Principles of Modern Radar - Volume 2 1891121537

754 CHAPTER 17 Advanced Processing Methods for Passive Bistatic Radar Systemsfurther approximation of the batches algorithm described already. To show its derivation,we first rewrite equation (17.2) and then replace the cross-correlation between the r-thportions of length n C of the surveillance and the reference signals, with its approximationevaluated in the frequency domain:χ[l,m] ∼ =∼=N∑C −1r=0N∑C −1r=0= 1mr− j2πen∑C −1N Cp=0mr− j2π 1Ne Cn∑C −1kl+ j2πen Ck=0s surv [rn C + p] · s ∗ ref [rn C + p − l]n∑C −1S (r)n Ck=0N∑C −1n Cr=0surv [k]S(r)∗ refkl+ j2πn[k]e Cmr− j2πNY [k,r]e C (17.10)Notice that by neglecting the border effects (additional approximation with respect tothe batches algorithm), the r-th subsequences are cyclically extended so that their crosscorrelationcan be evaluated in the frequency domain, as the IDFT of their individual DFTs[k] and S(r)∗[k] in the second line). By rearranging the terms of the summations, the(S (r)survrefthird line of equation (17.10) is easily obtained.The resulting algorithm (sketched in Figure 17-4b) can be subdivided into four steps:1. Evaluate the DFTs of surveillance and reference signals over the N C batches of n Csamples each.2. Compute the product of the DFTs outputs for each batch and arrange the results in the2-D sequence Y [k,r] = Ssurv (r) [k]S(r)∗ ref [k](k = 0,...,n C − 1; r = 0,...,N C − 1).3. Evaluate the DFT X[k,m] ofY [k,r] over the index r for each k (n C IDFTs); this steprepresents the parallel processing of the n C different frequency channels in which thesignals have been split.4. Evaluate the IDFT of X[k,m] over the index k for each m (N C DFTs) to obtain therange samples of the 2D-CCF at the m-th Doppler bin.Notice that evaluating the DFT of short subsequences (Step 1) separates each sequenceinto a number of frequency channels (channelization) that are independently processed(Steps 2 and 3) and then recombined to provide the final result (Step 4).The computational load for the channelization technique is reported in Table 17-1 asa function of the number n C of channels assuming n C N C = N. In this case, the numberof frequency channels is equal to the batches length; thus, the algorithm parameters mustbe selected so that N C ≥ N f . Moreover, since the range samples of the 2D-CCF areobtained from the last DFT stage performed over sequences of n C samples, assuming thatonly positive values of the differential bistatic range R B are included in the 2D-CCF map,n C ≥ 2N τ should be satisfied. Reducing n C within these limits allows a correspondingreduction in the computational load.The number of channels should be carefully selected since it yields a trade-off betweenthe two adopted approximations of equation (17.10) that determine the final SNR loss.Increasing n C yields a higher integration loss for fast-moving targets whose Doppler shiftis not properly compensated (as for the batches algorithm) [28]. However, in this case,reducing n C enhances the border effects in the evaluation of the batches cross-correlationsyielding additional loss for targets at higher bistatic ranges. Assuming a constant modulus