# Signal Correlation (Cross-Correlation) (G Dataflow)

Version:

Computes the cross correlation of two signals.

## x

The input signal.

This input supports the following data types.

• Waveform
• 1D array of waveforms
• 1D array of double-precision, floating-point numbers
• 1D array of complex double-precision, floating-point numbers
• 2D array of double-precision, floating-point numbers

## y

The second input signal, which you want to cross correlate with the first input signal.

This input supports the following data types:

• Waveform
• 1D array of waveforms
• 1D array of double-precision, floating-point numbers
• 1D array of complex double-precision, floating-point numbers
• 2D array of double-precision, floating-point numbers

## algorithm

The correlation method to use.

If x and y are small, the direct method typically is faster. If x and y are large, the frequency domain method typically is faster. Additionally, slight numerical differences can exist between the two methods.

Name Description
direct

Computes the cross correlation using the direct method of linear correlation.

frequency domain

Computes the cross correlation using an FFT-based technique.

Default: frequency domain

## normalization

The normalization method to use to compute the cross correlation between the two input signals.

Name Description
none

Does not apply normalization.

unbiased

Applies unbiased normalization.

biased

Applies biased normalization.

Default: none

## error in

Error conditions that occur before this node runs. The node responds to this input according to standard error behavior.

Default: No error

## Rxy

Cross correlation of the two input signals.

## error out

Error information. The node produces this output according to standard error behavior.

## Algorithm for Calculating the Cross Correlation

The cross correlation Rxy(t) of the sequences x(t) and y(t) is defined by the following equation:

$Rxy\left(t\right)=x\left(t\right)\otimes y\left(t\right)={\int }_{-\infty }^{\infty }x*\left(\tau \right)\cdot y\left(t+\tau \right)d\tau$

where the symbol $\otimes$ denotes correlation.

The discrete implementation of cross correlation is as follows. Let h represent a sequence whose indexing can be negative, let N be the number of elements in the input sequence x, let M be the number of elements in the sequence y, and assume that the indexed elements of x and y that lie outside their range are equal to zero, as shown by the following equations:
${x}_{j}=0,\text{\hspace{0.17em}}j<0\text{\hspace{0.17em}}\text{\hspace{0.17em}}orj\ge N$

and

${y}_{j}=0,\text{\hspace{0.17em}}j<0\text{\hspace{0.17em}}\text{\hspace{0.17em}}or\text{\hspace{0.17em}}j\ge M$

Then this node obtains the elements of h using the following equation:

${h}_{j}=\underset{k=0}{\overset{N-1}{\sum }}{{x}_{k}}^{*}\cdot {y}_{j+k}$

for $j=-\left(N-1\right),-\left(N-2\right),\text{\hspace{0.17em}}...\text{\hspace{0.17em}},-1,0,1,\text{\hspace{0.17em}}...\text{\hspace{0.17em}},\left(M-2\right),\left(M-1\right)$

The elements of the output sequence Rxy are related to the elements in the sequence h by

${Rxy}_{i}={h}_{i-\left(N-1\right)}$

for $i=0,1,2,...\text{\hspace{0.17em}},N+M-2$

Because you cannot index LabVIEW arrays with negative numbers, the corresponding cross correlation value at t = 0 is the Nth element of the output sequence Rxy. Therefore, Rxy represents the correlation values that this node shifts N times in indexing.

## How This Node Applies Unbiased Normalization

This node applies unbiased normalization as follows:

${R}_{xy}{\left(unbiased\right)}_{j}=\frac{1}{f\left(j\right)}{Rxy}_{j}$

for j = 0, 1, 2, ..., M + N - 2

where Rxy is the cross correlation between x and y with no normalization. f(j) is:

## How This Node Applies Biased Normalization

This node applies biased normalization as follows:

${R}_{xy}{\left(biased\right)}_{j}=\frac{1}{\mathrm{max}\left(M,N\right)}{Rxy}_{j}$

for j = 0, 1, 2, ..., M + N - 2

where Rxy is the cross correlation between x and y with no normalization.

Where This Node Can Run:

Desktop OS: Windows

FPGA: Not supported