Aerospace and Electronic Systems Magazine May 2017 - 48

Potential Transmit Beamforming Schemes for Active LPI Radars
each frequency component in far field. By applying a stepped frequency difference across the array elements, the beam can be made
to scan the space, with the beam pattern dictated by the interelement frequency component and amplitude distribution among the
elements [17]. The array beam focusing direction will change as a
function of the range, angle, time, and frequency increment [18],
[19], [20]. This contrasts with the conventional phased array providing a range-dependent beamforming direction in far field.
In a basic FDA system, the mth element radiated signal frequency is
f m  f 0  m · f ,

Figure 2.

Comparisons of a low-gain transmit basis pattern (dashed line) and
a formed high-gain scanning beam pattern (solid line) in the azimuth
angle domain, where the target range r0 = 20 km is assumed.

is transmitted with spoiled phases for a particular low-gain basis
pattern. Each subpulse return is processed through a matched filter and saved in a memory for subsequent joint beamforming to
produce a high-gain directional beam pattern. The transmitter and
receiver processing time slots are illustrated in Figure 1. Figure 2
demonstrates the high-gain patterns of a 32-element phased array
steered to the azimuth angle 0°, 10°, and 30°, respectively. Note
that the peak gain is only about 1 dB (otherwise, the peak gain for
the conventional methods using unspoiled frequency increments
is about 15 dB). Using the designed weighter coefficients, we can
form high-gain (about 14 dB) beam patterns by linearly combing
the 32 basis beam patterns. Other beamforming implementations
are also possible and could potentially be integrated with the radar
waveforms. As noted by one reviewer, because the power is already spread over the angular region of interest, the low-gain transmit basis pattern does not need to scan. In this case, the high-gain
scanning beam pattern can still be formed through beamforming in
the receiver with the known excitations and phase shift values αm
in the transmit array.
This scheme requires no extra scanning time when compared
with the traditional method of using N separate high-gain beams
that scan across the same region. Note that to ensure the phase
relationship between the basis patterns, the interested target must
remain coherent over the radar scanning time; thus, motion compensation may be required for a long scanning time. This requirement is similar to traditional Doppler processing over the same
scan time interval.

LPI RADAR USING RANGE-DEPENDENT TRANSMIT
BEAMFORMING
To overcome the disadvantage of a phased array antenna providing
only an angle-dependent beam pattern, we can use the FDA antenna
[16]. Each FDA element can be controlled individually to steer the
main beam of the array, relying on the time-domain relations of
48

m 0,1,, M  1,

(4)

where f0 is the carrier frequency and Δf is the frequency increment. Suppose the element factor can be factored out of the transmit field, when a uniform transmit weight vector is adopted in the
FDA, in a narrowband case, the array factor at the position (r, θ)
can be expressed as [21], [22]

AF  r , , t  

exp  j 0 
r



fr df 0 sin  fd sin   
sin  M   ft 


 
c0
c0
c0

 

, (5)
 
fr df 0 sin  fd sin   
sin   ft 


 
c0
c0
c0
 
 

where c0 is the speed of light. The additional phase factor Φ0 is

r

 0 2 f 0  t 
c
0



f 0 d sin 
fr
    M  1 c    M  1 c
0
0

fd sin 
   M  1
   M  1 ft.
c0

(6)

In (5), the maximum field is obtained when
Δft −

Δfr df 0 sin θ Δfd sin θ
+
+
= k,
c0
c0
c0

k = 0, ±1, ±2,...

(7)

This means that when only one parameter is fixed, there will be
multiple solutions for the unfixed parameters. On the other hand,
when two parameters are fixed, the pattern periodicity will depend
on the unfixed variables. Supposing M = 12, Δf = 30 kHz, and f0
= 10 GHz, Figure 3 shows the FDA transmit beam pattern, where
t = 0 is assumed in the simulations. It shows that the FDA creates
a range-dependent beam pattern whose amplitude and spatial distribution can be controlled by changing the frequency increments.
This provides the possibility to control the transmitted energy distribution and thus motivates the use of the FDA antenna for LPI
radar.
Figure 4 illustrates the FDA-based LPI scheme by using spoiled
frequency increments. Different from the basic FDA scheme using
stepped frequency increments, we make a judicious choice of the
spoiled frequency increments, namely, optimized nonlinear frequency increments. Consider an M-element FDA and suppose the
spoiled frequency increment for the mth element is Δfm:
f m = f 0 + Δf m ,

IEEE A&E SYSTEMS MAGAZINE

m = 0,1,..., M − 1.

(8)
MAY 2017



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