Principles of Modern Radar - Volume 2 1891121537

17.2 Evaluation of the 2D-CCF for the Passive Radar Coherent Integration 74917.2.1 Efficient Implementations of the 2D-CCF Based onthe Fast Fourier TransformTwo alternative efficient implementations can be obtained by rearranging the factors in(17.2) according to different criteria and exploiting the well-known fast Fourier transform(FFT) algorithm to evaluate the discrete Fourier transform (DFT).Correlation FFTThis algorithm is based on the observation that, at the m-th Doppler bin, the samplesalong time of the 2D-CCF correspond to the samples of the cross-correlation betweenthe reference signal and a Doppler shifted version of the surveillance signal s m [n] =s surv [n]exp(− j2πmn/N):N−1∑χ[l,m] = s m [n] · s ∗ ref [n − l] = C m[l] (17.4)n=0A saving in computation is obtained by evaluating such cross-correlation in the frequencydomain asC m [l] = IDFT{DFT { s m [n] }( DFT { s ref[n] }) ∗} = IDFT{Sm [k]S ∗ ref [k]} (17.5)For each one of the N f Doppler filters, an inverse DFT (IDFT) is evaluated and N − N τoutput samples are discarded. Figure 17-3a is a sketch of the resulting algorithm. Wenotice that, for a given m, the DFT S m [k] ofs m [n] can be simply obtained as a circularshift of the samples of S surv [k] = DFT{s surv [n]}. Consequently, the DFTs of the referenceand surveillance signals can be evaluated just once and, at each subsequent iteration, onlyS surv [k] = FFT{s surv [n]}S ref [k] = FFT{s ref [n]}for m = 1 : N fS m [k] = S surv [(k + m) modN]for l = 1 : N tx l [n] = s surv [n]s* ref [n − l]X l [m] = FFT{x l [n]}FIGURE 17-3Sketch of efficientalgorithms for the2D-CCF evaluation.(a) Correlation FFT.(b) Direct FFT.X [k] = S m [k]S* ref [k]Discard SamplesC m [l ] = IFFT{X[k]}Discard Samples(a)(b)