Toward Bandwidth Invariance of Spatial Processing in ... - IEEE Xplore

Toward Bandwidth Invariance of Spatial Processing in the Non-Cooperative Receiver Arnab Shaw Wright State University 3640 Colonel Glenn Highway Dayton, OH 45435 [email protected]

Moshin Jamali University of Toledo Toledo, OH 43606 [email protected]

Nathan Wilkins Air Force Research Laboratory, Sensors Directorate Bldg 620, SNRP 2241 Avionics Circle WPAFB, OH 45433 [email protected] Abstract - As the modern electro-magnetic environment in urban and battlefield scenarios becomes increasingly denser, system designers tend to gravitate toward wide bandwidth solutions. Not only do spread spectrum designs permit increased co-channel utilization, they also tend to be inherently more secure, decreasing the probability of intercept by non-cooperative receivers. This is in part due to the code that must be applied in order to modulate the information bandwidth into a much wider transmitted bandwidth. In addition the non-cooperative receiver must contend with issues of increased noise, decreased sensitivity, and increased signal processing resource requirements among others. The problem is further exacerbated for receivers that must perform emitter geo-location or other forms of spatial processing, since the conventional high-resolution techniques rely on phase measurements based on the monochromatic assumption. This paper will discuss some of the existing methods suggested to contend with wide bandwidth spatial processing for noncooperative receivers and their limitations. It will introduce a new method based on the bilinear transformation that overcomes some of these limitations. It will present simulation results that demonstrate the advantage of this technique.

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I. INTRODUCTION Several researchers have reported techniques for localization and source spectra estimation of wideband sources over the last four decades. The conventional approach for the case of a single source is to use a form of generalized correlator to estimate the Time-Difference-Of-Arrival (TDOA) of the signal at the sensors. Maximum likelihood based methods for single and multiple sources require the knowledge of source and noise spectra and are computationally expensive. Parameter estimation based methods assume Auto-Regressive Moving Average (ARMA) models for the received signals and the estimated ARMA parameters are utilized for TDOA estimation. Computational complexity of these methods is also high and the performance of these approaches depends on the assumed model for the unknown wideband signals. Extending existing ideas in the narrowband problem, Wax et al [1] proposed eigen decomposition-based approaches for wideband source localization. In this approach, the eigenvectors of the estimated spectral density matrix at each narrowband bin of the signal bandwidth were incoherently combined to estimate the TDOAs. Wang and Kaveh [2,3] presented a coherent signal subspace-

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based approach that avoids the rather expensive eigen decomposition of spectral density matrices at each frequency bin. In their approach, initial estimates of the angles of arrival are used to transform the signal eigenspaces at different frequency bins to generate a single coherent signal subspace and then a generalized eigen decomposition is used to obtain more accurate estimates. The algorithm iteratively estimates well-separated angles by focusing at different angles each time. More recently, Gelli and Izzo [4], among others have suggested enhancements largely based on the Wang and Kaveh formulation that exhibit improvements for specific signal types, but retain significant limitations.

angle that the kth wavefront makes with the line of the array, and ni(x) is the additive noise at the ith sensor. Representing both sides by respective Fourier coefficients, p

Ri(ZƐ) =

D

6 e-j Z (i-1) c sin X Sk(ZƐ)+ Ni(ZƐ) k=1 Ɛ

k

(2)

With ZƐ = (2S/T)Ɛ, Ɛ = Ɛ1,…,Ɛ1+nf, where ZƐ1 and ZƐ1+nf are the lowest and highest frequencies in B. In matrix notation, R(ZƐ) = A(ZƐ)S(ZƐ) + N(ZƐ)

(3)

where

II. DISCUSSION A. Problem Formulation The observed signal is assumed to be composed of p plane waves with an overlapping bandwidth of B Hz. They are sampled simultaneously at the output of a linear array of M (>p) equally spaced sensors. The signal received at the ith sensor is expressed as p

ri(t) =

6

D sin X ) + n (t) k i

s (t-(i-1) k=1 k c

(1) -T/2 = t = T/2, 1 = i = M Where sk(x) is the signal radiated by the kth source, D is the separation between sensor elements of a uniform linear array, c is the propagation velocity of the signal wavefront, Xk is the

(4a) (4a) (4a)

And A(ZƐ) is an M x p direction-frequency matrix (4d) A(ZƐ) =

1

…

e -j ZƐ W1

… e

e

…

-j ZƐ (M-1)W1

1 -j ZƐ Wp

…

In addition to simulation results we briefly discuss aspects of hardware implementations that can be used to implement key processing steps of the proposed method. The implementations incorporate hardware efficient techniques and utilize recent advances in available reconfigurable digital hardware.

R(ZƐ) = [ R1(ZƐ)…RM(ZƐ)]t N(ZƐ) = [ R1(ZƐ)…RM(ZƐ)]t S(ZƐ) = [ S1(ZƐ)…SM(ZƐ)]t

…

In this paper a simple bilinear transformation matrix and the approximation resulting from dense and equally spaced array structure assumption are utilized to combine the individual narrowband spectral density matrices for coherent processing. In a related problem, Henderson [5] used a bilinear transformation and dense array approximation for rank reduction of Hankel/Block-Hankel type data matrices. The method we propose is non-iterative and does not require preprocessing to obtain initial estimates of the angles of arrival. All the angles are estimated from a single step of coherent subspace computation [8]. The performance of the proposed method is characterized and compared to other methods using several simulation experiments. The results of these simulation experiments demonstrate the efficacy of the proposed method.

…

e

-j ZƐ (M-1)Wp

with Wi = (D/c)sinXi

(4e)

being the time difference of arrival of the ith source. The covariance matrix of the Fourier coefficient vector R(ZƐ) will approach the spectral density matrix if the observation time is large enough compared to the correlation time of the processes [6]. With this assumption, K(ZƐ) = A(ZƐ)Ps(ZƐ)AH(ZƐ) + Vn2Pn(ZƐ)

(5)

Where K(ZƐ), Ps(ZƐ) and Pn(ZƐ) are the spectral density matrices of the processes ri(x), sk(x) and ni(x), respectively. AH(ZƐ) stands for the transpose conjugate of A(ZƐ). The noise process is assumed to be independent of the sources and the noise spectral density matrix is assumed to be known except for a multiplicative constant Vn2. With the above model at hand, the problem is to estimate the ri’s from the estimated covariance matrices Kc(ZƐ) of the received signal plus noise. Estimates of

557

the angles of arrival, Xi ‘s, can then be computed using the relationship in (4e).

1

BA(ZƐ) =

…

Ɛ

(1+e-j Z

Ɛ

Ɛ

1

W1

-j ZƐ Wp M-1

1-e …( 1+e-j Z W )

M-1

)

…

(7)

( 1+e 1 =

-j ZƐ Wp

)

M-1

… 1

j tan( ZƐ2W1 )

… j tan( ZƐ Wp ) 2 … M-1

Z W ( j tan( Ɛ2 1 ))

E(ZƐ) M-1

ZƐ Wp … ( j tan( 2 ))

(8)

Where E(ZƐ) denotes the diagonal matrix in (7). If the antenna spacing D is small compared to all wavelengths in B, then tan(ZƐWi/2) | ZƐWi/2,  i, j. Using this approximation and premultiplying BA(ZƐ) by an MxM diagonal matrix D(Zc/ZƐ) whose (m,m)th term is geven by (2Zc/jZƐ)m-1, where Zc = 2Sfc and fc is the midband frequency in B, we obtain, 1

…

ZcW1

… …

( ZcW1 )

M-1

{ Ac(Zc)E(ZƐ)

1 Zc Wp

…

D(Zc )BA(ZƐ) | ZƐ

(6)

Now, since A(ZƐ) in (4a) has a Vandermonde structure, it can be shown that,

p

…

1 -1 1 -1

-j ZƐ W1 M-1

( 1-e ) 1+e-j Z W

…

BM=4 =

1 3 3 1 1 -1 1 -1 -1 1 -3 3

1-e -j ZƐ Wp … 1+e-j ZƐ Wp …

…

C. Proposed Method In order to avoid the requirement to perform an initial angle estimate, we propose using a bilinear transformation matrix similar to that presented in [5]. Let B be an MxM matrix constructed from the coefficients of the M-1th oder zpolynomials, pk(z) = (1-z)M-k(1-z)k-1, k=1, 2, …, M. The elements of the kth row of B are the coefficients of pk(z) taken in ascending order of z. The coefficients of pk(z) can be found by convolving the coefficients of the polynomials (1+z)M-k and (1-z)k-1. For example, a 4x4 B matrix will have the form,

1-e -j ZƐ W1 1+e-j ZƐ W1

…

B. Previous Methods The two major approaches which exploit the properties of the eigenspaces of K(ZƐ) to estimate the angles of arrival are those of Wax [1] and Wang/Kaveh [2, 3]. In [1], eigen decompositions are performed in all frequency bins and globally orthogonal direction vectors are obtained. The distance measures used by this method require eigen decomposition of Kc(ZƐ) for all Ɛ = Ɛ1,…,Ɛ1+nf, which is quite computationally expensive. Instead in [2 and 3], a transformation matrix was employed to reduce this burden. Using initial estimates of the angles of arrival, transformation matrices were formed such that direction-frequency matrices of all the frequency bins were transformed to the center frequency, Zc, in B. Then using the coherent sum of the transformed data, the coherent signal subspace theorem for the matrix pencil was utilized to estimate the angles of arrival. This is a more computationally efficient method for estimation of the angles of arrival than that of [1], however, it involves the limitation that the result is “focused” around the initial angle estimate.

… 1

… (ZcWp )

M-1

(9)

The matrix Ac(Zc) whose columns are the transformed direction-frequency vectors has dependence on Zc instead of ZƐ. Also note that Ac(Zc) has Vandermonde structure and its columns are linearly independent as long as Wi z Wj for i z j. The new transformation matrix that we define as Tc(ZƐ) { D(Zc/ZƐ)B, does not depend on the angles of arrival and since B is nonsingular and D(Zc/ZƐ) is diagonal with nonzero diagonal elements, Tc(ZƐ) is also nonsingular. Using these transformation matrices Tc(ZƐ), Ɛ = Ɛ1,…,Ɛ1+nf, all the spectral

558

density estimates can now be combined in the following manner, Ɛ1+nf

Gc {

6 Tc(ZƐ)Kc(ZƐ)TcH(ZƐ)

Ɛ=Ɛ1

= Ac(Zc)GscAcH(Zc) + Vn2Gnc

(10)

X1=9q and X2=12q, Figure 1 depicts the initial angle estimate using the Multiple Signal Classification (MUSIC) algorithm [7]. Note that the two sources are not resolved. Using the “focusing” method of [2 and 3], the plot of Figure 2 results, just resolving the sources, but the transformation matrices are dependent on the initial angle estimate.

where Ɛ1+nf

Gsc =

and

6 E(ZƐ)Ps(ZƐ)EH(ZƐ)

Ɛ1+nf

Gnc =

(11)

Ɛ=Ɛ1

6 TcH(ZƐ)Pn(ZƐ)TcH(ZƐ)

(12)

Ɛ=Ɛ1

Next, the coherent signal subspace theorem for the matrix pencil (Gc, Gcn) is utilized to estimate all the angles of arrival by computing the maxima of the measure, Jc(X) =

1 M

6 __a k=p+1

X

cH(Z

c)êck(Zc)

__2

(13) Figure 1. Initial estimate using MUSIC

where êcp+1(Zc), êcp+2(Zc), …, êcM(Zc) are the generalized eigenvectors of the matrix pencil (Gc, Gcn) corresponding to the smallest M-p eigenvalues and the aXc(Zc)’s are the new direction-frequency vectors defined as aXc(Zc) { [1 Zc(D/c)sinX … (Zc(D/c)sinX)M-1]t. III. SIMULATIONS The same simulation examples as presented in [2 and 3] were used to evaluate the performance of the proposed method. In all the simulations, a linear array of M=16 equally spaced antennas was used. The spacing between two consecutive antennas is D = c/(rfc) where r is the ratio of the wavelength at fc and the interelement spacing, D. The source signals are temporarily stationary bandpass white Gaussian processes with zero mean. The noise processes at each antenna are stationary, statistically independent, identical white Gaussian bandpass processes with zero mean and are independent of the source processes. The sources and the noise processes have the same bandwidth of B=100. The same sampling specifications and data segmentation as described in [2 and 3] were used. The received signal plus noise processes were sampled at each antenna at 80 Hz and then divided into 64 segments of 64 samples each and then each segment was transformed into frequency domain by unwindowed FFT to obtain nf+1=33 narrowband components. Using two uncorrelated sources at

Figure 2. Focusing matrix method Using the bilinear transformation matrix method as described here, we obtain the following results that are not dependent on an initial angle estimate. Figure 3 shows the results for SNR=10dB and r=4. The Figure shows overlapped plots of 5 independent runs. The source angles are clearly well resolved at this low SNR. In Figure 4, the results of the case of one signal being well separated from the two closely spaced sources is shown. Three independent sources with sinX1=0.15, sinX2=0.2 and sinX3=0.4 were used with SNR=10dB and r=3. The results of the five runs show that all three angles are well

559

estimated by one step of the modified coherent signal subspace processing method described above, whereas in [2] at least three iterations were required to ensure the angular positions of the three sources.

angle of arrival computation accuracy. Any algorithm for estimation of wide bandwidth emitter parameters typically involves several computational modules. For such algorithms to be appropriate for real time applications, we must efficiently implement these modules [9]. They include generation of the bilinear transformation matrices, covariance matrix estimation, and computation of eigenvalues/eigenvectors in field programmable gate array devices.

Figure 3. Spatial spectrum for two uncorrelated sources

Figure 5. Spatial spectrum for two correlated sources ACKNOWLEDGEMENT The authors wish to thank Dr. H. Hung, Dr. H. Wang and Dr. M. Kaveh for providing the algorithm and programs used in their simulations. REFERENCES Figure 4. Spatial spectrum for three uncorrelated sources The results for the case of two completely correlated sources [3] are shown in Figure 5 for SNR=10dB and r=4. The results show that the angles were resolved in all cases though there seems to be some variability in the estimate of the source at X2=12q. Studies are currently underway to help define efficient hardware implementations for the bilinear transformation algorithm. These will determine performance in terms of computation time and memory requirements for the computation of wideband angle of arrival estimation. Investigation of hardware implementations involves trade studies that will compare required computation time with the

1. M. Wax et al, “Spatio-Temporal Spectral …,” IEEE Transactions on ASSP, vol. ASSP-32, no. 4, Aug 1984 2. H. Wang and M. Kaveh, “Estimation of Angles …,” in Proc. ICASSP 84, pp. 7.5.1-7.5.4, Mar 1984 3. H. Wang and M. Kaveh, “Coherent Signal Subspace …,” IEEE Transactions on ASSP, vol. ASSP-33, no. 4, pp. 823-831, Aug 1985 4. G. Gelli and L. Izzo, “Cyclostationarity-Based Coherent Methods …,” IEEE Transactions on Sig. Proc., vol. 51, no. 10, Oct 2003 5. T. L. Henderson, “Rank Reduction for Broadband …,” 19th Asilomar Conf. On Circ., Syst. & Comp., Nov 1985 6. A. D. Whalen, Detection of Signals in Noise, Academic Press, 1971 7. Ralph O. Schmidt, “Multiple Emitter Location and Signal Parameter Estimation,” IEEE Transactions on Antennas and Propagation, Vol. AP-34, No. 3, March 1986 8. A. K. Shaw and R. Kmaresan, “Estimation of Angles of Arrivals of Broadband Signals,” ICASSP 1987, pp. 2296-2299 9. M. M. Jamali and S. C. Kwatra, “Design of Special Purpose Parallel Hardware for Real Time Applications, “Proceedings of the 1993 National Aerospace and Electronic Conference,” Vol. 2, ppl 54-60

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