elevation and azimuthal angles of an incoming signal generated by a source. The DOA process is then to extract these two frequencies from the data model.

In the previous two sections we discussed the expressions for data vector x using the array steering matrices of some commonly used antenna arrays. In the next section, we discuss the covariance matrix of these data vectors, an important quantity that is used in DOA algorithms.

3.3.3Covariance Matrices

The signals received by an array are noise-corrupted in the real world. These noises are normally uncorrelated while the pure signals received by different elements are correlated as they are originated from the same sources. By using this property, one may be able to extract effectively the DOA information. To do so, the concept of cross-covariance information among the noise-corrupted signals, spatial covariance matrix, is introduced and employed in finding DOAs. The spatial covariance matrix of the data (i.e., signals plus noises) received by an array is defined as

where E{ } denotes the statistical expectation.

Equation (3.35) quantifies the degree of correlation of the data signals received by array elements. The higher values of its elements, the higher degree of correlations among the signals.

By substituting (3.10) into (3.35), we have

where Rss = E{s(t)sH(t)} is the signal covariance matrix and σ 2N is the com-

mon variance of the noises.

In practice, the exact covariance matrix of Rxx is difficult to find due to the limited number of data sets received and processed by an array. Therefore, an estimation is made. By assuming that all underlying random noise processes are ergodic, the ensemble average (or statistical expectation) can be replaced by a time average. Let X be denoted as the noise corrupted signal (or data) matrix composed of N snapshots of x(tn), 1 ≤ n ≤ N,