【正文】 olution requires a long coherent signal. To minimize eclipsing requires a short transmission time (unless the radar is especially designed to transmit and receive simultaneously, as in CW radar). A solution that meets both requirements is a coherent train of pulses. The basic type of such a signal was discussed in Sections and , where the train was constructed from identical constantfrequency pulses. In many practical cases the pulses are modulated and are not identical. Modulation produces wider bandwidth, hence pulse pression. The identicalness is violated by even the simple introduction of interpulse weighting, used to lower Doppler sidelobes. In some signals, significant diversity is introduced between the pulses in order to obtain additional advantages, such as lower delay sidelobes or lower recurrent lobes.In this and the following chapters we extend the discussion in two directions: adding modulation and adding diversity. Adding modulation but keeping the pulses identical still allows us to use the theoretical results regarding the periodic ambiguity function (Section ) and to obtain analytic expression for the ambiguity function (within the duration of one pulse, ., |τ| ≤ T ). Adding diversity in amplitude (., weighting) or by different modulation in different pulses within the coherent train will usually require numerical analysis, except for a few simple cases in which theoretical analysis is available. Such is the case for the subject of this chapter—a coherent train of LFM pulses—probably the most popular radar signal in airborne radar (Rihaczek, 1969。 Stimson, 1983。 Nathanson et al., 1991). COHERENT TRAIN OF IDENTICAL LFM PULSESA train of identical linearFM pulses provides both range resolution and Doppler resolution—hence its importance and popularity in radar systems. Its ambiguity function still suffers from significant sidelobes, both in delay (range) and in Doppler. Thus, modifications are usually applied to reduce these sidelobes. In this section we consider the basic signal without modifications. The coherency is reflected in the expression of the real signal, given by ()where the plex envelope is a train of N pulses with pulse repetition period Tr, ()The uniformity of the pulses is expressed by assuming that un(t) = u1(t). The LFM nature of a pulse of duration T is expressed in its plex envelope, ()An example of a real signal is shown in Fig. , where all the pulses begin with the same initial phase. This is not a mandatory requirement for coherence. Coherency can be maintained as long as the initial phase of each pulse transmitted is known to the receiver.Changes in phase from pulse to pulse will be expressed in the plex envelope of the nth pulse as ()As long as T Tr /2 (which will be assumed henceforth), the additional phase element has no effect on the ambiguity function for |τ| ≤ T . It will only affect the recurrent lobes of the AF: namely, over |τ 177。 nTr| ≤ T (n = 1, 2, . . .). The additional phase term can already be considered as some sort of diversity, but one that affects only recurrent lobes.To get an analytic expression for the partial ambiguity function (AF) of our signal, we start with the AF of a constantfrequency pulse, apply AF property 4 to it, and obtain the AF of a single LFM pulse (as done in Section ): (single pulse) ()where ()The first equality |χ(τ, ν)| = |χT (τ, ν)| stems from the fact that T Tr /2. To describe the AF of the train, for the limited delay |τ| ≤ T , we now apply the relationship of the periodic AF: (train of pulses) ()Using () in () yields the ambiguity function of a train of N identicalLFM pulses: ()We will demonstrate the signal and its AF using a train of eight LFM pulses. The time–bandwidth product of each pulse is 20 and the duty cycle is T/Tr = 1/9。 The phase and frequency history are given in Fig. . The partial ambiguity function, plotted in Fig. , is restricted in delay to 177。1 pulse width, and in Doppler to 10/8 = the pulse repetition frequency (PRF, 1/Tr). Note the first null in Doppler that occurs at ν = 1/(NTr) = 1/(8Tr) = 1/Tc, where Tc = 8Tr is the coherent processing time. This improved Doppler resolution is the main contribution of the pulse train. Note also the first recurrent Doppler peak at ν = 1/Tr. It is difficult to note from the plot, but the recurrent Doppler lobe is slightly lower and slightly delayed pared to the main lobe. The zeroDoppler cut of Fig. (., the magnitude of the normalized autocorrelation function) is identical to the cut that would have been obtained with a single LFM pulse. This is a property of all trains of identical pulses. An AF plot extending beyond the first delay recurrent lobe is plotted in Fig. , and an AF plot extending over the entire delay span appears in Fig. . The Doppler span displayed was doubled in Fig. , showing two Doppler recurrent lobes. FILTERS MATCHED TO HIGHER DOPPLER SHIFTSThe ambiguity function displays the response of a filter matched to the original signal, without Doppler shift. As shown in Fig. , a coherent pulse train achieves good Doppler resolution, and a matched filter will produce an output only when the Doppler shift of the received signal is within the Doppler resolution. A typical radar processor is likely to contain several filters, matched to several different Doppler shifts. In a coherent pulse train, implementing such a processor is relatively simple, especially if the intrapulse modulation is Doppler tolerant, as LFM is.The principle of Doppler filter implementation is summarized in Figs. and . Figure shows that each pulse is processed by a zero Doppler matched filter. The N outputs from the N pulses are then fed to an FFT. The first output of the FFT is equivalent to a zeroDoppler filter. For that first output the FFT c