2146
PROCEEDINGS
OF
THE
IEEE,
DECEMBER
1967
With respect to the midpoint between the antenna ele-
ments, the relative time delays of the noise at the two an-
tenna elements are
f
[
1/(4fo)] sin
46
=
-t
1/(8fo)
=
&
&/(8~),
which corresponds to phase shifts of
-t
a/4 at frequency&.
The array output due to the incident noise at
0
=
a/6
is then
N
[wl
sin
(coot
-
i)
+
w2
sin
(oat
-
;)
+
w3
sin
(coot
+
i)
+
w4
sin
(coot
-
;)]e
(4)
For this response to equal zero, it is necessary that
w1
+
w4
=
0
w2
-
w3
=
0
Thus the set of weights that satisfies the signal and noise
response requirements can be found by solving
(3)
and
(5)
simultaneously. The solution is
1
w1
=
3,
w2
=
2,
w3
=
T,
Wq
=
-
T.
With these weights, the array will have the desired proper-
ties in that it will accept a signal from the desired direction,
while rejecting a noise, even
a
noise which is at the same
frequency
fo
as the signal, because the noise comes from
a
different direction than does the signal.
The foregoing method of calculating the weights is more
illustrative than practical. This method is usable when there
are only a small number of directional noise sources, when
the noises are monochromatic, and when the directions of
the noises are known
a
priori.
A practical processor should
not require detailed information about the number and the
nature of the noises. The adaptive processor described in
the following meets this requirement. It recursively solves
a sequence of simultaneous equations, which are generally
overspecified, and it finds solutions which minimize the
mean-square error between the pilot signal and the total
array output.
1
1
1
(6)
CONFIGURATIONS
OF
ADAPTIVE
ARRAYS
Before discussing methods of adaptive filtering and signal
processing to
be
used in the adaptive array, various spatial
and electrical configurations of antenna arrays will
be
considered. An adaptive array codguration for processing
narrowband signals is shown in Fig. 4. Each individual
antenna element is shown connected to a variable weight
and to
a
quarter-period time delay whose output is in
turn connected to another variable weight. The weighted
signals are summed, as shown in the figure. The signal,
assumed to
be
either monochromatic or narrowband,
is
received by the antenna element and is thus weighted by a
complex gain factor
A&.
Any phase angle
4
=
-tan-
(w2/w1)
can
be
chosen by setting the two weight values, and
the magnitude of ths complex gain factor
A=
,/-
can take on a wide range of values limited only by the range
limitations of the two individual weights. The latter can
assume a continuum of both positive and negative values.
""Kv
-
4f.
x.
Fig.
4.
Adaptive
array
configuration for receiving narrowband signals.
1/11
L
I
I
Fig.
5.
Adaptive
array
configuration for receiving broadband signals.
Thus
the
two
weights and the 1/(4f0) time delay provide
completely adjustable linear processing for narrowband
signals
received by each individual antenna element.
The full array
of
Fig. 4 represents a completely general
way of combining the antenna-element signals in an ad-
justable
linear
structure when the received signals and noises
are narrowband. It should be realized that the
same
generality (for narrowband signals) can be achieved even
when the time delays do not result
in
a
phase shift
of
exactly
742
at
the center frequency
fo.
Keeping the phase shifts
close to
n/2
is
desirable for keeping required weight values
small,
but
is
not necessary in principle.
When one
is
interested in receiving signals over a wide
band
of
frequencies, each of the phase shfters in Fig.
4
can
be
replaced
by
a
tappeddelay-line network as shown in
Fig.
5.
This
tapped
delay line permits adjustment of gain
and
phase
as
desired at a number of frequencies over the
band of interest. If the tap spacing is sufficiently close, ths
network approximates the ideal filter which would allow
complete control of the gain and phase at each frequency
in the
passband.
ADAPTIVE
SIGNAL
PROCESSORS
'
Once
the form of network connected to each antenna
element
has
been
chosen, as shown for example in Fig. 4
or Fig.
5,
the next step is to develop an adaptation procedure
which
can
be
used
to adjust automatically the multiplying
weights to achieve the desired spatial and frequency filtering.