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Figure 2.19 Comparison of the time‐domain pulse waveforms: (a) single‐tone pulse, (b) LFM (Chirp) pulse.

      The common waveform is the LFM pulse, also known as the chirp pulse, whose waveform is shown in Figure 2.19b. In practice, this waveform is repeated in every T_PR intervals for most common radar applications, especially for localization of targets in the range. T_PR is called the pulse repetition interval (PRI) or pulse repetition period. The inverse of this interval gives the pulse repetition frequency (PRF), defined as

(2.60)

      The mathematical expression of the upward chirp signal whose frequency is increasing as time passes along the pulse is given as

      (2.61)

      where n is an integer, τ is the pulse width, and K is the chirp rate. The instantaneous frequency of the pulse is f_i(t) = f_o + Kt. It is also possible to form another LFM pulse by decreasing the frequency along the pulse width as shown below:

      (2.62)

      For the downward chirp pulse, the instantaneous frequency is then equal to f_i(t) = f_o − Kt.

To demonstrate the broad spectrum of the LFM waveform, the FT of single‐tone and LFM pulse signals in Figure 2.19 is taken and plotted in Figure 2.20. It is clearly seen from this figure that chirp signal provides much wider bandwidth when compared to constant‐frequency pulse.

      In radar applications, LFM pulse waveforms are mainly utilized in finding range profiles, and also for synthetic aperture radar (SAR) and ISAR processing as will be discussed in Chapters 3 and 6, respectively.

Figure 2.20 Comparison of the spectrum of (a) single‐tone pulse and (b) LFM pulse. Although both signals use the same time duration, frequency bandwidth of the Chirp waveform is much wider than the single‐tone waveform.

2.7 Pulsed Radar

Pulsed radar systems are commonly used especially in SAR and ISAR systems. They transmit and receive a sequence of modulated pulses. Therefore, the same type of pulse is repeated in every T_PR interval, or, as it is called, PRI as depicted in Figure 2.21. The range information can be gathered from the two‐way trip time (or time delay) between the transmitted and received pulses. Pulsed radar systems have the ability to measure both the range (the radial distance) and the radial velocity of the target.

      2.7.1 Pulse Repetition Frequency

As pulses are repeated in T_PR, the corresponding PRF of the radar is as given in Eq. 2.60. PRF gives the total number of pulses transmitted in every second by the radar. In radar applications, PRF value can be quite critical as it is linked to maximum range of a target, R_max, and the maximum Doppler frequency, f_D,max (so the maximum target velocity v_max of the target), that can be detectable by the radar. The use of PRF in ISAR range‐Doppler processing will be explored in Chapter 6.

      2.7.2 Maximum Range and Range Ambiguity

As calculated in Eq. 2.58. the range resolution is proportional to the pulse duration as Δr = c·τ/2. Therefore, the smaller the pulse duration, the finer the range resolution we can get. On the other hand, maximum range is determined by time delay between the transmitted and received pulses. Since the pulses are repeated for every T_PR seconds, any received pulse that is reflected back from a target at R distant on the range should arrive before the next pulse is transmitted to avoid the ambiguity in the range, that is,

      (2.63)