Principles of Modern Radar - Volume 2 1891121537

10.2 Space-Time Signal Representation 469The space-time covariance matrix describes all possible covariances among the elementsof x k . Formally, its definition isR k = E [ x k xkH ] − μk μk H ; μ k = E[x k ] (10.37)E[·] is the expectation operator and μ k is the mean vector characterizing the mean valueof each element of x k . We define the outer product operation in (10.3). Consider the casewhere μ k = 0 (a typical, practical assumption). Then (10.37) takes the form⎡E [ |[x k ] 1 | 2] E [ [x k ] 1 [x k ] ∗ ]2 ··· E [ [x k ] 1 [x k ] ∗ ] ⎤MNE [ [x k ] 2 [x k ] ∗ ]1 E [ |[x k ] 2 | 2] ··· E [ [x k ] 2 [x k ] ∗ ]MNR k =⎢. (10.38)⎣ . ··· .. .⎥E [ [x k ] MN [x k ] ∗ ]1 E [ [x k ] MN [x k ] ∗ ]2 ··· E [ |[x k ] MN | 2] ⎦The diagonal entries correspond to the variance of each element, E [ |[x k ] m | 2] =E [ [x k ] m [x k ] ∗ m] , and the off-diagonal entries are the autocorrelations of varying lag [2,9].Directly interpreting the significance of the covariance matrix—other than the diagonalentries, which serve as a measure of power—is difficult. The power spectral densityis a more useful view of the properties of the signal environment. An important resultrelating the covariance matrix and PSD isI 2( )R k↔S fsp , ˜f d (10.39)where I p is the p-dimensional Fourier transform (p = 2 for the space-time case), andS ( f sp , ˜f d) is the PSD. The result in (10.39) follows from the Wiener-Khintchine theorem[9-11].The periodogram is a biased estimate of the PSD [11]. Given the data cube of Figure10-6, we compute the periodogram by averaging the 2-D Fourier Transform ofindividual snapshots. The estimate for S( f sp , ˜f d ) is thus given asŜ( f sp , ˜f d ) = 1 KK∑|I 2 {X m }| 2 . (10.40)Due to the presence of receiver noise, the covariance matrix is positive definite, viz.m=1d H R k d > 0 (10.41)for arbitrary d ≠ 0. We then can express the covariance matrix as [9]R k =NM∑m=1λ k (m)q k (m)q H k (m); λ k(1) ≥ λ k (2) ≥···λ k (NM)>0 (10.42)λ k (m) is the m-th eigenvalue corresponding to eigenvector q k (m). The collection of alleigenvectors forms an orthonormal basis. An individual eigenvector describes a modeof some linear combination of the various signal sources embedded in the covariancematrix; each eigenvector is a subspace of the overall signal characterization. The eigenvaluedescribes the power in a particular subspace. An eigenbasis can be used in amanner similar to a Fourier basis to describe the properties of a stochastic space-time