Virtual Array Processing Using Wideband Cyclostationary Signals
Transcript of Virtual Array Processing Using Wideband Cyclostationary Signals
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Virtual Array Processing Using Wideband Cyclostationary Signals
Mari lynn P. Wylie
*
WINLAB
Rutgers University
P.O. Box 909
Piscataway,
N J
08855
Abstract
A new spatio-temporal array signal processing tech-
nique for wideband array data is presented that im-
proves spatial resolution and increases the number of
resolvable sources. T he method is predicated on digi-
tal communication signals that are temporally cyclo-
stationary. We show that th e (frequency-dependent)
array manifold has a
separable
representation in the
directions of arrival (DOAs) and the array geometry
and exploit the structure of the cyclic cross spectral
density matrix in order to obtain virtual array ‘obser-
vations’
wzthout
a-priori knowledge of the DOAs.
1
Introduction
Frequency focusing techniques for direction of ar-
rival estimation (DOA) of temporally wideband sig-
nals is
a
problem of considerable continuing interest
motivated, for example, by developments in mobile
communications. One approach to this problem for
wide-band array data as will be shown) is to design
frequency-dependent transformations which focus the
DOA information to
a
reference frequency and, simul-
taneously, generate virtual. array ‘observations’. Wit h
proper design, the new, virtual array geometry may
be selected
so
as
to
increase the aperture and hence
obtain improved DOA estimates.
An important contribution t o wide-band DOA es-
timation can be traced to the subspace focusing tech-
nique introduced by [l] termed
as
Coherent Signal
Subspace Method (CSSM). In this method, stationary
wide-band array d at a is decomposed i nto narrowband
components followed by focusing of the narrowband
components to a reference frequency. Subsequent to
frequency focusing, the DOA estimates are obtained
using conventional signal subspace methods (such as
MUSIC) for narrowband signals. Some new interpre-
tations and results related to CSSM were proposed
[2]-[3]. Unfortunately, performance o the CSSM is
critically dependent on some a-priori initial estimate
of the DOAs close to the true value, in order to con-
struc t the focusing matrix. An import ant contribu-
tion to resolution of the above dilemna was provided
by [GI, who demonstrated the
separabi l i t y
of the array
manifold into two functions: one of the DOAs and an-
ot,her of the array geometry and temporal frequencies,
*Author for
all
correspondence; e-mail:
mwyIieQ winlab Tu gers edu
Sumit
Roy
Divn.
of
Engineering
University of Texas
San A ntonio , T X 78249
respectively. Th e resulting array manifold interpola-
tion matrix (focusing matrix) is then computed wzth-
out
any a-priori knowledge of the DOAs of the sources.
In this work, we develop a new technique for DOA
estimation of wide-band signals for 2-D (planar) arrays
by combining the separability of the array manifold
with exploitation of th e wide-sense cyclostationarity
of the signals of interest (SOIs). It is shown that the
array manifold has
a
separable representation which
may be exploited in order to focus its information to
a reference frequency bin
without
a-priori knowledge
of the DOAs. We also introduce a spatial extrapo-
lation parameter which, if judiciously selected during
the frequency focusing procedure, generates ‘virtual’
array observations at the reference frequency.
i)
A-priori DOA estimates are not required to gener-
ate
the ‘virtual’ array observations;
ii) Rejection of wide-sense stationary or cyclostation-
ary interference and noise;
iii) Increased resolution by spatial extrapolation.
The salient features of this new method are:
2 Problem Formulation
The signal observed by an M-element planar array
is assumed to be the supe rposition of L a cyclostation-
ary waveforms received in the presence of interference
and noise. Th e signal measured at the output
of
the
mth sensor can be described by
I=1
over the observation interval 0 5 t 5 TOand for
m = 1, . . . ,M .
{ ~ ~ ( t ) } ~ ~re the radiated signals,
and {n,(t)}Z=, is interference. Th e delay T ~ B I ) =
, where 01 denotes the DOA of the
c m sin h + y m c os )
lth radiated waveform
as
measured from the array
broadside and (x m , m) denotes the 2-D coordinates
of the mth array element normalized
relatzve
to the
wavelength,
A 0
= c/fo.
The Fourier coefficient of (1)
at
frequency 6 s de-
fined as Z m f k )
=
?
zm( t )e- l”fk tdt
In vector
f o
1058-6393/96 5.00 1996 EEE
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notation, we have
L
I=1
where
.
4:)
2 T a z m
s n ~~+y,,, cos^^
[GIl,,l ( f k ) = e fo
Sl fk)
nd
;V,,,(fk)
are the lcth Fourier coefficients of
the Ith radiated waveform and the noise/interference
present at the mth sensor, respectively.
s / t ) is a cyclostationary signal and therefore ex-
hibits spectral correlation at frequencies separated by
multiples of the cycle frequency, a ,
{fk
+ 5 , k
-
5 )
[7], where Q. typically corresponds to the bit rate, chip
rate, or twice the carrier frequency. Let us define the
cyclic autocorrelation of the I t h waveform (in discrete
time) to be
(where
i
is the discrete time variable).
We assume that the signals of interest are mutually
cyclically uncorrelated with each other and with the
noise, and that the noise is cyclically uncorrelated with
itself at cycle frequency
cr,
i.e.,
~ { s l [ i ] s : , [ ; j e - j ~ ~ ~ ~ }
P S ~ ~ (6)
E {
SI [;In;
[ i ] e - j?=ai} =
0
( 7 )
E {
Invt[i] T a i
} = O
(8)
where
E { . }
s the expecta.tion operator.
3
The
Virtual Array Processing Algo-
rithm
Th e first step in generating the virtual arr ay obser-
vation is to operate on the vector pair, z ( f k + a / 2 )
and
e ( f k
a/2)
in
order t o generate
a
set of obser-
vations that are commensurate with the narrowband
model of signal recept ion, which (in general) , is given
bY
In the sections to follow, it will be shown that is
selected
as a
spatial extrapolation parameter.
In general, t he focusing matrices Tm fk) m
=
1,2)
are functions of the unknown {Oi}f:,. However, we
show that by exploiting the separability of the depen-
dence of
(f)
on the array geometry and frequency
versus the direction of arrival, that we can construct
Tl fk) and Tz fk) without a priori knowlege
of
the
source directions of arrival.
3.1 Separability of (f)
The focusing matrices Tl fk) and T? fk)are cle-
rived by exploiting the separability of (f) using the
Jacobi-Anger expansion of a plane wave in a series of
cylindrical waves [9]
m
r = - m
where J,.(.) s the r%horder Bessel function of the first
kind. Using this expansion, we approximate
z e f ) )
~ ( f ) $ e
(15)
where G f) and
2R
+
1 vector given, respectively, by
are the M x 2R + 1 matrix and
[G f)],, =
j r - R- 1
e
- j r -R- l )Lrt
Jr -
R-
1 (2r f /fO rra 1
n
= l , . . . , M , T = 1 , . . . , 2 R +
1.
T = y, + j ~ , ,
and R is the highest order Bessel function used
in
the
approximation.
Using (15), we define
Tl fk)
and
T? fk) as
the
M
x M matrices which satisfy
Tl fk)G fk + a / 2 )
= G ( ( 1 -
7 ) f o )
(17)
T~ fk)G* fk
a / 2 )
= G* -yfo). (18)
3.2
Virtual Array Observations
(13) at each frequency, we have th at
Applying the transformations indicated in (12) and
=
a'e(f)s(t)
+ n- t).
(9)
L ,
@l(fk) = Czer((l ) f O ) S I ( f k
/ a )
Assuming t,hat the L , cyclostationary arrivals are
characterized by the common cycle frequency
a,
we
I 1
define + Qd f k +ala )
(19)
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Now, taking the inverse DFT of (19) and (20) , the
nth snapshot
is
given by
l O
+
.i[nl (21)
La-1
7 i i [ 7 l ]
=
4,
-7fo)s;
[ i ] e - ~ ~ *
l O
.i[n]. (22)
Because of the focusin operations (using Tl (f k) and
T ? ( f k ) , the vectors &fi] and
&[i]
can be modeled as
narrowband. The final step is to take the expected
value of the random vectors:
2
= E
{ w l [ i ]
@ w z [ i ] )
(23)
where
@
represents the Kronecker product.
(In this
case, the Kronecker product is just the element-by-
element product of the vectors).
Substituting (21 and (22) into (23
,
assuming un-
correlated SOIs (61 and using the i ntity
AB) @
CD)
=
A ) B
)
[SI, it follows that
La
*
C s s , ( f O ) C P
(24)
1=1
where c y = E {
sr i]sy i ] e - - 3 2 T a r }
nd the M2-vector
( f a ) = G 1 - *Ofo) @ G,(-rfo). (25)
j j ~ ,
f o ) is the M2 x 1 vzrtual array manifold vector
generated by taking the Knonecker product and is the
concantenation of M subvectors each of length M x 1.
The
mth
such subvector of Yo fo) (for m = 1, . ,M )
is given by
(26)
{~,(fo)
thus corresponds to a virtual array with el-
ements places at locations
2, +
y(z1
z m ) , y m +
Y( YI
- ~ m ) )
. .
~ ~ + Y ( ~ M - - ~ ) , Y ~ + Y ( Y M
~ m . ) )
for m =
1, . . . ,
M . We note that for
y
=
0,
the vir-
tual array sensor geometry coincides with the original
array geometry (with redundancy).
The vector of virtual array 'observations' given in
(24) may be used in a conventional narrowband direc-
tion finding algorithm in order t o estimate the source
directions,
{6}fg1.
Note the attenuation of noise and
interference since they are assumed to be cyclically un-
correlated a t cycle frequency
CY.
Also, the parameter
may be used in order to increase the virtual array
aperture vis-a-vis the true (physical) aperture.
e32 T f o
k m -r zi l m
I sinel+[ym+-r Y1 - Y ) ] co s
0
f o
2 T f o =m+-r =M - = m I
s i n
@ I + [ Y ~ + ~ Y M - Y ~ ) ] ol 8
f0
In practice, we estimate
R
using N snapshots to
form the statistic
The conventional beamforming algorithm employed in
the next section operates on the vector in (27) (assum-
ing the virtual array geometry) in order to achieve
enhanced resolution vis-a-vis the original array geom-
etry.
4
Simulatioiis
The algorithm described above was implemented
using simulated array da ta and assuming uniform cir-
cular arrays of various radii.
M =
15 uniform circular
array with a (normalized) radius of unity. The SOIs
and cyclostationary interferers were all direct sequence
spread spectrum observed over Nb observation inter-
vals. The P h transmitted waveform is characterized
by the chip rate, CY. The frequency band of interest
in each simulation, (fo
5
: fo + .5)/fo, with fo nor-
malized to unity. In each example, the cutoff for the
approximation in (16) is chosen
as
R was chosen to
equal the largest integer not exceeding ( M 1)/2. In
all examples, a maximal length code was used for the
chipping sequence.
4.1 Simulation
Examples
Example 1: The simulation indicates the increased
resolution obtained using the virtual array process-
ing algorithm compared to results obtained using the
narrowband focusing technique discussed in [3]. The
array has
M
= 15 elements, with y = -0.5. There are
7
SOIs present spaced uniformly every 30 from 0 to
180'.
Colored noise with variance (a t each element) of
unity and correlation of
0.5
with adjacent sensors was
used. 50 bit intervals were observed and the number
of chips per bit was 6. A 15 chip maximal length code
was used. Th e cycle frequency,
CY
=
116.
Example 2: Thi s simulation demonstrates the ability
the anti-aliasing feature of th e virtual array processing
algorithm. A
9
element array was used with
=
-0.5
and one source present at 81 = 100'. Length 7 chip-
ping code was generated. Whi te noise was generated
and th e signal to noise ratio was 10 dB.
As
is evident
from Figure 2, the virtual array processing algorithm
is capable of estimating the source location although
the information is aliased using the original array (and
narrowband focusing).
Example
3:
This simulation demonstrates the sup-
pressio n of cyclostationary interference using a 15el-
ement array with 7 = -0.5. The
SO1
has unit energy
and arrives at l oo , while t he cyclostationary interfer-
ers are present at
02
= 120°,
6
= 150 and
64 = 90°.
30 bit intervals of the SO1 are observed. The SIR is 10
dB for each interferer and the noise is white with unit
variance. Th e cycle frequency of the SO1 is CY
=
1/6
while the interferers share the common cycle frequency
Y
=
1/7.
As
is demonstrated in Figure 3 , the virtual
array processing algorithm suppresses the undesired
interference.
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5 Conclusion
In this paper, we have
a
described
a
new method
for obtaining a virtual array geometry for wide-band
array signal processing. Th e struc ture of the array
manifold was exploited in order to refocus
a
vector
pair of frequency shifted measurements without a-
priori knowledge of the DOAs and to generate new,
virtual array ’observations’
at
the final operating fre-
quency, fo We have shown the improved resolution
after application of the virtual array processing algo-
rithm vis-a-vis that of the original array geometry. I[n
addition, this algorihm is useful for the cancellation
of
unwanted cyclostationary and wide-sense stationary
interference (noise) present
at
the array.
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Es-
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Actual Array (M=9)
90,
270
27
Simulation 1
901
Simulation
0
80
270
70
510