Principles of Modern Radar - Volume 2 1891121537

3.2 Optimum MIMO Waveform Design for the Additive Colored Noise Case 91is given by [7]:H w = R − 1 2 (3.5)The reader should verify the whitening properties of (3.5) (see problem 2 and [9]).The output of the linear whitening filter, z ∈ C N , will consist of signal and noisecomponents, z s , z n , respectively, given byz = z s + z n= H w y s + H w n (3.6)= H w H T s + H w nwhere y s ∈ C N denotes the target echo as shown in Figure 3-2 (i.e., the output of H T ).Since the noise has been whitened via a linear—in this case full-rank—transformation[7]), the final receiver stage consists of a white noise matched filter of the form (to withina multiplicative scalar)w z = z s ∈ C N (3.7)The corresponding output SNR is thus given bySNR o =∣ ∣w ′ zz s 2var (w ′ zz n )=∣ ∣z ′ sz s 2var (z ′ sz n )=∣ ∣z ′ sz s 2E{z ′ sz n z ′ nz s }∣ ∣∣z ′ sz s 2=z ′ s E{z n z ′ n}z s=∣ ∣z ′ sz s 2z ′ sz s= ∣ ∣z ′ ∣sz s (3.8)where var(·) denotes the variance. Note that due to the whitening operation E { z n z ′ n} = I .In words, the output SNR is proportional to the energy in the whitened target echo.This fact is key to optimizing the input function: Chose s (the input) to maximize theenergy in the whitened target echo:max{s}∣ z ′ sz s∣ ∣ (3.9)Substituting z s = H w H T s into (3.9) yields the objective function that explicitly dependson the inputmax{s}∣ s ′ ( H ′ H ) s ∣ ∣ (3.10)