Principles of Modern Radar - Volume 2 1891121537

748 CHAPTER 17 Advanced Processing Methods for Passive Bistatic Radar Systemsof filters covers all the possible target velocities. The filter where the target is detectedprovides the estimates of the bistatic Doppler shift of each target echo. Similarly, theestimate of the bistatic range is given by the sample along range where it is detected.Assuming that the signal s ref (t) collected at the reference antenna is a perfect copy (or atleast good enough as discussed in a later section) of the transmitted signal, the 2D-CCFfor a PBR is evaluated as∫ ∞χ(τ,f D ) = s surv (t) s ∗ ref (t − τ)e− j2π f Dt−∞dt (17.1)wheres surv (t) = the complex envelope of the signal collected at the surveillance antennaτ = R B /c = the bistatic time difference of arrival of interest, where R B is the relativebistatic range, or the difference between the two-way path length and the baselinebetween the transmitter (Tx) and the receiver (Rx)f D = v B /λ = the Doppler shift of interest, v B , being the bistatic velocity (i.e., the rateof change of Tx – target – Rx range)The integral is limited in practice to the coherent processing interval (CPI) T int .Assuming that the signals are sampled at frequency f s , equation (17.1) can be easilyexpressed in discrete time notation aswhereN−1∑χ[l,m] = s surv [n] · s ∗ mnj2πref[n − l] · e− N (17.2)n=0N = ⌊T int f s ⌋ = the number of integrated samplesl = the time bin corresponding to time delay τ = l/f sm = the Doppler bin corresponding to Doppler shift f D = mf s /NThe evaluation of the 2D-CCF for a PBR represents one of the most costly operations interms of computational burden [4,27,28]. In fact, the exploited waveform of opportunitytypically has a low power level for radar purposes, so a very long integration time isusually required to obtain an acceptable signal-to-noise ratio (SNR). Moreover, large2-D maps might be required depending on the desired surveillance region extent in bothrange (0, R Bmax ) and Doppler dimensions (–|v B | max , |v B | max ), where R B max and |v B | maxare the maximum relative bistatic range and the maximum bistatic velocity of interest,respectively. This implies that a huge amount of data has to be managed and a largenumber of complex operations has to be performed that might require very fast hardwarefor real-time processing. As a reference, the direct evaluation of (17.2) over N τ time binsand N f Doppler bins requires N(N τ + 1)N f complex multiplications and (N − 1)N τ N fcomplex additions, where, assuming that the 2D-CCF is not oversampled, we haveN τ =⌈RB maxcf s⌉⌈ 2|vB |and N f = maxλ⌉Nf s(17.3)