RADAR EQUATIONS FOR CLUTTER AND JAMMING
3.4 DETECTION IN VOLUME CLUTTER
3.4.6 Volume Clutter in CW and PD Radars
Volume clutter in CW radars that use phase modulation to obtain range resolution
n is described by the expressions given in previous section for pulse radar. The cross-range dimension of the resolution cell in azimuth is determined by the beamwidth, and in elevation either by the beamwidth or the cloud thickness, as in (3.46).
In unmodulated CW radar, the range dimension of the clutter is the length of the path that lies within the cloud. Blake [3, p. 300, Eq. (7.18)] presents an expres-sion using a triple integral (in range, azimuth, and elevation) to give the clutter energy for pulsed radar where v remains constant over a hemisphere, but the ele-vation limits of the cloud are, in general, range-dependent.
3.4.6.1 Volume Clutter Energy for CW Radar
Useful insight into CW radar performance in volume clutter may be gained under the assumption that the cloud extends beyond the radar beam in both angle coor-dinates, for which case the surface-clutter expressions (3.29)–(3.33) can be modi-fied for volume clutter to yield
2
Consider again the X-band radar described in Table 3.1, operating in the rain-fall rate of 1 mm/h assumed for Figure 3.7, but with a CW waveform and circular polarization. The clutter density resulting from (3.55) is shown in Figure 3.8. The peak density occurs at range Rff, and most of the clutter energy comes from the volume within 6Rff in front of the antenna.
The clutter reflectivity v is usually assumed constant within the cloud, and is included in Kcv, but may be moved within the integral of (3.56) if it varies in some
Radar Equations for Clutter and Jamming 83
known way with range. Atmospheric attenuation is included within the integral because it may become significant for large Rcmin. As with the surface-clutter analysis, the factors Frdr and Lcc are assumed unity for CW radar and omitted from the equations. volume clutter pattern-propagation factor can be expressed as
c
Redefining the terms Lc and Frdr to be weighted averages over the limits of integration, they may be placed outside the integral, which becomes
0 200 400 600 800 1000 1200 1400 1600 1800 2000 0
the radar. The atmospheric attenuation Lc 1, and input clutter energy is then
antenna parameters other than the polarization factor.3.4.6.2 Detection Range for CW Radar in Volume Clutter
Combining (1.26) and (3.58), the signal-to-clutter energy ratio can be written for the general case as Note that the terms within the brackets are independent of target range R.
As with surface clutter, there are two special cases for which closed-form ex-pressions can be written. Where clutter residue is dominant, C0e >> N0, we solve for the range Rmc at which E/C0e = Dx to obtain
Radar Equations for Clutter and Jamming 85
The final approximation in (3.62), for clutter enveloping the antenna, can be expressed directly in terms of the radar parameters by substituting (3.57) to obtain
combination of changing clutter volume and far-field range, the latter affecting the response to nearby clutter.The closed-form solution for the second case is based on the assumption that C0 e = N0:
3.4.6.3 Example of CW Radar in Rain
For a surface-based X-band CW radar with parameters listed in Table 3.1, with circular polarization in 1-mm/h rainfall rate, the received clutter energy given by (3.60) is C0 = 9 1014 J. An improvement factor Im = 60 dB would reduce the effective clutter spectral density to C0 = 9 1020 W/Hz, compared to thermal noise with density N0 = 1.4 1020 W/Hz. The resulting C0 e/N0 = 8.2 dB is high enough to permit (3.63) to be used to find an approximate range:
2 4 2
4 rdr 18
2 2 2
15.3 p m 5.6 10 (m); 49 km
mc mc
a e v x pc mc
F F F I
R R
D F L R
The accurate range is slightly less, because 8.2 dB is not quite high enough to allow noise to be neglected. The result is independent of the beam elevation, be-cause the clutter arises within a few hundred meters from the antenna. The results given above are based on Im = 60 dB, F2pc = 0.01, and F2p = 0.5, for circularly po-larized antennas having the same sense (e.g., right-hand polarization for both transmitting and receiving). Much larger Im values are generally available in CW radar designed for use against aircraft targets: values in excess of 100 dB were shown necessary in Section 3.3.5 for rejection of surface clutter. That level of Doppler performance would preserve the thermal-noise detection performance when operating in rain, even without the use of circular polarization.
3.4.6.4 Volume Clutter Energy for PD Radar
Volume clutter energy for pulsed Doppler radar may be found using (3.49), but when several range ambiguities are occupied by clutter it may be easier to apply the CW radar equations with a correction for the effects of duty cycle and pulse compression. The clutter, when averaged over all range gates in the pulse repeti-tion interval (PRI), is given by adding duty cycle Du as a factor in the numerator of Kcv in (3.28) and (3.56)–(3.58). The pulse compression ratio /n is applied to the denominator of Kcv when a modulated pulse is used. The factor of 2 in the nu-merator of (3.57) is omitted in calculating the far-field range of PD radar anten-nas, because the transmitting and receiving beams are formed by the common antenna at a range Rff = wh/ and the beams coincide at all ranges. The resulting expression for average PD volume clutter energy, when the cloud envelopes the radar, becomes:
Radar Equations for Clutter and Jamming 87
This average clutter energy is added to the noise spectral density to obtain I0e for use in calculating an average detection range from (3.12).
The clutter density as a function of range is shown in Figure 3.7 with adjust-ment for duty cycle, pulse compression, and aliasing at the unambiguous range Ru. The clutter competing with the target varies about that average, as shown in Fig-ure 3.9. The maximum value lies in the first gate that includes the range Rff where the beam is fully formed. Clutter in that gate is 65 dB above noise level, requir-ing an improvement factor Im > 75 dB to avoid reduction in target detectability.