Tracking Performance of Adaptive Array Feed Algorithms for 70-Meter DSN Antennas TMO Progress Report 42-143
November 15, 2000
Tracking Performance of Adaptive Array Feed
Algorithms for 70-Meter DSN Antennas
R. Mukai,
1
V. Vilnrotter,
1
and P. Arabshahi
1
This article describes computationally intelligent neural-network and least-
squares algorithms for precise pointing of NASAs 70-meter Deep Space Network
(DSN) antennas using the seven-channel Ka-band (32-GHz) array feed compen-
sation system (AFCS). These algorithms process normalized data from the seven
horns of the array in parallel and thus are more robust and more accurate than in-
herently serial conventional processing techniques (CONSCAN) currently used by
the DSN. A previous article discussed the use of new algorithms for acquisition and
estimation of relatively large pointing errors [1] while addressing only briey the is-
sue of ne tracking near the source. However, neural networks designed specically
for ne-tracking operations yield better ne-pointing performance and signicantly
lower complexity than those designed for coarse acquisition, and large reductions
in complexity may be achieved by using a low-complexity ne-pointing neural net-
work in conjunction with a very simple coarse-acquisition algorithm. In addition to
complexity reduction, we also demonstrate the ability to update parameters of the
radial basis function (RBF) network in near-real time in response to changes in the
antenna, highlighting a useful characteristic of RBF neural networks for antenna-
pointing control. The ability to update an RBF network in near-real time without
complete restructuring or redesign of the network permits ecient operation even
in the presence of frequent changes in the antenna surface.
I. Introduction
A. Tracking in the Presence of Antenna Distortions
An accepted technique for recovering losses due to gravitational deformations, thermal distortion,
and wind is by means of a real-time compensation system employing an array of feeds in the focal
plane, as described in [1,2]. A seven-element focal-plane array feed compensation system designed to
recover gravitational losses on large DSN antennas has been constructed and evaluated at the Goldstone
Deep Space Communications Complex (GDSCC) [2]. This system, called the array feed compensation
system (AFCS), was developed to demonstrate real-time gravity compensation and closed-loop tracking
of spacecraft and radio-source signals at Ka-band frequencies (nominally 32 GHz).
1
Communications Systems and Research Section.
The research described in this publication was carried out by the Jet Propulsion Laboratory, California Institute of
Technology, under a contract with the National Aeronautics and Space Administration.
1 In the absence of antenna distortions, a single properly designed receiving horn collects virtually all
of the focused signal power [2]. Antenna surface distortions often lead to a shift in the peak of the
signal distribution as well as a redistribution of the signal power in the focal plane. This leads to loss
of power in the central channel, which can be eectively recovered by the outer horns of an array placed
in the focal plane. When the horn signals are multiplied by complex combining weights matched to the
instantaneous magnitude and phase of the signal in each channel, the signal-to-noise ratio (SNR) of the
combined channel can be improved, approaching that of an undistorted antenna under ideal conditions.
Distortions also aect the pointing of the antenna by introducing shifts in the signal peak. Antenna-
pointing errors can degrade the received SNR of both single-horn and array receivers, particularly at
Ka-band frequencies. Our intent here is to demonstrate that properly designed neural-network or least-
squares algorithms eectively remove the time-varying pointing errors and keep the antenna pointed in
the direction of maximum SNR even in the presence of signicant antenna distortions.
As in [1], it is convenient to represent the received samples as K dimensional vectors, r(i) = s(i)+n(i),
where r(i) = (r
1
(i), r
2
(i),
· · · , r
K
(i)) is the vector of samples over a K-element array at time i. In order
to reduce the eects of noise, a large number of consecutive received vectors may be averaged, yielding:
r
a
(jL) = (r
a,1
(jL), r
a,2
(jL),
· · · , r
a,K
(jL)) = 1
L
jL
i=jL
L+1
r(i)
(1)
where
var r
a,k
(jL) =
2
L
(2)
and r
a,k
refers to the kth component of the averaged vector,
2
is the variance of the additive white
Gaussian noise, and r
a
(j) is the averaged vector output at time jL, where j = 1, 2, 3,
· · · .
B. Using AFCS Data for Tracking Purposes
It has been shown [1,4] that data from the AFCS can be used to estimate and to correct pointing errors
to achieve maximum SNR at the horns. The instantaneous pointing error vector (XEL, EL) is a two-
dimensional error vector. The incremental pointing error in cross-elevation is XEL, and the incremental
pointing error in elevation is EL. Both are measured in millidegrees (mdeg).
We seek to compute the mapping from the K-dimensional voltage vector r
a
to the two-dimensional
error vector (XEL, EL). This mapping may be represented as
XEL
EL
= f (r
a
)
(3)
Residual errors in the voltage vector r
a
due to noise cause errors in the estimate of (XEL, EL) even if
f (r
a
) is known exactly. However, f (r
a
) is also aected by the physical structure of the antenna, which
is not always precisely known and which changes as the antenna ages or is bueted by wind. The noisy
and time-varying nature of f (r
a
) poses an additional challenge.
Our objective is to develop the processing function f (r
a
). Both interpolated least-squares algorithms
and radial basis function (RBF) networks have been shown to provide good approximations to the under-
lying function, allowing pointing osets to be accurately estimated using the complex voltage data from
the horns.
2 II. Algorithm Descriptions
Two algorithms, a radial basis function network and a quadratic interpolated least-squares algorithm,
were developed to synthesize the function f (r
a
) described by Eq. (3), and theoretical descriptions of both
are given in the following subsections.
A. Algorithm 1: Radial Basis Function Network
A radial basis function (RBF) neural network was developed and used to estimate antenna pointing
errors [3,5]. As shown in Fig. 1, this radial basis algorithm takes 12 real inputs at a time, which are
the real and imaginary parts of the six normalized outer horn voltages. The RBF network consists of
two layers: a nonlinear radial basis function layer and a linear combiner layer. Each radial basis unit
implements a Gaussian function of the form
G(r
a
(jL); c
i
) = exp
(b r
a
(jL)
c
i
)
2
= exp

b
12
k=1
(r
a
(jL)
k
c
ik
)
2


(4)
where r
a
(jL) is the 12-element averaged input voltage vector at time jL [previously dened in Eq. (1)],
c
i
denotes the ith radial basis center, and b = 0.8326/spread controls the width of the units region of
response. The scalar b is dened so that G(r
a
(jL); c
i
) = 0.5000 when r
a
(jL)
c
i
= spread.
Dene the matrix G as [5]
G =


G(r
a
(L); c
1
)
G(r
a
(L); c
2
)
· · ·
G(r
a
(L); c
M
)
1
G(r
a
(2L); c
1
)
G(r
a
(2L); c
2
)
· · · G(r
a
(2L); c
M
)
1
.
.
.
.
.
.
. ..
.
.
.
.
.
.
G(r
a
(N L); c
1
)
G(r
a
(N L); c
2
)
· · · G(r
a
(N L); c
M
)
1


(5)
where M is the number of radial basis units and N is the number of consecutive input voltage vectors
applied to the network during the course of operation.
BIAS
TERM
LINEAR
COMBINER
INPUT
LAYER
RADIAL BASIS
(HIDDEN) LAYER
å
Fig. 1. The RBF neural network.
3 The column of ones on the right side of G is necessary since the network contains a bias weight in its
linear combiner. A bias weight c
bias
may simply be thought of as a weight connected to a xed input of
+1 rather than to a radial basis unit.
Dening the weight vector as [5]
w =


w
1
w
2
.
.
.
w
M
c
bias


(6)
the output of the radial basis function network in response to the N input voltage vectors r
a
(L) through
r
a
(N L) as computed by the linear combiner is
y =


y(L)
y(2L)
.
.
.
y(N L)

= Gw
(7)
where y(jL) is the RBF networks response to the jth averaged input vector r
a
(jL) dened in Eq. (1)
for j = 1, 2, 3,
· · · .
Therefore, each of the radial basis units receives a 12-element input vector and computes the squared
Euclidean distance from the input vector to its 12-dimensional basis center vector. The output of the
radial basis unit is a number that depends on this squared distance. It