We propose a new algorithm for direction finding using a spinning Direction Finding (DF) antenna and an omnidirectional antenna. In the proposed algorithm we use all information about when a target is received using either antennas, whereas in literature only situation when the signal is received using the spinning DF antenna and the signal with the spinning DF antenna is stronger than with the signal received with the omnidirectional antenna. Simulated results show that the proposed algorithm is more accurate in direction finding and target positioning than the algorithm found in literature.
Directional antenna Omnidirectional antenna
In [
The assumption of a constant transmission amplitude does not usually hold as radar systems incorporate scanning patterns and the transmission direction changes. The constant amplitude assumption is valid approximation only if the transmission BW is high and the radar scanning speed is low compared to the receiver spinning DF antenna BW.
In this paper, we propose an algorithm that uses information from all received pulses and does not assume anything about the transmission pattern. Rest of this paper is organized as follows. Section 2 describes the models used in this paper. Section 3 shows how the probability density function (pdf) of DoA can be determined using all information. In Section 4 we show how the mean and variance of the estimates are computed. Section 5 contains example applications of the proposed algorithm and Section 6 concludes the article and discusses future work.
We assume here that the emitter is a pulsed radar, and we can associate pulses from a single radar together. Received pulses can be divided into three groups based on the PAs
Ar(t) = fr (θ (t)) ft (t) + εr, (1) where Ar(t) is the received amplitude, fr (θ (t)) is the receiver gain at time t to the direction of the transmitter, ft (t) is the field strength of the transmitted signal at the receiver at time and ε is a noise term, which is assumed to be zero mean normal with variance σ r2. We assume t, that the receiver antenna pattern fr (θ (t)) is known. We do not make any assumptions of the shape of transmission field strength
ft (t). Transmission field strength can have complex shapes
especially when the transmitter is an Active Electronically Scanned Array (AESA) radar. In
AESA radars, the scanning pattern can be steered electronically from one direction to other
via phase control practically without delay [
For the omnidirectional receiver, the antenna gain is constant go and the received PA is:
Ao(t) = goft (t) + εo, pro(Ar(ti), Ao(ti)|ϕ ) = pN ⃓⃓ fr (ϕ − θ i) Ao(t) (︃ fr (ϕ − θ i) )︃ 2 where εo ∼
N(0, σ o2). We assume that both antennas are connected to a multichannel receiver so that the sensitivity for both channels is same and we use same threshold Alimit for detecting a pulse for both antennas.
The model is more general than those found in [
When we have detected pulse with both antennas at time ti we can solve the unknown field strength ft (t) from (2) and substitute it into (1) to get:
Ao(ti) = fr (ϕ − θ i) ︃( Ao(ti) − εo )︃ go + εr, where ϕ is the direction of the target in geographic coordinates. After this we compute the likelihood for target being at angle ϕ using the normal pdf (2) (3) (4) (5) (6) ⃓ 1 = √︃(︂ fr(ϕ − θ i) )︂ 2 go σ o2 + σ r2 e ︄(
Ao(ti)− fr(ϕ − θ i)Ao(t) )︄2
go 2︄( fr(ϕ − θ i) )︄2 go σ o2+2σ r2 as accurate evaluation of the DoA as in the first case. However, as detecting a pulse and we neglect the noise in the inequality we get
Alimit ≥ goft (t) ⇒ where subscript ro refers to the case where pulse was received with both antennas.
If the pulse is received only with the directional antenna, we can deduct that the signal source is in the direction where the gain of the spinning DF antenna is higher than the gain of the omnidirectional antenna and the amplitude of the received PA is below the detection threshold of the omnidirectional antenna. This area corresponds to the area marked with thick blue line in Figure 1. Because the transmitting gain (and power) is unknown we cannot make Alimit is the threshold for
pN (︁ Ar(ti)|ft (ti) fr (ϕ ) , σ r2)︁ dft (ti), and to get the pdf we use uninformative prior and normalize using Bayes’ rule to get p(ϕ |Ar, Ao) =
p(Ar, Ao|ϕ ) ∫︁ϕ 3=600 p(Ar, Ao|ϕ )dϕ .
In practice, the computation of the probabilities above encounter often numerical errors and instead of using likelihoods the computations should be done using log-likelihoods until the normalization.
ft (ti) ≤
Alimit .
go If we had a known value of ft (ti) and (1) the likelihood is
p(Ar(ti))|ϕ, f t (ti)) = pN (︁ Ar(ti)|ft (ti) fr (ϕ ) , σ r2)︁ .
By integrating the possible values of ft (t) using (7) we get (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)
The pdf of DoAs is not always practical and it is often required to be transformed to a mean
and variance of DoA in degrees. To obtain the mean of DoA first a weighted Cartesian mean
µ is computed [
ϕ =0 p(ϕ ) cos ϕ µ ϕ = ∠µ c = atan2(µ c,1, µ c,2), σ ϕ2 = ∫︂ 180
Transmission field strength Spinner gain
Omnidirectional gain 90 180 270 90 180
270 0 (t)
Ar(t) Ao(t) Pulses detected with both antennas Pulses detected with spinner only Pulses detected with omnidirectional antenna only Detection threshold 90 180 270 0 90 180 270
In this section, we show examples of the proposed algorithm. The antenna gain and transmission efild strengths are simulated. The noise is simulated from a normal distribution with standard deviation 0.1 and the detection threshold for a pulse is 1.
In the first example radar is located at 270 ◦ from the receiver. The radar is making a circular scanning pattern so that it’s mainlobe is directed towards the receiver when receiver has direction 0◦ . The DoA of radar pulses is solved using pulses received of two revolutions of the spinning DF antenna. In this situation the radar mainlobe is directed towards the receiver when the receivers spinning DF antenna’s first sidelobe is directed towards the radar. Top plot of Figure 2 shows the gains as the function of spinning DF antenna angle. In the bottom plot the noisy signals and detected pulses are shown. All pulses that are received with both antennas come from the rfist sidelobe. Then there are few pulses detected with spinning DF antenna’s mainlobe that come from sidelobes of the transmitter and few pulses detected with the omnidirectional antenna only.
In real world situations the spinning antenna could be rotating 1200 degrees in second and in our simulation there are 100 pulses for 720 degrees. This would make the pulse repetition frequency of the transmitting radar 167 Hz, which is a low value. We have chosen this small number of pulses to show how the proposed algorithm can perform in dificult situations with a small number of received pulses.
Figure 3 shows the likelihoods computed from the received pulses. Because all pulses received
with both antennas are in the first sidelobe and the spinning DF antenna has two almost
identical sidelobes the likelihood pro has two peaks and due to the noise in measurements
the wrong likelihood peak is the stronger one. Algorithms in [
Likelihood computed from pulses received only with the spinning DF antenna is directed approximately to the correct direction but has much larger variance than the peaks that are computed with both antennas. But when these two are combined we get an accurate peak in the correct direction. In this example, the likelihood computed using only omnidirectional antenna does not provide more information to other measurements.
The mean and standard deviation estimates computed using diferent data are given in Table 1. The estimates were computed using a 1◦ resolution. Results show how the combined estimate is much more accurate than any of estimates computed alone. Results also show that all standard deviations are reasonable and can be used to determine the goodness of the DoA.
Transmission field strength Spinner gain Omnidirectional gain 90 180 270 90 180
270 0 (t)
Ar(t) Ao(t) Pulses detected with omnidirectional antenna only
Detection threshold 90 180 270 0 90 180 270
The second example shows how the proposed method can be used to determine direction even
without any pulses detected with the spinning DF antenna. Algorithms in literature [
In our third example, we show how the number of received samples afect to the directionifnding accuracy. The simulated situation provides a lot of information for DoA finding task. The situation is simulated 1000 times with varying the number of pulses and the exact time when the mainlobe of a slowly scanning radar is directed approximately towards the receiver. Figure 6 shows an example of the situation.
The average DoA estimation error as the function of total pulses is shown in Figure 7. In this test, we can see that the estimation accuracy using all information is better than using only pulses that are received with both antennas when number of pulses is less than 13 and with 7 pulses the average DoA error with all information is less than 1 degree, while the error with using only information when both antennas receive the pulse is more than 8 degrees. In these results the DoA estimation with omnidirectional antenna only is more accurate than the estimation with spinning antenna only. This can be explained looking to Fig. 6 where one can see that there are only a few pulses received with the directional antenna only compared to the pulses received with the omnidirectional antenna.
In this paper, we studied how to use all available information from a spinning DF antenna and an omnidirectional antenna to determine the DoA of a set of pulses. We showed in examples that the proposed method can determine the DoA of a signal source and the use of all information. The use of all information gives estimates better accuracy and more importantly helps to determine estimate DoA in dificult situations where a traditional DoA estimation methods would not have provided any information about the DoA of the radar pulse.
In this paper, all data was simulated. In future, the algorithm should be tested with real data and radiation patterns of antennas. The algorithm could also be extended to be able to take into account imperfections in the radiation pattern of the real-world omnidirectional antennas, that is, to use a radiation pattern of the omnidirectional antenna instead of a constant go. Furthermore, the use of radiation patterns could be extended to take into account the elevation angle and the algorithm could incorporate the complex amplitude instead of the absolute amplitude.
90 180 270 90 180
270 0 (t) Pulse amplitudes
Field strength Spinner gain
Omnidirectional gain Ar(t) Ao(t)