sensorcov

Sensor spatial covariance matrix

Syntax

  • xcov = sensorcov(pos,ang) example
  • xcov = sensorcov(pos,ang,ncov) example
  • xcov = sensorcov(pos,ang,ncov,scov) example

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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Covariance Matrix for Two Signals without Noise

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

N = 3;     % Elements in array
d = 0.5;    % sensor spacing half wavelength
elementPos = (0:N-1)*d;
xcov = sensorcov(elementPos,[30 60]);
xcov =

   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 represent the sum of the two signal powers.

Covariance Matrix for Two Independent Signals with 10 dB SNR

Create a spatial covariance matrix for a 3-element, half-wavelength-spaced line array. Assume there are two incoming unit-power signals and there is a noise value of –10 dB. By default, scov is the identity matrix.

  N = 3;     % Elements in array
  d = 0.5;    % sensor spacing half wavelength
  elementPos = (0:N-1)*d;
  xcov = sensorcov(elementPos,[30 35],db2pow(-10));
xcov =

   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.

Covariance Matrix for Two Correlated Signals with 10 dB SNR

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 a noise power value of –10 dB.

N = 3;     % Elements in array
d = 0.5;    % sensor spacing half wavelength
elementPos = (0:N-1)*d;
scov = [1, 0.8; 0.8, 1];
xcov = sensorcov(elementPos,[30 35],db2pow(-10),scov);
xcov =

   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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pos — Positions of array sensor elements1-by-N real-valued vector | 2-by-N real-valued matrix | 3-by-N real-valued matrix

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. If pos is a 2-by-N matrix, then it represents the (y,z)-coordinates of the sensor elements of a planar array which is assumed to lie in the yz-plane. The x-coordinates are assumed to be zero. If pos is a 3-by-N matrix, then the array has arbitrary shape.

Example: [0, 0, 0; .1, .2, .3; 0,0,0]

Data Types: double

ang — Arrival directions of incoming signals1-by-M real-valued vector | 2-by-M real-valued matrix

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

ncov — Noise spatial covariance matrix0 (default) | non-negative real-valued scalar | 1-by-N non-negative real-valued vector | N-by-N positive definite, complex-valued matrix

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

scov — Signal covariance matrix1 (default) | non-negative real-valued scalar | 1-by-M non-negative real-valued vector | N-by-M positive semidefinite, complex-valued matrix

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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xcov — Sensor spatial covariance matrixcomplex-valued N-by-N matrix

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.

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