Input Energy of Surface Clutter

3.3.4.1 Input Surface Clutter Energy in Pulsed Radar

For pulsed radar, the surface clutter energy at the output of the antenna from each ambiguous area A_{c i} is found from (3.4) using c i from (3.23). These contributions are summed as in (3.5) to yield the total input clutter energy:

 

The problem in applying this expression to a calculation of detection range R_mc is that the range dependence of the clutter pattern-propagation factor F_{c i} and the grazing angle _i preclude writing of a closed-form expression for R_mc, even when the target pattern-propagation factor is constant.

For example, Figure 3.2 shows a plot of signal and clutter energy levels for an LPRF radar with a pulsewidth _n = 1 s. The radar achieves detection in noise at R_m = 93.2 km, but as the target, as it flies along the beam axis near zero elevation angle. Reflection lobing and possible diffraction loss is omitted from the calcula-tion. As the target comes within the clutter horizon at 15 km the clutter rises, re-ducing the ratio of signal to interference below the required D_x = 20 dB. Within 1-km range, the clutter falls below the elevation beam, restoring the necessary detec-tion margin. However, the range interval within which detecdetec-tion is impaired

can-Radar Equations for Clutter and Jamming 69

not be found from any simple equation, and reading from the graph or an equiva-lent root-finding algorithm is necessary. The required improvement factor to avoid impairment, I_m  18 dB, can be determined from the graph at R = 5 km.

3.3.4.2 Input Surface Clutter Energy in CW Radar

Although unmodulated continuous-wave (CW) radar is less widely used than in the past, equations developed in this section also serve as starting point for simpli-fied clutter calculations in HPRF pulsed Doppler radar. Surface clutter in HPRF radar and in phase-modulated CW radar can be described by the expressions given in Section 3.3.4.1 for pulsed radar, with _n representing the width of the range resolution cell resulting from a modulated waveform. When several such ambigui-ties contain clutter, the expressions developed here, modified as in Section 3.3.4.3, provide an alternative method that is often more convenient.

Surface clutter in unmodulated CW radars is beamwidth-limited, according to (3.19). At low grazing angles the reflectivity, pattern-propagation factor, and im-provement factor vary with range within the beam footprint in a way similar to those in different ambiguities of pulsed radars. The following expression for clut-ter energy density as a function of range replaces C_i in (3.5), as noted by Blake

Figure 3.2 Input energy levels for the example radar using LPRF waveform: target signal E on beam axis (heavy sold line), interference I0e (dashed line), detection threshold 20 dB above interference (dash-dot line), and noise (light solid line at N0 = 198.6 dBJ).

where dc is the RCS density in m² per m of range, which varies with R_c, along with the term F_c⁴ in brackets. In the CW radar case, the range-dependent response factor F_rdr and attenuation L_c are taken as unity and omitted from (3.28), since STC cannot be used, the transmission does not eclipse the signal, and short-range clutter is dominant. The clutter RCS density is found from (3.19) with csc  1:

sin 2

Figure 3.3 shows the variation in the term dC for a typical CW radar charac-terized by: P_av = 100W, t_f = 0.01s, beamwidths a = e = 1.7, height hr = 12m above a land surface for which _h = 1m and  = 12 dB. The solid line applies to a horizontally directed beam. Ranges shown by vertical dotted lines are R₁, the limit of the near region, and R_a, the range at which the two-way antenna gain for a hori-zontal beam is 1/e at the surface:

8ln 2 ^r

a

e

R  h

 ^(3.31)

In the example, Ra = 1,177m. The major contribution to clutter input comes from a region 0.5 < R_c/R_a< 2, and contributions for R_c/R₁ > 1 are negligible.

0 1000 2000 3000 4000 5000 6000

150 example CW radar: horizontal beam (sold line), and beam elevation to e/3 (dashed line).

Radar Equations for Clutter and Jamming 71 For a low-sited radar, where the flat-Earth approximation is valid over most of the near region R_c < R₁, the clutter pattern-propagation factor is

2

where b is the upward tilt of the beam axis. The summation for input clutter ener-gy C0 in (3.27) is replaced by integration:

Only terms within the integral are functions of clutter range R_c. The rapid reduc-tion in f_c as R_c  0 makes the result insensitive to the lower limit of the integra-tion, but for convenience it is taken here as the height h_r above the surface on which the antenna is mounted.

For b = 0 (horizontal beam), the integral in (3.33) reduces to  4R_a³,

This can be simplified for equal transmitting and receiving gains given by

4 10.75 simple expression for CW radar clutter energy:

 

clutter-to-noise ratios define the challenge in design of surface-based CW radars for air

de-fense applications. Equation (3.36) shows that the clutter input, for a beam di-rected at the horizon, can be reduced only by increasing the height h_r of the anten-na above the surface, decreasing the ratio of elevation to azimuth beamwidth or transmitting less energy (which also reduces signal energy).

Clutter input can be reduced by tilting the antenna axis upwards. The integral in (3.33) for arbitrary beam-axis elevation _b reduces to

2 2 2 clutter energy for an elevated beam to that for a horizontal beam, given for clutter by (3.36), may be expressed by including a beam elevation factor F_b⁴, plotted in Figure 3.4 for both the Gaussian beam and the beam pattern of an aperture with cosine taper. Also shown is the factor F⁴ for targets at elevation _t = 0. The clutter is slightly larger for the Gaussian beam with small elevation because the mainlobe pattern decays more slowly, but the more realistic cosine illumination generates sidelobes that govern the factor for axis elevations above one beamwidth. The improvement in signal-to-clutter ratio for elevated beams is the difference be-tween the clutter and target factors (e.g., 6.6 dB for _b = 0.33_e).

The beam elevation factor in Figure 3.4 for the cosine taper was calculated by numerical integration. It shows that the clutter energy expressions developed for the Gaussian beam are adequate for beams elevated less than about one

Figure 3.4 Beam elevation factor Fb4 for surface clutter (heavy lines) and F⁴ for targets at the horizon (light lines), with Gaussian beam (solid lines), and beam from cosine-illuminated aperture (dashed lines).

Radar Equations for Clutter and Jamming 73

3.3.4.3 Input Surface Clutter Energy in Pulsed Doppler Radar

In pulsed Doppler radar, a range gate matched to the pulse width  and centered at range R_c from the transmitted pulse passes clutter energy given by integration of the clutter density over ambiguous ranges (R_c + iR_u)  c/4. Figure 3.5 shows the clutter energy in each of nine 1-s gates between successive pulses at t_r = 10 s, including the first four range ambiguities and using the clutter density shown in Figure 3.3. The rectangular gate, sampling over 75m from its center, passes clut-ter energy given by

C₀ (dBJ) = density dB(J/m) + 10 log(150m) = density dB(J/m) + 21.8 dB(m) at each ambiguous range. The clutter energy varies by 6 dB over the nine gates (the potential tenth gate is totally eclipsed by the transmitted pulse). The average over the gates is 10 dB below the level for the CW radar, as expected for a duty cycle D_u = /t_r = 0.1. For gates 1 and 2 the beam does not reach the surface until the second range ambiguity, giving lower clutter than in gates 3–9.

3.3.5 Detection Range of Surface-Based CW and HPRF Radars

In document Radar Equations for Modern Radar (Page 88-93)