R_{xy}[n]=\Sigma_{-\infty}^{\infty}x[n]y^*[k-n]$

$R_{yx}[n]=\Sigma_{-\infty}^{\infty}y[n]x^*[k-n]$

Auto Correlation: Autocorrelation, also known as serial correlation, refers to the degree of correlation of the same variables between two successive time intervals.

Auto correlation for real and discrete signals x[n] and y[n] :

$R_{xx}[n]=\Sigma_{-\infty}^{\infty}x[n]x[k-n]$

$R_{yy}[n]=\Sigma_{-\infty}^{\infty}y[n]y[k-n]$

Auto correlation for complex and discrete signals x[n] and y[n] :

$R_{xx}[n]=\Sigma_{-\infty}^{\infty}x[n]x^*[k-n]$

$R_{yy}[n]=\Sigma_{-\infty}^{\infty}y[n]y^*[k-n]$

Example : x(n) = { 1 , 0 , 1 , 2 , 1 , 0 , 3 , 1 , 1 , 2}

y(n) = { 0 , 0 , 1 , 1 , 2 , 0 , 3 , 0 , 1 , 0}

The cross correlation of a[n] and b[n] is :

z(n) = {0 , 0 , 1 , 1 , 3 , 3 , 8 , 5 , 9 , 10 , 12 , 7 , 14 , 7 , 6 , 7 , 1 , 2 , 0}

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Correlation of 2 signals

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