Radar Losses
5.6 PROCESSOR AND DETECTION LOSSES
The final set of losses we discuss are those associated with the matched filter, the signal processor, and the constant false alarm rate (CFAR) circuitry (Table 5.7).
The mismatch loss associated with the matched filter mainly applies to matched filters for unmodulated pulses or for the chips of phase coded pulses. (See Chapter 10.) This loss occurs because the ideal rectangular pulse generated by the transmitter becomes distorted because of the bandwidth limiting that takes place as the pulse travels through the transmitter and antenna, to and from the target and back through the antenna and receiver to the matched filter. The estimate provided in Table 5.7 was derived by considering rectangular pulse that has been passed through different types of bandlimiting devices. A summary of the results of the analysis is shown in Table 5.8.4 In that table, the N-stage tuned filters are filters of different orders that have a bandwidth equal to the reciprocal of the pulsewidth. As can be seen, the nominal loss is about 0.5 dB.
Table 5.7
Processor and Detection Losses
Source Typical Values (dB)
Matched filter loss
Mismatch loss 0.5
Sidelobe reduction weighting loss 1.5
MTI loss with staggered waveforms 0–1
Doppler filter sidelobe reduction loss 1–3
Range straddle loss 0.3–1.0
Doppler straddle loss 0.3–1.0
CFAR loss 1–2.5
Table 5.8
Matched Filter Mismatch Loss
Input Signal Filter Mismatch Loss (dB)
Rectangular pulse Gaussian 0.51
Rectangular pulse 1-stage single-tuned 0.89
Rectangular pulse 2-stage single-tuned 0.56
Rectangular pulse 3-stage single-tuned 0.53
Rectangular pulse 5-stage single-tuned 0.50
Rectangular pulse Matched 0.00
The sidelobe reduction weighting loss applies to waveforms that use linear frequency modulation (LFM) for pulse compression (see Chapter 10). It is an amplitude taper used to reduce the range sidelobes of the compressed pulse. Since it is an amplitude taper, it also reduces the peak of the matched filter output. The amount of reduction generally depends on the type of weighting and the desired sidelobe levels. A list of various types of amplitude tapers and the associated SNR loss is shown in Table 5.9.5 (For a summary of some common weighting functions, see Appendix B.)
The table also contains the peak sidelobe level associated with the weighting, along with the associated straddle loss. Straddle loss will be discussed later in this chapter. Some common weightings used with LFM are Hamming, Hann, and Gaussian. The other amplitude tapers are often used for sidelobe reduction in antennas and in Doppler processors. Amplitude weighting is not used with phase coded waveforms because the phase coding sets the sidelobe levels. In fact, if amplitude weighting were used with a phase coded waveforms (see Chapter 10), it is likely that the compression properties of the waveform would be destroyed.
With the increasing use of digital signal processors, renewed attention is being given to the use of phase weighting with LFM waveforms to produce nonlinear LFM waveforms [31–34].
These waveforms have a desirable property of reduced sidelobes without the attendant weighting loss. They have the disadvantages of being difficult to generate and process.
Nonlinear LFM is discussed further in Chapter 10.
As is discussed in Chapter 13, for radars that use moving target indicator (MTI) processors, it is common practice to use waveforms with staggered PRIs [35, 36]. That is, waveforms with PRIs that change from pulse to pulse. The reason is that radars that use MTI processors and constant PRIs have frequency responses that have nulls in the range of expected target Doppler frequencies. The range rates corresponding to these nulls are termed blind velocities.
Table 5.9
Amplitude Weighting and Associated Properties
a The parameter k controls pedestal height.
b The parameter α is inversely proportional to sidelobe level.
c The parameter controls the extent of constant level sidelobes, specified in dB, nearest the main lobe.
With a staggered PRI waveform, the nulls are “filled in” by the stagger so that the nulls move out of the range of expected target Doppler frequencies. With staggered PRIs, the MTI frequency response will vary quite a bit (5–10 dB) over the range of velocities. However, for reasonable sets of PRIs, the average response will be close to 0 dB across the frequency range of interest. Thus, the average SNR loss across the frequency range is between 0 and 1 dB, and most of the time is closer to 0 dB than to 1 dB. If the output of an MTI is noncoherently integrated, the noise correlation effect of the MTI will cause an additional loss [6, 7]. Barton
indicates that this loss is approximately 1.5 and 2.5 dB for two- and three-pulse MTIs, respectively [9, p. 384].
As with LFM waveforms, amplitude weighting is also used to reduce the sidelobes of Doppler signal processors. In this case, the sidelobe reduction is needed in order to increase the clutter rejection capability of the Doppler processor. This topic is discussed further in Chapter 13. As with LFM weighting, use of amplitude weighting in Doppler processors causes a loss in SNR (and spectral broadening) relative to the case of no weighting (rectangular weighting in Table 5.9).
A common amplitude weighting in modern radars that use digital signal processing and FFTs is the Chebyshev with a sidelobe level determined by the cutter rejection requirements.
However, Blackman and Blackman-Harris are also used. These amplitude weightings are attractive because of the low sidelobe levels that can be obtained with them.
For illustration, the Doppler response of a 45-dB Chebyshev-weighted FFT processor is presented in Figure 5.13. As discussed previously, we note Figure 5.13 indicates a 1.4 dB-Doppler weighting loss. The Doppler filter responses are dotted, and the straddle loss, which we discuss below, is represented by the heavy black line. The scalloped shape of the straddle loss is why the term scalloping loss is sometimes used. A single Doppler filter centered as 10 kHz is shown by a solid line. For radar range equations purposes, we use the average of the straddling loss (see Table 5.9).
Detection decisions in radars are made by sampling the output of the matched filter or signal processor in range, and sometimes, in Doppler. Generally, the range samples are spaced between ½ and 1 range resolution cell width apart and the Doppler samples are spaced
½ to 1 Doppler resolution cell apart. Because of this finite spacing, it is likely that the samples will not occur at the peak of the range or Doppler response. The result is a loss in SNR.
Fig ure 5.13 Straddle (scalloping) loss—45 dB Chebyshev weighting—N = 16, Fs = 20 kHz.
Fig ure 5.14 Range straddle loss.
Representative curves for this loss, which is called straddle loss, are indicated in Figure 5.14. The dashed curve applies to Doppler straddle loss and to range straddle loss when the radar uses LFM pulses. Nominal values of loss for these cases vary from about 0.3 to 1 dB for typical sample spacings of 0.5 to 1 resolution cell. For unmodulated pulses, or pulses with phase modulation, the loss is somewhat more severe and ranges from about 1 to 2.4 dB.
The final loss in Table 5.9 is CFAR loss. In modern radars, the detection threshold is computed by a CFAR because this circuit or algorithm can easily adapt to different noise (and jammer) environments. The CFAR attempts to determine the desired threshold-to-noise (TNR
—See Chapter 6) ratio based on a limited number of samples of the noise at the output of the signal processor. Because of the limited number of samples used, the threshold will not be precisely set relative to theory. This impreciseness is accommodated by adding a CFAR loss to L.
The precise CFAR loss value depends upon the type of CFAR and the number of noise samples (number of reference cells)6 used to determine the threshold. It also depends upon the desired false alarm probability (Pfa), the detection probability (Pd) (though minimally), and the type of target (Swerling model—0 through 5—see Chapter 3) [37]. The analysis of CFAR loss for particular parameters can become quite involved [8, 38–43].
For preliminary designs, we choose a simpler expression that is applicable in general. One such expression is provided by Hansen and Sawyers for the CFAR loss of a greatest of (GO) cell averaging (CA) CFAR, given a square law detector and a Swerling 1 target [8] is
where Pfa is the desired probability of false alarm, Pd is the desired probability of detection, and M is the number of reference cells used to form the noise estimate. Equation (5.7) can be approximated by
where x is obtained from
It turns out that this equation is also a reasonably good approximation when considering linear and log detectors, SO-CFAR (smallest-of CFAR) and CA-CFAR (cell-averaging CFAR), as well as the other Swerling targets. An example of the dependency of CFAR loss upon various parameters is shown in Figure 5.15 for a CA-CFAR.
Perhaps the simplest approximation for CFAR loss (valid for M > 16) commonly used is provided by Nitzberg [41]
Fig ure 5.15 Loss for a cell averaging CFAR, Pd = 0.9.
To complete our example loss table, we will add processor and detection losses. For the L-band search radar we assume that the radar is using LFM pulses with Hamming weighting to reduce the range sidelobes. Since it is a search radar, we assume that the range samples are spaced one range resolution cell apart. The radar has the ability to use MTI processing, but for the long-range search uses only the LFM pulses (because the targets are expected to be beyond the horizon and we are not considering rain). The radar uses a CA-CFAR with a reference window of 18 range cells. The desired Pfa is 10-6.
The S-band radar also uses LFM with Hamming weighting. Since this radar may need to operate in ground clutter, it uses an MTI processor with a staggered PRI waveform. Analyses
of the frequency response of the MTI indicates that the average SNR loss across the range rates of interest is about 0.2 dB. During search, the radar spaces the range samples one range resolution cell apart. It uses a GOCFAR designed to provide a Pfa of 10–8. The CFAR uses are reference window of 22 cells.
The X-band radar uses phase coded waveforms and a pulsed-Doppler signal processor.
Since the radar has a stringent clutter rejection requirement, the pulsed-Doppler processor uses 100-dB Chebyshev weighting. The radar samples in range at one range resolution cell and in Doppler at ½ Doppler resolution cell. The radar uses a GO-CFAR with 32 reference cells and a Pfa of 10–4. Even though the X-band radar uses Doppler processing, it performs CFAR and detection in only the range direction. Specifically, it performs CFAR and detection on each Doppler cell.
Table 5.10 contains the total losses with the processor and detection losses included.
Table 5.10
Total Losses for the Example
The losses introduced in this chapter are what we consider representative of those one would use in a preliminary radar design or analysis. We did not attempt to present an exhaustive list of losses, as that would require hundreds of pages instead of the few devoted to this chapter. For more detailed expositions of the many loss terms that would need to be considered in a final radar design, the reader is directed to [1, 2, 4, 6–8, 26, 35, 39–48]. A very good reference is Barton’s 2013 text [9], which contains approximately 200 pages dedicated to the discussion of losses. Another notable reference is Blake [28].