Principles of Modern Radar - Volume 2 1891121537

5.2 CS Theory 165as , where is a basis for C N , and is a random matrix of one of the mentionedclasses [21]. The proofs of these results rely on probabilistic arguments involving concentrationof measure, see for example, [46]. Intuitively, the idea is that random vectorsdrawn in a very high-dimensional space are unlikely to have large inner products.For random matrices, we have seen that we need to collect on the order of s log(N/s)measurements. A simple analog to bit counting may provide some intuition about thisbound. If we have N coefficients with s nonzero values, then there are ( Ns) possible setsof nonzero coefficients. If we allow c 0 quantization levels for encoding each nonzerocoefficient, then the total number of required bits for this information is log{ ( N) s + c0 s}.If we neglect the bits for encoding the values, we can obtain{( ) } ( )N Nlog + c 0 s ≈ logs s{( ) Ne s }≤ logs(5.16)= s log(N/s) + s log e (5.17)Thus, a simple calculation of the required coding bits yields a leading-order term of thesame form as the number of required measurements predicted by CS theory. Intuitively,randomization of the A matrix ensures with high probability that each measurement providesa nearly constant increment of new information bits. This result in no way constitutesa proof but is presented to provide some intuitive insight about the origin of this expression.We should mention that randomization has a long history in radar signal processing.For example, array element positions can be randomized to reduce the sidelobes in sparselypopulated arrays [48–50]. It is also well understood that jittering or staggering the pulserepetition frequency can eliminate ambiguities [13]. The transmitted waveform itself canalso be randomized, as in noise radar [51,52], to provide a thumbtack-like ambiguityfunction. From a CS perspective, these randomization techniques can be viewed as attemptsto reduce the mutual coherence of the forward operator A [17].5.2.5.4 Mutual CoherenceAs pointed out already, estimating and testing the RIC for large M is impractical. Atractable yet conservative bound on the RIC can be obtained through the mutual coherenceof the columns of A defined asM(A) = max |Ai H A j |i ≠ jMutual coherence can be used to guarantee stable inversion through l 1 recovery [53,54],although these guarantees generally require fairly small values of s. Furthermore, the RICis conservatively bounded by M(A) ≤ R s (A) ≤ (s − 1)M(A). The upper bound isvery loose, as matrices can be constructed for which the RIC is nearly equal to the mutualcoherence over a wide range of s values [55].The mutual coherence is of particular importance in radar signal processing. Recallfrom Section 5.2.2.2 that entries of the Gramian matrix A H A are samples of the radar ambiguityfunction. The mutual coherence is simply the maximum off-diagonal of this matrix.Thus, the mutual coherence of a radar system can be reduced by designing the ambiguity