Principles of Modern Radar - Volume 2 1891121537

10.2 Space-Time Signal Representation 4712520Eigenspectrum FIGURE 10-8Eigenspectrum forthe three space-timesignals shown inFigure 10-5, withadditive receivernoise of 1watt/channel.15dB10500 5 10 15 20Ranked Eigenvalue Numberwhere s m ∈ C NM×1 are the various space-time signal sources s 1 ,s 2 ,...,s P impingingon the array. Equation (10.46) indicates each eigenvector is a linear combination of thedifferent sources. If the sources are largely separated and of sufficient SNR, then a singlesource can dominate an eigensubspace, as seen in Figure 10-7.A plot of the rank-ordered eigenvalues is known as the eigenspectrum. Identifyingthe colored noise subspace and relative interference-to-noise ratio (INR) or clutter-tonoiseratio (CNR) are common uses of the eigenspectra (the interference-to-noise ratiois given as the ratio of the interference and noise eigenvalues). The dominant subspaceis also known as the signal or interference subspace, depending on the context, whilethe eigenvectors corresponding to the smaller eigenvalues comprise the noise subspace.Figure 10-8 shows the eigenspectrum for the first 20 eigenvalues of the covariance matrixused to generate Figure 10-5, with the addition of receiver noise; we find three dominanteigenvalues corresponding to the three signal sources. Since the three space-time signalsare largely separated in angle and Doppler with unity amplitude weighting (σs2 = 1) andthe output noise is set to 1 watt (σn 2 = 1), the three eigenvalues of 25.85 dB are expectedand correspond to the signal power times the space-time integration gain plus the noisepower; that is, the eigenvalue measures power in a given subspace and is given in decibelsas 10 ∗ log 10 (σs 2NM + σ n 2 ) when the sources are largely separated in angle and Doppler. Theeigenspectrum is valuable when determining the number of DoFs required to cancel theinterference: each eigenvalue, or signal source, requires one DoF for effective cancellation.Some important properties of one-dimensional covariance matrices are given in[9–11]. Many of these properties apply to the space-time scenario. Among these variousproperties, it is worth mentioning two important characteristics: (1) the space-timecovariance matrix is Hermitian; and (2) it is nonnegative definite and usually positivedefinite (as indicated by (10.41)). A Hermitian matrix equals its conjugate transpose; thisapparent symmetry of the covariance matrix also applies to its inverse and is useful in