# sensorcov

Sensor spatial covariance matrix

## Syntax

``xcov = sensorcov(pos,ang)``
``xcov = sensorcov(pos,ang,ncov)``
``xcov = sensorcov(pos,ang,ncov,scov)``

## Description

example

````xcov = sensorcov(pos,ang)` returns the sensor spatial covariance matrix, `xcov`, for narrowband plane wave signals arriving at a sensor array. The sensor array is defined by the sensor positions specified in the `pos` argument. The signal arrival directions are specified by azimuth and elevation angles in the `ang` argument. In this syntax, the noise power is assumed to be zero at all sensors, and the signal power is assumed to be unity for all signals.```

example

````xcov = sensorcov(pos,ang,ncov)` specifies, in addition, the spatial noise covariance matrix, `ncov`. This value represents the noise power on each sensor as well as the correlation of the noise between sensors. In this syntax, the signal power is assumed to be unity for all signals. This syntax can use any of the input arguments in the previous syntax.```

example

````xcov = sensorcov(pos,ang,ncov,scov)` specifies, in addition, the signal covariance matrix, `scov`, which represents the power in each signal and the correlation between signals. This syntax can use any of the input arguments in the previous syntaxes.```

## Examples

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Create a covariance matrix for a 3-element, half-wavelength-spaced uniform line array. Use the default syntax, which assumes no noise power and unit signal power.

```N = 3; d = 0.5; elementPos = (0:N-1)*d; xcov = sensorcov(elementPos,[30 60])```
```xcov = 3×3 complex 2.0000 + 0.0000i -0.9127 - 1.4086i -0.3339 + 0.7458i -0.9127 + 1.4086i 2.0000 + 0.0000i -0.9127 - 1.4086i -0.3339 - 0.7458i -0.9127 + 1.4086i 2.0000 + 0.0000i ```

The diagonal terms of the matrix represent the sum of the two signal powers.

Create a spatial covariance matrix for a 3-element, half-wavelength-spaced uniform line array. Assume there are two incoming signals with unit power and there is additive noise with –10 dB power.

```N = 3; d = 0.5; elementPos = (0:N-1)*d; xcov = sensorcov(elementPos,[30 35],db2pow(-10))```
```xcov = 3×3 complex 2.1000 + 0.0000i -0.2291 - 1.9734i -1.8950 + 0.4460i -0.2291 + 1.9734i 2.1000 + 0.0000i -0.2291 - 1.9734i -1.8950 - 0.4460i -0.2291 + 1.9734i 2.1000 + 0.0000i ```

The diagonal terms represent the two signal powers plus noise power at each sensor.

Compute the covariance matrix for a 3-element half-wavelength spaced line array when there is some correlation between two signals. The correlation can model, for example, multipath propagation caused by reflection from a surface. Assume an additive noise power value of –10 dB.

```N = 3; d = 0.5; elementPos = (0:N-1)*d; scov = [1, 0.8; 0.8, 1]; xcov = sensorcov(elementPos,[30 35],db2pow(-10),scov)```
```xcov = 3×3 complex 3.7000 + 0.0000i -0.4124 - 3.5521i -3.4111 + 0.8028i -0.4124 + 3.5521i 3.6574 + 0.0000i -0.4026 - 3.4682i -3.4111 - 0.8028i -0.4026 + 3.4682i 3.5321 + 0.0000i ```

## Input Arguments

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Positions of the elements of a sensor array specified as a 1-by-N vector, a 2-by-N matrix, or a 3-by-N matrix. In this vector or matrix, N represents the number of elements of the array. Each column of `pos` represents the coordinates of an element. You define sensor position units in term of signal wavelength. If `pos` is a 1-by-N vector, then it represents the y-coordinate of the sensor elements of a line array. The x and z-coordinates are assumed to be zero. When `pos` is a 2-by-N matrix, it represents the (y,z)-coordinates of the sensor elements of a planar array. This array is assumed to lie in the yz-plane. The x-coordinates are assumed to be zero. When `pos` is a 3-by-N matrix, then the array has arbitrary shape.

Example: `[0,0,0; 0.1,0.4,0.3;1,1,1]`

Data Types: `double`

Arrival directions of incoming signals specified as a 1-by-M vector or a 2-by-M matrix, where M is the number of incoming signals. If `ang` is a 2-by-M matrix, each column specifies the direction in azimuth and elevation of the incoming signal `[az;el]`. Angular units are specified in degrees. The azimuth angle must lie between –180° and 180° and the elevation angle must lie between –90° and 90°. The azimuth angle is the angle between the x-axis and the projection of the arrival direction vector onto the xy plane. It is positive when measured from the x-axis toward the y-axis. The elevation angle is the angle between the arrival direction vector and xy-plane. It is positive when measured towards the z axis. If `ang` is a 1-by-M vector, then it represents a set of azimuth angles with the elevation angles assumed to be zero.

Example: [45;0]

Data Types: `double`

Noise spatial covariance matrix specified as a non-negative, real-valued scalar, a non-negative, 1-by-N real-valued vector or an N-by-N, positive definite, complex-valued matrix. In this argument, N is the number of sensor elements. Using a non-negative scalar results in a noise spatial covariance matrix that has identical white noise power values (in watts) along its diagonal and has off-diagonal values of zero. Using a non-negative real-valued vector results in a noise spatial covariance that has diagonal values corresponding to the entries in `ncov` and has off-diagonal entries of zero. The diagonal entries represent the independent white noise power values (in watts) in each sensor. If `ncov` is N-by-N matrix, this value represents the full noise spatial covariance matrix between all sensor elements.

Example: [1,1,4,6]

Data Types: `double`
Complex Number Support: Yes

Signal covariance matrix specified as a non-negative, real-valued scalar, a 1-by-M non-negative, real-valued vector or an M-by-M positive semidefinite, matrix representing the covariance matrix between M signals. The number of signals is specified in `ang`. If `scov` is a nonnegative scalar, it assigns the same power (in watts) to all incoming signals which are assumed to be uncorrelated. If `scov` is a 1-by-M vector, it assigns the separate power values (in watts) to each incoming signal which are also assumed to be uncorrelated. If `scov` is an M-by-M matrix, then it represents the full covariance matrix between all incoming signals.

Example: [1 0 ; 0 2]

Data Types: `double`
Complex Number Support: Yes

## Output Arguments

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Sensor spatial covariance matrix returned as a complex-valued, N-by-N matrix. In this matrix, N represents the number of sensor elements of the array.

## References

[1] Van Trees, H.L. Optimum Array Processing. New York, NY: Wiley-Interscience, 2002.

[2] Johnson, Don H. and D. Dudgeon. Array Signal Processing. Englewood Cliffs, NJ: Prentice Hall, 1993.

[3] Van Veen, B.D. and K. M. Buckley. “Beamforming: A versatile approach to spatial filtering”. IEEE ASSP Magazine, Vol. 5 No. 2 pp. 4–24.