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z
41/~4 ~
- ~ 4~/
FIGURE 8.1: Linear array of uniformly spaced sensors.
8.3 Using Linear Arrays of Uniformly Spaced
Sensors
The most widely studied case is the one with a linear array of equally spaced
sensors (see Figure 8.1) and a narrow-band planewave signal with a known
carrier frequency, which may be represented by
skits = Resects exp[-iw~t—rj)] = Reaj(?,b)sett) exp(-i~t), (~.3)
where Re denotes the real part of what follows, {sect)' O < t < T) is the
complex signal envelope function, and
~ = the carrier (angular) frequency such that AT ~ 2;r,
Tj = (;— 1~4 cos ¢/c
= the time delay of the signal planewave arrival at the Josh sensor
relative to the first,
= the distance between two adjacent sensors,
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the velocity of sound propagation,
= the assumed direction of the signal planewave arrival, and
aj(~) = expti~j—1)—tacos ¢],
j = 1, e ~ ~ ~ J. ~ (~3~4)
Then, in the absence of the interference z, the logarithm of the likelihood
ratio becomes
log,dpl~r) = (s,r)—Use = Re ~a1~t,b)
sets) drj~t) - 2 So , (8 5)
where drj is the envelope function of the narrow-band representation of the
data, namely,
drift) = Re alrj(~) exp(-i~t) .
Suppose a planewave arrives in the direction of jot Then the signal part of
the data-dependent term, the first term of (~.5), is proportional to
Re a(?~la(?~'O) = Re 1—exptit7~a?(cos To - cos ¢~/c] (~.6)
1 - exp[~wcl~cos To - cos ?,b)/c]
where Ably = fad (I), . . ., a, is the direction (or steering) column vector
in the Redirection and ~ denotes the complex conjugate transpose. Equation
(~.6) is in the form of a "main beam" centered at ~ and "side lobes" on each
side of the main beam as To varies from -~r/2 to ~r/2. Hence, the process-
ing (of the data r) described by (8.6) is called "beam forming" (Steinberg,
1976~. On the other hand, if the actual direction of the planewave arrival To
is fixed and the assumed direction fib is varied, (~.6) attains the maximum
at ~ = To, namely, when the beam is steered at the signal source. Thus,
detection of the planewave signal and estimation of its direction of arrival
are done by varying ~,b (steering the beam) from—~r/2 to ~r/2 to find the
maximum of (~.6) and comparing it to a preassigned threshold determined
by the false-alarm probability (according to the Neyman-Pearson criterion)
(Helstrom, 1986~. Instead of steering a single main beam, one can place
many beams to fill the angular sector ~—~r/2, ~r/2) by providing many direc-
tion vectors a(?,km), m = 1,...,M. Then by comparing the magnitude of
(~.6) for each lbm, instead of varying fib, we effectively accomplish the same
task of detection and estimation.
In the presence of the interference z, some modification to the beam-
forming is necessary. For any interference to be effective against the signal,
it must have energy at or near the carrier frequency (otherwise, it can be
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simply filtered out). For the sake of simplicity, suppose we have one sinu-
soidal interference arriving in the ,,6' direction with a Gaussian distributed
amplitude, namely,
Aft) = Reuexp(-u~- (j- l~cicosl,b'/c]) = Reuzjexp(-i~t), (~.7)
where the second equality defines z; and u is a complex Gaussian variable
with mean O and variance i32. This represents a planewave arriving through
a multipath medium causing the Rayleigh fading. By carrying out the expec-
tation with respect to z in (~.2) (i.e., with respect to u), the data-dependent
term of the log likelihood ratio becomes
Re (s, r - EO{z~r)) = Re (Se, art—I + 22 ably) a(¢ babe ~ ~ r)
J /
=
where
Redid+ R)-is,r`),
sj(~) = aj(~)seLt),
and Eo~z~r) is the conditional expectation of z given r under Ho, and
R = ,(32a(¢')a(¢')~. The first member of (~.~) has an obvious interpre-
tation: the optimum processing is to make the least-mean-square-error es-
timate of the interference and subtract the estimate from the data before
the beamforming. The second member, on the other hand, shows how the
conventional beamformer is to be modified due to the presence of the inter-
ference. By recalling that a(¢') is the steering vector in the direction of the
interference, the modified beamformer has a considerably reduced output
in the direction of the interference, thus acquiring the term, null-steering
(Steinberg, 1976; Gabriel, 1976; Friediander and Porat, 1989~.
In practice, the interfering source is not known a priori and its covari-
ance matrix R must be estimated. The estimation may be done beforehand
or simultaneously with the detection operation, assuming that the direction
of the signal arrival is known (which is the case with the fixed multibeam
scheme). This simultaneous method is referred to as the adaptive beam-
forming and is implemented by attaching a variable gain (or weight) and a
variable time-delay (which are adjusted as data are obtained) to the out-
put of each sensor. Of course, such an adjustment must be done rapidly
so that accurate signal detection and direction-of-arrival estimation can be
accomplished. The iterative methods of adjusting and their convergence
characteristics have been extensively studied. Monzingo and Miller (1980)
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is one of the comprehensive textbooks on the subject. A benchmark paper
series edited by Haykin (1980) has many important papers, including those
of Gabriel, Applebaum, Widrow, Griffiths, and Owsley.
3.4 Using General Arrays of Sensors
The results presented above can be generalized as follows (Kadota en cl Rot
main, 1977~: first, instead of a linear array of equally spaced sensors, we
can consider a general three-dimensional array (or configuration) with the
coordinates if;, A, (j ), j = 1, . . ., J. Then the planewave arriving in the
(8, ~b)-direction, where ~ and i,) are the elevation and the azimuthal angles,
incurs at the JO sensor the phase shift expressed by
aj(w, 8, ¢) = exp pi—(A cos ~ cos ~ + 77j cos ~ sin ~ + (j sin 8~] . (~.9)
Next, instead of a single planewave, we consider a signal consisting of Ii
planewaves, each having a different frequency ~k, k = 1,...,](, and each
arriving in M different directions (69m,?/)m), m = 1,. . . ,M. For convenience,
we assume that cokT is an integral multiple of Or for every k. Also, rather
than a "slowly varying" envelope function sets), we consider a complex Gaus-
sian variable (independent of time) as the amplitude of each planewave.
Thus, the signal at the JO sensor is now given by
K M
sj(t) = ~ ~ Re Ukmaj(Wk,(9m'?/)m) exp(—inlet), (8.10)
k=1 m=1
where (nkm), ukm = ukm+inkm, are complex, zero-mean, Gaussian variables
with
E UkmUk~m' = E Ukmuk~m' = Pkk~mm, ~ k, k = 1, . . ., h; m, m' = 1, . . ., M .
We allow some As to be zero since not all frequency components have all M
arrival directions. The interference is now generalized to
vj(t) = ~ Re vje exp (-i T )
, (8.11)
where {v;e}, vj' = vje + iVje, are complex, zero-mean, Gaussian variables.
That is, for each j, vj(t) is a-discretized version of the spectral representation
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of a general stationary noise. Then the data-dependent part of the log-
likelihood ratio takes the following quadratic form in the data:
(x, (! + V)-~S(! + V + S)-ix), (~.12)
where
X —
(X1 In,. ,X(K_l)J+l, ~XKJ, X1. Id,. ,X(K_l)J+l, ~XKJ)'
X(~l)J+i = (T) | cos~ktdrj(t), ~(~l)J+j = (T) | sinwktdrj(t)'
[ ~ A ]
with the ((k—l)J + j' (k' - 1)J + j,)th elements of S and S given respectively
by the real and the imaginary parts of
M
~ Pkk~mm~ai(Wk'6tm'(m)a:~(wki~f~mi,lbmI), j,; =1~, J; k,k =1,,X,
m,m'=1
and
V=[O Vat
(V) (k—l ) J+ j,(k'—l ) J+ jl = E Vjl Vj'tikk' = E vJevj~e~kkl,
i_ ANT
21r
The J sensors constitute spatial samplers of the available (acoustic) data and
their configuration specifies the pattern of spatial sampling. This sampling
pattern is incorporated into the covariance matrices S and V to influence
the detection statistic (8.123 which specifies the data-processing algorithm.
Although the linear array (with or without the equal spacing) is the most
common configuration, due primarily to the ease of implementation, the
sampling pattern can be considered as a factor with respect to which the
detection and estimation performance can be optimized. In fact, we show
next an interesting example of this application.
So far, we have assumed that the sensor positions are rigidly fixed and
their coordinates are known a priori. Although this is the case with the
phased-array radars and seismographic sensors, for underwater-acoustic sen-
sors the exact positions in the ocean are difficult to determine and calibra-
tion of the array becomes necessary. One way to deal with this problem
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is to mode} the deviation (or fluctuation) of the sensor position (from the
presumed value) as an additional noise and incorporate into the optimum
processor (for detection and estimation) the sensitivity of the performance
to this noise. For example, the array gain (the output signal-to-noise ratio of
an array processor for a given direction of signal arrival) may be maximizer!
under the constraint that the array-gain sensitivity to the sensor-position
noise be kept below a given level (Cox et al., 1987~. An alternative is to
devise the sensor configuration so as to make these test statistics immune to
the sensor-position fluctuation. The ESPRIT (Estimation of Signal Param-
eters by Rotational Invariance Techniques) method (Roy and Kailath, 1989)
forms pairs of sensors to create an array of doublets such that it consists
of two identical subarrays where one is a translate of the other. Suppose
the signal consists of M planewaves with complex, zero-mean, Gaussian am-
plitudes, having the same frequency ~ arriving from M directions (O. tbm),
m = 1, . . ., M. We further assume for simplicity that the interference is ab-
sent. Suppose we have already detected the signal and our goal is to estimate
the M arrival directions ~m' m = 1,,M, which are specified relative to
the axis of the doublet (the displacement vector). Denote the data from the
two subarrays of sensors by two (~/2~-vectors x and y, assuming ~7 to be
even,
x = (~1, - ~ -, :~7/2 ), pi = ( T ) 1; exploit ~ drj (~ ~ ' be. ~3)
~ 2 ~ 2~( expticot) dr~+j~t), j = 1, . . ., J/2 .
Y=(Y~,.~.,Y7/2), Ye=
Then
R== =Exx* =AUA*+!
R=y = E by* = AURA* ~
where A' U. and 4} are ~7 x M, M x M, and M x M matrices respectively
and specified by
(A)jm = a;~77 O7 Abe) 7 (U)mm' = 2Pmm, 7 (q>)mm' = exp (i c sin (m) [mm'
j = 1, ,J, m = 1, ,M,
where ~ is the distance between the two paired sensors. Assuming U to be
nonsingular, we observe that the determinant of
R=X -·—yR=y = AUNT- ~ *)A* (8.14)
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vanishes if and only if
y=exp~—~ singes), m=l,...,M.
This fact can be used to estimate hem, m = I, . . ., M, as follows: regard Rex
and Ray as measured covariance matrices. For example, we might subdivide
the observation interval T into N equal subintervals ((n—1)T/N,nT/N),
n = 1,..., N. where N = ~T/~2;r), replace the integration limits in (~.13)
by (n- 1~)T/N and nT/N, n = I,...,N, and denote the integrals by own)
and yj~n), j = l, . . ., J; n = 1, . . ., N. Then, put
N N
RXX = N ~ x~n)x*(n), Ray = N ~ x(n)y*(n)
n=1 r~=1
Now substitute these empirical matrices into the left-hand side of (~.14) and
find M minima of the absolute value of the determinant as ~ moves on the
unit circle centered at the origin of the complex plane. Substitute the M
i- v~;u~ Cal 1 ~ ~QlL~;ll~ L~ L11~= 111;1~;111~ ~1~= ~1 v ~ 1V1 vim ~ ~ ~ ~—1 ~ . . . ~ M.
Observe that the knowledge of the sensor positions incorporated into
A and of the signal powers Pmm' is not required. Thus, this method of
estimating the signal arrival directions is free of the costly array calibration.
The price to be paid for this is that the two subarrays must be identical,
with one being a translate of the other.
~~ ~~ I ~ ~~ +~ +~ ~~— —~^ ~1~^ the— ~/~ ~ — I
S.5 Future Research Considerations
The assumption that both the signal and the interference plus noise be
Gaussian fields is primarily for mathematical convenience since the problem
then is completely treatable by linear operators in Hilbert spaces, and Gaus-
sian fields are the simplest class of the second-order random fields. How-
ever, there are evidences, especially in the case of the ocean acoustics, that
the probability distributions of the interference fields considerably deviate
from the Gaussian distribution (Middleton, 1987~. Some simple analytical
examples, such as the "contaminated Gaussian" distribution (Martin and
Schwartz, 1971), have been proposed for the one-dimensional i.i.~. time se-
ries. Although the non-Gaussian interference makes the analytical solution
to the optimum processing problem infeasible, some suboptimum processing
methods are explored in special cases (Monzingo and Miller, 1980~. Since it
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156
is unrealistic to completely specify the probability distribution of the inter-
ference, a robust method, such as the min-max solution (Huber, 1981), has
been sought. The results so far are restricted to the one-dimensional time
series having independent identical distributions (Kassam and Poor, 1985),
and generalization to the higher dimensional case with dependent distribu-
tions should be sought. Another area of investigation is the case where the
interference is a nonstationary and inhomogeneous random field, such as a
transient disturbance. In this case, one might use a semideterministic cri-
terion rather than the totally probabilistic Neyman-Pearson criterion, and
estimate (maximum likelihood) the interference z in (~.2) rather than av-
erage with respect to its probability distribution. One practical problem in
dealing with multidimensional data is computational complexity. Even if
there is an explicit algorithm for the optimum signal-processing, the com-
plex~ty may be too prohibitive to justify its use. Thus, a trade-off between
the detection-estimation performance and the computational complexity, or
the cost of processing the data, must be considered. Study of this trade-off
is another area of useful research in the future.
Bibliography
[1] Cox, H., R. M. Zeskind, and M. M. Owen, Robust adaptive beamform-
ing, IEEE Trans. Acoust., Speech, Signal Process. 35 (1987), 1365-
1375.
[2] FriedIander, B., and B. Porat, Performance analysis of a nub steering al-
gorithm based on direction-of-arrival estimation, IEEE Trans. Acoust.,
Speech, Signal Process. 37 (1989), 461-466.
[3] Gabriel, W. F., Adaptive arrays An introduction, [EKE Proc. NatI.
Aerosp. Electron. Conf. 64 (1976), 239-272.
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157
[4] Haykin, S., ea., Array Processing Applications to Radar, Benchmark
papers in electrical engineering and computer science 22, Dowden,
Hutchinson and Ross, Stroudsburg, PA, 1980.
[5] Helstrom, C. W., Statistical Theory of Signal Detection, 2nd ea., Perg-
amon Press, Oxford, Englan(l, 1986, 87-95.
[6] Huber, P. J., Robust Statistics, John Wiley and Sons, New York, 1981.
[7] Kadota, T. T., and D. M. Romain, Optimum detection of Gaussian
signal fields in the multipath-anisotropic noise environment and nu-
merical evaluation of detection probabilities, IEEE Trans. Inf. Theory
23 (1977), 167-178.
[~] Kassam, S. A., and H. V. Poor, Robust techniques for signal processing:
A survey, IEEE Proc. NatI. Aerosp. Electron. Conf. 73 (1985), 433-481.
[9] Martin, R. D., and S. C. Schwartz, Robust detection of a known signal
in nearly Gaussian noise, IEEE Trans. Inf. Theory 17 (1971), 50-56.
[10] Middleton, D., Channel modeling and threshold signal processing in
underwater acoustics: An analytical overview, [EKE ]. Oceanic Eng.
12 (1987), 4-28.
[11] Monzingo, R. A., and T. W. Miller, Introdluction to Adaptive Arrays,
John Wiley and Sons, New York, 1980.
[12] Roy, R., and T. Ka~lath, ESPRIT Estimation of Signal Parameters via
Rotational Invariance Techniques, IEEE Trans. Acoust., Speech, Signal
Process. 37 (1989), 984-995.
[13] Steinberg, B. D., Principles of Aperture and Array System Design, John
Wiley and Sons, New York, 1976.
[14] Wegman, E. J., S. C. Schwartz, and J. B. Thomas, Topics in Non-
Gaussian Signal Processing, Springer-VerIag, New York, 1989.
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Representative terms from entire chapter:
signal processing
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