The Santa Barbara Channel Experiment (SBCX)

Cross Spectral Density Matrices

One of the prime ingredients in source localization is the cross spectral density matrix (otherwise known as the covariance matrix). A covariance matrix is estimated as follows:

Obtain a set of time-series data from a set of N sensors.
Determine the array transit time.
Set your snapshot length to be at least twice the array transit time
Obtain one snapshot's worth of data from your time-series data set
Multiply the snapshot by a normalized Hanning window
Fourier Transform the windowed snapshot>
Zip out the appropriate FFT bin for each of the N sensors. Arrange in an Nx1 vector.
Take the outer product of this vector
Proceed to the next snapshot by advancing half the length of a single snapshot down the time series data (50% overlap)
Repeat the windowing, Fourier Transform, and outer product. Sum the result to the previous outer product
Do this until the selected data set has been exhausted (M snapshots total).
Divide the summed outer product matrix by M (As a rule of thumb, M should be greater than 3N).
Congratulations! You now have your estimated covariance matrix.

Of course, if you do not want to go through all the trouble, then there are pre-packaged covariance matrices available here. Ranges and bearings are given from the center of the FFP array.

The original data which made up these matrices is available here.

Tape 34 File 15 (first 70 seconds):

This is not a stationary event. Rather, there is a high bearing rate change during this time. From the A/X 2 master timesheet:

Date	Time	Range (m)	Bearing (radians)	Bearing (degrees)	Speed, relative
YYMMDD	(UTC)	from array center	(east is 0)	(north is 0)	to array (m/sec)
980411	22:47:05	1205.0383	-1.572409	180.0924	0.3853
980411	22:47:20	1210.4598	-1.609104	182.1949	0.3614
980411	22:47:35	1218.5559	-1.644620	184.2298	0.5397
980411	22:47:50	1227.3850	-1.682698	186.4115	0.5886
980411	22:48:06	1238.0834	-1.720919	188.6014	0.6687
980411	22:48:20	1249.9575	-1.755517	190.5837	0.8481

155 snapshots

Freq/Array	1	2	3	4	5	all	Freq/Array	1	2	3	4	5	all
64.0	34.15	34.15	34.15	34.15	34.15	34.15	67.0	34.15	34.15	34.15	34.15	34.15	34.15
79.0	34.15	34.15	34.15	34.15	34.15	34.15	82.0	34.15	34.15	34.15	34.15	34.15	34.15
94.0	34.15	34.15	34.15	34.15	34.15	34.15	97.0	34.15	34.15	34.15	34.15	34.15	34.15
112.0	34.15	34.15	34.15	34.15	34.15	34.15	115.0	34.15	34.15	34.15	34.15	34.15	34.15
130.0	34.15	34.15	34.15	34.15	34.15	34.15	133.0	34.15	34.15	34.15	34.15	34.15	34.15
148.0	34.15	34.15	34.15	34.15	34.15	34.15	151.0	34.15	34.15	34.15	34.15	34.15	34.15
166.0	34.15	34.15	34.15	34.15	34.15	34.15	169.0	34.15	34.15	34.15	34.15	34.15	34.15
201.0	34.15	34.15	34.15	34.15	34.15	34.15	204.0	34.15	34.15	34.15	34.15	34.15	34.15
235.0	34.15	34.15	34.15	34.15	34.15	34.15	238.0	34.15	34.15	34.15	34.15	34.15	34.15
283.0	34.15	34.15	34.15	34.15	34.15	34.15	286.0	34.15	34.15	34.15	34.15	34.15	34.15
338.0	34.15	34.15	34.15	34.15	34.15	34.15	341.0	34.15	34.15	34.15	34.15	34.15	34.15
388.0	34.15	34.15	34.15	34.15	34.15	34.15	391.0	34.15	34.15	34.15	34.15	34.15	34.15

Tape 35 File 17:

Range: 7101 meters. Bearing: -2.837426 radians. Depth: 60.65 meters. Snapshots: Approximately 700

Freq/Array	1	2	3	4	5	Freq/Array	1	2	3	4	5
64.0	35.17	35.17	35.17	35.17	35.17	67.0	35.17	35.17	35.17	35.17	35.17
79.0	35.17	35.17	35.17	35.17	35.17	82.0	35.17	35.17	35.17	35.17	35.17
94.0	35.17	35.17	35.17	35.17	35.17	97.0	35.17	35.17	35.17	35.17	35.17
112.0	35.17	35.17	35.17	35.17	35.17	115.0	35.17	35.17	35.17	35.17	35.17
130.0	35.17	35.17	35.17	35.17	35.17	133.0	35.17	35.17	35.17	35.17	35.17
148.0	35.17	35.17	35.17	35.17	35.17	151.0	35.17	35.17	35.17	35.17	35.17
166.0	35.17	35.17	35.17	35.17	35.17	169.0	35.17	35.17	35.17	35.17	35.17
201.0	35.17	35.17	35.17	35.17	35.17	204.0	35.17	35.17	35.17	35.17	35.17
235.0	35.17	35.17	35.17	35.17	35.17	238.0	35.17	35.17	35.17	35.17	35.17
283.0	35.17	35.17	35.17	35.17	35.17	286.0	35.17	35.17	35.17	35.17	35.17
338.0	35.17	35.17	35.17	35.17	35.17	341.0	35.17	35.17	35.17	35.17	35.17
388.0	35.17	35.17	35.17	35.17	35.17	391.0	35.17	35.17	35.17	35.17	35.17

Tape 36 File 7 (second half):

Range: 15342 meters. Bearing: 3.112526 radians. Depth: 60.65 meters. Snapshots: Approximately 350

Freq/Array	1	2	3	4	5	all	Freq/Array	1	2	3	4	5	all
64.0	36.7	36.7	36.7	36.7	36.7	36.7	67.0	36.7	36.7	36.7	36.7	36.7	36.7
79.0	36.7	36.7	36.7	36.7	36.7	36.7	82.0	36.7	36.7	36.7	36.7	36.7	36.7
94.0	36.7	36.7	36.7	36.7	36.7	36.7	97.0	36.7	36.7	36.7	36.7	36.7	36.7
112.0	36.7	36.7	36.7	36.7	36.7	36.7	115.0	36.7	36.7	36.7	36.7	36.7	36.7
130.0	36.7	36.7	36.7	36.7	36.7	36.7	133.0	36.7	36.7	36.7	36.7	36.7	36.7
148.0	36.7	36.7	36.7	36.7	36.7	36.7	151.0	36.7	36.7	36.7	36.7	36.7	36.7
166.0	36.7	36.7	36.7	36.7	36.7	36.7	169.0	36.7	36.7	36.7	36.7	36.7	36.7
201.0	36.7	36.7	36.7	36.7	36.7	36.7	204.0	36.7	36.7	36.7	36.7	36.7	36.7
235.0	36.7	36.7	36.7	36.7	36.7	36.7	238.0	36.7	36.7	36.7	36.7	36.7	36.7
283.0	36.7	36.7	36.7	36.7	36.7	36.7	286.0	36.7	36.7	36.7	36.7	36.7	36.7
338.0	36.7	36.7	36.7	36.7	36.7	36.7	341.0	36.7	36.7	36.7	36.7	36.7	36.7
388.0	36.7	36.7	36.7	36.7	36.7	36.7	391.0	36.7	36.7	36.7	36.7	36.7	36.7

CSDM file format

Now that you have a covariance matrix, you probably want to do something with it. These matlab scripts (read_cov.m and read_complex_matrix.m) should help. The format of the file is listed below.

data type description

long long int (8 bytes) Byte-swapping flag. When reading this flag in, this should equal 1. If it does not, then you must close the file, re-open it with byte-swapping turned on, and verify the value is 1.

IEEE double (8 bytes) Frequency (Hz)

long long int (8 bytes) Number of arrays (num_aco) represented by this covariance matrix

long long int (8 bytes * num_aco) Arrays (1, 2, 3, 4, or 5) represented in this file.

long long int (8 bytes) Another byte-swapping flag, identical to the first. (Sorry, long story...)

long long int (8 bytes) Number of rows in matrix

long long int (8 bytes) Number of columns in matrix

What follows this is the complex matrix element data. This is written in IEEE double (8 bytes) format. First the real part (8 bytes), then the imaginary (another 8 bytes). This is repeated for each element, in columnwise order (the first N elements written to the file are the first N rows from column 1.

Last updated: 990917
Comments/Questions: pmd@alum.mit.edu

Ships in the SBCX logo (left to right, from the top) Lockheed P-3 ("Redhook"), Acoustic Explorer, FLIP, Danny C, USS Dolphin.

data type	description
long long int (8 bytes)	Byte-swapping flag. When reading this flag in, this should equal 1. If it does not, then you must close the file, re-open it with byte-swapping turned on, and verify the value is 1.
IEEE double (8 bytes)	Frequency (Hz)
long long int (8 bytes)	Number of arrays (num_aco) represented by this covariance matrix
long long int (8 bytes * num_aco)	Arrays (1, 2, 3, 4, or 5) represented in this file.
long long int (8 bytes)	Another byte-swapping flag, identical to the first. (Sorry, long story...)
long long int (8 bytes)	Number of rows in matrix
long long int (8 bytes)	Number of columns in matrix