The specifics of functioning of the active homing head exposed to ground surface clutter

Description of ground clutter signal

Radar reflections from the ground surface are characterised by specific radar cross-section σ0 (RCS of ground surface area unit) and illuminated segment area S.

During operation of an active HH (AHH), the ground clutter spectrum coincides in form with the square of AHH antenna radiation pattern (ARP) by power [1] and is characterised by Doppler shift of the echo signal frequency and bandwidth magnitude. The Doppler shift depends on the missile speed, the angle between velocity vector and the direction of ARP maximum, whereas the spectrum bandwidth also depends on the ARP main beam width. In certain cases the clutter spectrum may comprise components associated with reflections from the ground surface segment which is below the missile (altimeter reflections), whose level is determined by the level of ARP side lobes (background). The Doppler frequency shift is determined by the magnitude of missile velocity vector projection on the altimeter segment direction.

Let us consider in more detail an estimate of clutter signal spectrum using a typical pattern of missile closing in on target, as shown in Fig. 1.

In estimation of ground clutter effect, of major importance is the distance of clutter spectrum from Doppler frequency shift of the wanted signal f_дц determined by the formula

 (1)
where λ - wavelength of radiated oscillations, V_p, V_ц - speeds of missile and target.

Since during ground surface illumination the scattering elements are basically motionless, Doppler shift of the centre frequency of clutter signal spectrum f_дп depends on missile motion and ARP centre direction, which may not coincide with the direction to target,

 (2)
where Δφ - angle of ARP centre deflection from the “missile – target” line.

The bandwidth of the main lobe of clutter signal spectrum is mainly determined by the maximum difference between V_p velocity vector projections on the directions of scattering elements, distanced from one another, that are inside or on the edges of ARP main beam, this ARP being conditioned by dissimilar magnitude of the angle between V_p velocity vector and directions towards scattering elements on the ground surface.

Fig. 2 shows the shaping pattern of the maximum difference ΔV between V_p velocity vector projections for a typical situation when Δφ=0 and φ>θ/2.

Considering the small value of angle θ/2, difference ΔV at the ARP edges is calculated by the formula

 (3)

The difference ΔV between V_p velocity vector projections is matched by the difference between Doppler frequency shifts Δf, determining clutter signal bandwidth,

 (4)

Taking into account the ARP width ratio, expressed through λ and antenna reflector diameter d,

 (5)

we obtain a simpler expression for the clutter signal bandwidth in the considered case

 (6)

At small values of angle φ, when φ<θ/2, the bandwidth diminishes, but it does not tend to zero at φ=0, when the difference between V_p velocity vector projections at the ARP edges becomes equal to 0.

For the case of φ=0 the maximum value of the difference between V_p velocity vector projections is observed between scattering elements located in the ARP centre and at the edges, i.e.

 (7)

Accordingly, the clutter signal bandwidth becomes equal to

 (8)

As an example for clutter spectrum estimation, let us take typical values V_p=2000 m/s, d=3×10^-1m, λ=3×10^-2m и θ=0,1 rad (5,73°), Then, depending on angle φ clutter signal’s Doppler shift f_дц, calculated by formula (2) at Δφ=0, and bandwidth Δf, calculated by formula (6) for the case of φ>θ/2 and (8) for the case of φ=0, will have the following values (see Table 1).

Here, the case of φ=0 is matched by the least clutter signal bandwidth, but this may only occur in case of straight-line flight of the missile and zero values of AHH bearing angles, which can hardly be achieved in practice. When firing at high-speed targets, the real values of angle φ may reach 60° and more, with the clutter signal bandwidth increasing simultaneously with decrease of the level of its power spectral density

When firing at low-speed low flying targets and shaping the most energetically favourable missile flight trajectories, there is also an increased value of angle φ at the moment of target seeking by AHH. When using a comb of narrow-band filters, e.g., such whose bandwidth is acceptable for a low-speed target, as specified in [2], i.e., 10 Hz or less, this enables to considerably improve the conditions for target detection against the background of clutter. Using a filter comb under the conditions of example being considered (Table 1), given a uniform clutter spectrum, will lead to clutter power reduction in the passband of each one of the 10-Hz filters by 13 dB at φ = 0 and more than 25 dB at φ > 15°.

An estimate of radar reflections power is drawn based on determining RCS of the illuminated ground surface segment, which is characterised by its specific value σ₀, depending on surface type and glancing angle ψ.

A typical variation of specific RCS for the centimetre wavelength range [2] is given in Fig. 3.

For the surface at angle ψ≈20°, σ₀≈10^-2 (-20 dB) and with angle value reduced to ψ=7° its value becomes equal to 10^-3 (-30 dB), with further decrease along with angle ψ diminishing. Therefore, to eliminate clutter signal effect, it is desirable that missile trajectory at the final leg of the flight is shaped below the target, which is not always feasible.

Clutter effect depending on signal type

The ratio of signal and clutter powers at the input of AHH receiver is proportional to the ratio of the RCS of target σ_ц and that of the illuminated ground surface segment σ_зп and inversely proportional to the fourth power of the ratio of ranges to the ground surface segment R_зп and the target R_ц

 (9)

The RCS of the segment is equal to σ_зп=S·σ₀, where S – segment area calculated depending on the sensing signal type.

For a continuous signal, the illuminated segment within the main beam limits is approximated by a tetragon with length AB and width EF≈PC·θ= H_p θ/sin ψ (Fig. 4).

Length AB is calculated through missile flight altitude H_p by the formula

 (10)

Due to smallness of the ARP width value, we take sin sinθ≈θ, cosθ≈1, then

 (11)

The illuminated segment area within the main beam limits is approximately equal to

 (12)

As an example, let us calculate the input signal/clutter ratio for the considered case under the following conditions: H_p=20 km, target altitude Н_ц=5 km, θ=10^-1 rad, ψ=30°, σ₀=2·10^-2 (-17 dB), σ_ц=20 m².

Under such conditions R_зп/R_ц=H_p / (H_p - Н_ц) = 4/3.

Therefore, the signal/clutter ratio will be

 (13)

To detect signal against the background of clutter and receiver’s intrinsic noise at the processing system output, an output signal/(clutter + noise) ratio of at least 15 dB is normally required. Therefore, in such conditions, the factor of signal/ clutter ratio improvement by the input signal processing system shall be no less than 55 dB

Let us estimate a possibility to obtain such a factor. In addition to the initial data used for the calculation of ratio (13), let us assume that the target has speed V_ц = 200 m/s, target signal spectrum W_ц( f ) has bandwidth Δf_ц=10 Hz, and the ARP, by power in the sum channel as a function of deflection angle β, under uniform distribution of field across the aperture, is specified by the formula [3]

 (14)

and angles φ and γ are equal, accordingly, φ = 15° and γ = 30°. It follows from Table 1: f_дп = 128,8 kHz, Δf=3,4 kHz. Using formula (1), we have f_дц=11,5+f_дп=11,5+128,8=140,3 kHz.

The clutter spectrum coincides in form with the ARP square and is defined by the formula

 (15)

The clutter spectrum width by zero level in the main beam is equal to 2Δf=6,8 kHz, and the width of each side lobe is equal to Δf=3,4 kHz. Fig. 5 shows spectra of the clutter and target signals W_ц(f).

Target spectrum is offset in frequency by 11,5 kHz, relative to the clutter spectrum, therefore target spectrum occurs on the third right-hand side lobe of the clutter spectrum, which occupies frequency range from f_дп + 3Δf = f_дп + 10,2 kHz до f_дп + 4Δf = f_дп + 13,6 kHz. It is known that the level of ARP third side lobe G(β) is equal to –20.8 dB. Since the clutter spectrum is equal in form to ARP square, the level of the third side lobe of the clutter spectrum is equal to –41.6 dB.

Signal processing system, in the form of a comb of narrow-band filters with bandwidth of 10 Hz each, will discriminate target signal and attenuate clutter signal, firstly, by 10 lg (Δf / Δf_ц) = 10 lg (3400 /10) = 25,3 dB, and secondly, due to the difference between Doppler frequencies of the clutter and target signals, the clutter spectrum level on the target frequency is attenuated by at least 41.6 dB. Hence, the processing system will improve signal/noise ratio by at least 25.3 + 41.6 = 66.9 dB, which is higher than the required improvement factor of 55 dB. Under the conditions of the considered example, a system of narrow-band Doppler filters will ensure target signal discrimination against the background of reflections from the ground surface, because target spectrum, by frequency, lies beyond the main lobe of clutter signal spectrum.

When using a quasi-continuous pulse (QCP) signal with pulse repetition period T and range resolution δ_R the area of illuminated segment S_кн within the main beam limits is calculated as a sum of the areas of strips with length δ_R/cos ψ and width EF ≈ H_p θ/ sin ψ each, filling segment AB with increment cT/(2cos ψ) (c - radio propagation velocity)

 (16)
where S_н - area of illuminated segment for a continuous signal (12).

Let us take for the conditions of the previous example the following input parameter values: δ_R = 150 m, T = 3,33·10^-6 s (300 kHz), c = 3·10⁸ m/s. Then, under quasi-continuous signal, the illuminated area will change by the factor of

 (17)

as compared with the continuous signal, and the input signal/noise ratio will be

 (18)

In this case the factor of signal/clutter ratio improvement by the input signal processing system shall be no less than 50.2 dB.

In this way, for quasi-continuous signal the requirements to the input signal processing system are somewhat eased, and in the case when target spectrum is located outside of the clutter spectrum main lobe, processing with the use of a narrow-band filter comb will ensure target signal discrimination against the background of clutter from the underlying surface.

For a signal with unambiguous range and range resolution δ_R three cases should be considered.

A case when range to target R_ц is greater than missile altitude (R_ц > H_p) is given in Fig. 6.

The length of illuminated segment AB is calculated by the formula

 (19)

The illuminated segment width is calculated, same as in case of a continuous signal, by the formula EF ≈ H_p θ/sin ψ.

The power of clutter signal from the segment should be calculated considering antenna gain factor square G²(β) in its direction, where angle β between ARP maximum directed to target and the line directed to the centre of illuminated segment is calculated by the formula

 (20)

If R_ц < H_p the resolution element in which target resides does not contain ground surface, so clutter signal power P_{п вх}= 0. 

If R_ц = H_p the AHH operates in the presence of altimeter reflections normal to the ground surface segment with area S,

  (21)

The power of clutter signal from the altimeter segment should be calculated considering antenna gain factor square G²(β) in its direction, where angle β = 90° - ψ.

Let us calculate the input signal/noise ratio for the most typical case of R_ц > H_p under conditions of the considered example, with H_p = 20 km, R_ц = 30 km, δ_R = 150 m, θ = 10^-1 rad, ψ = 30°, φ = 15°, σ_ц, = 20 m².

Angle β = 90° - 30° - arccos(20/30) = 11,8° corresponds to the second side lobe of ARP, having the level of –17.8 dB. Therefore, we assume that gain factor square at the side lobes G²(β) = 3,2·10^-4 (-35 dB).

For glancing angle ∠PCO = 90° - ψ - β = 90° - 30° - 11,8° = 48,2° (Fig. 6), specific RCS σ₀=3,2·10^-2 (-15 dB) (Fig. 3).

For a signal with unambiguous range, the area of illuminated segment in this case equals to

 (22)

The signal/clutter input ratio will be

 (23)

In this case the factor of signal/clutter ratio improvement by the input signal processing system shall be no less than 11.1 dB. Such improvement factor can be ensured by a processing system in the form of a comb of narrow-band filters even when clutter spectrum overlaps target spectrum with its main lobe, since the filtering yields a gain of 10 lg (Δf/Δf_ц) in signal/clutter ratio, which in the considered example is equal to 25.3 dB.

Target detection specifics

Detecting targets against the ground surface background, under target designation by angles in the ARP zone, can be done using signals with unambiguous range or quasi-continuous signals with target selection by Doppler frequency.

When using unambiguous-range signals for target detection, it is possible to use short pulses with high pulse ratio so as to reduce the illuminated area. In this case, however, detection of target, especially one with a small RCS and flying at low altitudes, will be hampered, making it necessary to add Doppler filtering.

To detect a signal by speed, normally used are quasi-continuous pulse trains with high repetition period F_п, which exceeds the value corresponding to possible maximum speed of missile closing-in on target. At the same time, in case of a low-speed small-RCS target, it is often difficult to detect a wanted signal with Doppler shift frequency f_дц, overlapped by the main lobe of ground clutter spectrum (Fig. 7).

An interesting approach to detecting a target whose spectrum is overlapped by ground clutter band, when using a monopulse antenna, i.e. antenna with both sum and difference radiation pattern, is described in papers [1], [4] dealing with the use of radars with synthetic aperture.

As shown in Fig. 7, target and clutter have different Doppler frequency shifts, even though they lie on the same line of sight. However, with the Doppler frequencies aligned, they will be in different angular directions, which actually is taken as a fundamental principle of the method proposed in the paper.

Proceeding from expressions (1) and (2), let us determine a required deflection of ARP maximum from the direction to target Δφ, at which f_дп will coincide with f_дц,

 (24)

Using relationship f (х₀ - Δх) ≈ f(х₀) - f '(х₀)Δх, let us represent cos (φ - Δφ) as

 (25)

then, substituting it in (24), we have

 (26)

From (26), the value of required angular deflection Δφ, rad, is determined

 (27)

The value of Δφ can be calculated on board the missile using the inertial system data and target coordinates and velocity components transmitted over the radio correction channel.

When ARP deflection by a required value of Δφ is implemented, the spectrum of clutter signal in the difference channel with Doppler shift f_дп, contained in the same filter with f_дц, will repeat the square form of the difference ARP by power, having zero intensity on frequency f_дп, and intensity of the wanted signal will then correspond to the difference ARP in the direction to target, offset relative to zero by angle Δφ.

However, said offset must not exceed half the ARP width, which considerably restricts the possibilities for applying this method. According to formula (27), the required condition Δφ < θ/2 can be implemented for low-speed targets at sufficiently large value of angle φ, which can be ensured at the start of target search when shaping respective trajectories of missile flight.

Let us assess feasibility of the method applying a monopulse antenna for detecting target in the most complicated case, when a continuous signal is used and the target signal spectrum is overlapped by the main lobe of ground clutter spectrum.

Let the clutter characteristics, sum channel ARP, and the value of target signal spectrum bandwidth be the same as in the example with calculation of the improvement factor for a continuous signal.

It should be noted that for the difference channel, under uniform field distribution across antenna aperture, the power-normalised ARP looks like

(28)

The clutter spectrum in the difference channel coincides in form with the ARP square of the difference channel and is defined by the formula

 (29)

When considering overlapping of target spectrum by clutter spectrum, we assume that Doppler frequency offset of the target signal spectrum relative to the clutter spectrum maximum is equal to 1 kHz. Then, in accordance with formula (1) and the data of Table 1 for φ = 15°, we have 

In Fig. 8, blue colour designates the spectra of target and clutter signals in the sum channel, expressed in decibel, for a case when the ARP maximum is directed to target. For convenience, in this and the following figures the frequency axis is offset by the value f_дп = 128,8 kHz, such that in new coordinates this frequency is matched by zero frequency.

Let us determine Δφ correction under which an offset by 1 kHz of the clutter spectrum is ensured through offsetting angular position of the sum channel ARP maximum and, accordingly, of the difference channel ARP zero, based on formula (27)

 (30)

Given such angular offset of the ARP (Δφ = 1,66°) the target will not lie in the maximum of sum channel ARP any more, and target signal power will decrease by the magnitude 

In that case the clutter signal spectrum will shift by frequency and its maximum will occur on the Doppler frequency of target signal, which is shown by red curve in Fig. 8.

Fig. 9 shows the spectra of target and clutter signals in the difference channel for a case when zero of the difference ARP is offset relative to the direction to target by angle Δφ.

As can be seen from Fig. 9, clutter spectrum in the difference channel turns to zero on the Doppler frequency of the target. In consideration of clutter signal passage through a narrow-band filter with passband Δf_φ = 10 Hz, tuned to the target frequency, we should note that the maximum value of clutter spectrum in the difference channel will occur at the filter edges

 (31)

Target signal level in the difference channel will be attenuated by the magnitude

Since ARP width in the difference channel is approximately twice as large as in the sum channel, clutter signal input power will be greater by 3 dB, and clutter bandwidth in the difference channel will be twice as large, amounting to 2Δf = 6,8 kHz. Considering target signal attenuation in the difference channel by 9.3 dB, signal/ clutter ratio, calculated by formula (13) for the sum channel in case of continuous signal, will diminish in the difference channel by 3 + 9.3 = 12.3 dB. The signal/clutter input ratio in the difference channel will amount to −40 – 12.3 = −52.3 dB, and the required signal/clutter ratio improvement factor shall be no lower than 52.3 + 15 = 67.3 dB.

Processing with the use of a comb of narrow-band filters will improve signal/clutter ratio by

and considering the clutter spectrum form in the difference channel on target frequency with the level of –99.9 dB, the improvement factor will theoretically amount to 28.3 + 99.9 = 128.2 dB, which is significantly higher than the required factor for a large-size target.

In this way, due to a stronger suppression of clutter signal as compared with wanted signal attenuation, a processing system in the form of a comb of narrow-band filters in the difference channel, with ARP offset by Δφ, makes it possible to ensure target signal detection against ground surface background also when the target spectrum in the sum channel is overlapped by the clutter spectrum.

Conclusion

The ground clutter signal spectrum has been assessed.

The effect of ground clutter signals upon signal/clutter ratio for continuous and quasi-continuous signals, as well as signals with unambiguous range, is considered.

The expediency of considering an AHH operation mode with an offset of its antenna direction for detecting target with the use of a difference ARP is shown.

An estimate is given of the magnitude of required angular deflection of the monopulse antenna for support of AHH operation under conditions when the wanted signal being received is overlapped by the ground clutter spectrum.

It is shown that for the typical characteristics of a large-size target and ground surface clutter, in case of a signal with unambiguous range, it can be possible for AHH receiver to detect target in the sum channel equipped with a system of narrow-band Doppler filters. For continuous and quasi-continuous signals, target detection is possible in the sum channel if the target spectrum lies outside of the clutter spectrum main lobe.

If target frequency spectrum lies within the clutter spectrum main lobe, target detection is possible as well, but by means of difference ARP offsetting.