Principles of Modern Radar - Volume 2 1891121537

40 CHAPTER 2 Advanced Pulse Compression Waveform Modulationsa wideband LFM waveform on receive and converts relative time delay into frequency. Afilter limits the frequencies passed to the ADC and defines the processing bandwidth. Thefilter bandwidth is chosen to be less than the transmit bandwidth. The sampled waveformis compressed using a DFT, and the resultant range resolution is inversely proportional tothe transmit bandwidth. The filter does limit the size of the range window to a fraction ofthe pulse width; however, in high-resolution systems the range to the target or imaged areais known with sufficient precision to position the window about the region of interest.2.3 STEPPED CHIRP WAVEFORMS2.3.1 IntroductionAs the name implies, a stepped chirp waveform [7–15] consists of LFM pulses that arestepped in frequency to reduce the waveform’s instantaneous bandwidth. On transmit,the waveform’s reduced intrapulse bandwidth lessens dispersion effects in some systems(e.g., a wideband, phased array radar that does not employ time delay units). On receive,the stepped chirp waveform supports a lower sampling rate, similar to stretch processing,but does not limit the size of the range window. The waveform is composed of N sc LFMpulses, each with swept bandwidth, β, that are separated in time by the pulse repetitioninterval (PRI) and that partially overlap or are contiguous in frequency. Successive pulsesare separated in frequency by a constant offset, f . In practice, the number of pulses usedto construct the waveform is relatively small (e.g., 1 < N sc ≤ 10).On receive, each pulse is mixed to baseband or to a common intermediate frequencyand sampled a rate sufficient to support the intrapulse bandwidth. The composite waveformis reconstructed in the signal processor by shifting the returns by the appropriate frequencyoffset and stitching them together to form a coherent wideband signal. The bandwidth ofthe composite waveform, β sc ,isβ + (N sc − 1) f . The requirements placed on the ADCsample rate are driven by the single pulse bandwidth, β, and the range window extentis limited by the PRI and not the low pass filter as in the case of stretch processing. Asexpected, the waveform’s reconstruction does increase the computational requirementslevied on the signal processor. With a stepped chirp waveform, the pulse repetition frequency(PRF) must be increased by a factor of N sc to maintain the effective rate requiredto support multiple-pulse, coherent processing (e.g., SAR or Doppler processing) [10].2.3.2 Transmit WaveformThe stepped chirp waveform consists of N sc LFM pulses, each with swept bandwidth β,that are shifted in frequency from pulse-to-pulse by a constant frequency offset f wheref ≤ β as illustrated in Figure 2-8. The n-th transmit pulse is modeled as(x tx (t) = cos 2π ( f 0 + (−(N sc − 1)/2+n) f ) t + π β (t − τ ) ) 2(2.53)τ 2where 0 ≤ t ≤ τ, and 0 ≤ n ≤ (N sc − 1), and f 0 is the transmit center frequency. Timein equation (2.53) is referenced to the beginning of each pulse, but the individual pulsesare separated by the PRI, T , where T >τ. In the text, an odd number of pulses is usedto develop a mathematical model, but a similar analysis applies for an even number ofpulses.