Principles of Modern Radar - Volume 2 1891121537

44 CHAPTER 2 Advanced Pulse Compression Waveform Modulationsare shifted by their respective frequency offsets, pf . Shifting the n-th spectrum by2π pf /F s()()∣2π pfY n ω − = exp (− j2π f 0 t d ) exp (− j2π pft d )F ∣ X 2π pf ∣∣∣2ω −s F sand after reducing terms()(2π pfY n ω − = exp (− j2π f 0 t d )F ∣ X ω −sexp (− j (ωF s − 2π pf ) t d ) (2.69)2π pfF s)∣ ∣∣∣2exp (− jωF s t d ) (2.70)where (−π + 2π pf /F s ) ≤ ω ≤ (π + 2π pf /F s ). Note that the phase term 2πpft din equation (2.68) has been removed by the shifting operation. Its removal is necessary tocoherently combine the pulses. The match filtered and frequency shifted spectra containa constant phase term that is a function of f 0 and t d and a linear phase term proportionalto t d . The linear phase term positions the response in time after the inverse transform isapplied.The spectra in equation (2.70) are a result of applying the DTFT. To compute thespectra inside a signal processor, the DFT is applied. The K -length DFT is constructedby sampling the spectrum in equation (2.70) at ω k = 2πk/K , −K /2 ≤ k ≤ K/2 − 1 toyieldY n(2π k K−2π pfF s)(= exp (− j2π f 0 t d )∣ X 2π k K)∣2π pf ∣∣∣2− exp(− j2π k )F s K F st d(2.71)To align the pulse spectra, the frequency step size must be an integer multiple of the DFTbin size. The size of a DFT bin is found by computingδ f DFT = F sKThe frequency step size, f , is therefore constrained to be(2.72)f = Pδ f DFT (2.73)where P is an integer. Substituting equations (2.72) and (2.73) into equation (2.71)(Y n 2π k )(pP− 2π = exp (− j2π f 0 t d )K K∣ X 2π k )∣pP ∣∣∣2− 2π exp(− j2π k )K KK F st d(2.74)where (−K/2 + pP) ≤ k ≤ (K/2 − 1 + pP). Shifting the spectra has increased theK + (N sc − 1) Psize of the DFT to K + (N sc − 1) P, and the effective sample rate is F s .KZero padding may be applied in the frequency domain to force a power of 2 size FFT orto further interpolate the match filtered time-domain response.At this point, the spectra have been shifted, and a matched filter has been appliedto each pulse. The next step is to properly stitch together the shifted spectra to form acomposite spectrum that achieves the desired main lobe and sidelobe response when theinverse transform is applied. The stitching process is illustrated in Figure 2-9 using three