An Augmented Smoothed Rank Profile Algorithm for ...

IEEE TRANSACTIONS ON ACOUSTICS. SPEECH. AND SIGNAL PROCESSING, VOL. 37. NO. 7. JULY 1989

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a sinusoid, provides two options for spectrum analysis. The first of these, via the clipped ACF, requires a sampling rate many times the Nyquist rate in order to match the performance of a conventional DFT analyzer [10]. The second of these, of which the Kay and Sudhaker analyzer is a particular example, requires a similarly high clock rate. The Kay and Sudhaker analyzer does not therefore appear to offer any advantage in bandwidth over either the 1 bit ACF analyzer or a conventional analyzer. Addition of a triangular auxiliary waveform, rather than a sinusoid, generates naturally sampled pulse duration and pulse position modulation T(XJ). Direct Fourier transformation of the intervals T will lead to spectral distortion except in the limiting case of small signals. However, various algorithms (e.g., Hostetter [11]) may be used to yield the 2/Vdiscrete complex Fourier transform coefficients as in a conventional analyzer with lower computational complexity than the Kay and Sudhaker method. REFERENCES

[1] S. Weinreb, "A digital spectral analysis technique and its application to radio astronomy," Mass. Inst. Technol. Res. Lab. Electron., Tech. Rep. 412, 1963. [2] J. H. van Vleck, "The spectrum of clipped noise," Proc. IEEE, vol. 54. pp. 2-19. Jan. 1966. [3] B. P. Th. Veltman and H. Kwakernaak, "Theorie und Technik der Polaritatskorrelation fur die dynamische Analyse niederfrequenter Signale und Systeme," Regelungstechnik, vol. 9, pp. 357-364, 1961. [4] P. Jespers, P. T. Chu, and A. Fettweis, "A new method to compute correlation functions," IRE Trans. Inform. Theory, vol. IT-8, pp. 106-107, Sept. 1962. [5] J. Ikebe and T. Sato, "A new integrator using random voltages," Electroiech. J. Japan, vol. 7, no. 2, pp. 43-47, 1962. |6] H. Berndt, "Correlation function estimation by a polarity method using stochastic reference signals," IEEE Trans. Inform. Theory, vol. IT-14, pp. 796-801, 1968. |7] J. B. H. Peek, "The measurement of correlation functions in correlators using 'shift invariant independent' functions," Phillips Res. Rep., suppl. 1. |8J P. J. Kindlemann and E. B. Hopper, "High speed correlator," Rev. Sci. lustrum., vol. 39, no. 6, pp. 864-872, 1968. [9] I. Bar-David, "An implicit sampling theorem for bounded band-limited functions," Inform. Contr., vol. 24, pp. 36-44, Jan. 1974. [10] P. J. Edwards, "Zero crossing-based spectrum analysis," in Proc. lasted Int. Symp. Signal Processing Appl., ISSPA:87, vol. 1, 1987, pp. 217-220'. [11] G. H. Hostetter, "Recursive discrete Fourier transformation with unevenly spaced data," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-31, pp. 206-209, 1983.

shows that for determining the source coherency structure, the number of elements required is greater than or equal to the total number of sources plus the highest degree of signal coherency present. In the modified approach suggested here, we show how the SRP method can be augmented by the incorporation of "forward-backward" averaged covariance to increase the likelihood of successful determination of source coherency structure (SCS) with a limited number of sensors. We would like to caution, however, that for some situations, the modified approach may not yield the desired increase in the array aperture. I. INTRODUCTION

The eigenstructure method has proven to be an effective approach to bearing estimation when the sources are partially coherent. In the presence of fully coherent sources, a spatial smoothing technique to preprocess the array covariance matrix for decorrelating the sources has been proposed by Evans et al. [6] and more fully developed by Shan et al. [2]. They have shown that a condition for solvability using spatial smoothing method is that the number of sensors required is greater than or equal to the total number of sources plus the highest degree of signal coherency present. More recently, Shan et al. have also suggested a procedure that can be applied to the array covariance matrix for determination of source coherency structure (SCS). A so-called smoothed rank profile (SRP) of a telescoping series of matrices obtained by averaging smaller and smaller principal diagonal submatrices of the original covariance matrix is generated and used to deduce the SCS. The purpose of this correspondence is to propose a modification of the SRP procedure which is motivated by the "forward-backward" spatial smoothing technique proposed by Evans et al. [6] and investigated in detail by Williams et al. [4], It is shown here that a slightly modified procedure, called the "augmented smoothed rank profile," may relax the requirements on the number of sensors to a significant extent. However, under certain situations, this approach may not be able to yield the desired increase in array aperture. II. PROBLEM FORMULATION

Consider a linear array with M equally spaced omnidirectional sensors. Assume that D narrow-band sources impinge on the array as planar wavefronts with incident angles 9 , , • • • , Q,,. Using the standard and narrow-band assumptions, we can write the sensor signal vector as r(t)

r{t)

An Augmented Smoothed Rank Profile Algorithm for Determination of Source Coherency Structure SURENDRA PRASAD AND BINDU CHANDNA

Abstract—This correspondence deals with the problem of determining the source coherency structure and estimation of directions of arrival in the presence of coherent signals. The method is closely related to the smoothed rank profile (SRP) algorithm presented in [3], which

= As(t)

+ n(t)

(1)

where =

[r,(0.

•••

.r

M

(t)]\

A = [a(0,), ••• ,«(9D)] s(t) = [S](t), • • •

n(t) =

, sD(t)]T,

and

, nM(t)\T.

A is the array manifold comprised of the array steering vectors for the directions of the D sources; s(t) represents the vector of complex envelopes of the sources, and n(t) denotes the additive noise vector. [ • ] T denotes the matrix transpose. The array covariance matrix can be written as

R = ASA+ + a1! Manuscript received July 20, 1987; revised October 3, 1988. This work was supported by the Naval Physical and Oceanographic Laboratory (N.P.O.L.) and the Department of Electronics (D.O.E.), Government of India. S. Prasad is with the Department of Electrical Engineering, Indian Institute of Technology, New Delhi 110 016, India. B. Chandna is with the Centre for Applied Research in Electronics, Indian Institute of Technology, New Delhi 110 016. India. IEEE Log Number 8928133.

where [ • ]+ denotes the conjugate transpose, S is the source covariance matrix, and a 2 / is the covariance matrix of the additive noise, assumed to be uncorrelated from sensor to sensor. If the source covariance matrix is nonsingular, i.e., no two sources are fully coherent and M > D, then it is well known that [1] 1) the smallest eigenvalue of R is a2 which has a multiplicity of M - D, and

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IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING. VOL. 37. NO. 7, JULY 1989

2) the space spanned by the eigenvectors corresponding to the minimal eigenvalue is orthogonal to the columns of the direction matrix A. However, if the sources are coherent with respect to each other, the S matrix becomes singular, and the above properties are not available in the eigenstructure of R. A solution of this problem was proposed by Evans et al. [6] and developed further by Shan et al. [5] and is referred to as spatial smoothing. Their solution is based on a preprocessing scheme that essentially decorrelates the signals by averaging the covariance matrices across several identical overlapping subarrays in order to restore the rank of a smaller subarray covariance matrix to D. When there are several groups of coherent signals, the minimum number of subarrays required equals the size of the largest group of coherent signals. Shan et al. [3] have also suggested a procedure called smoothed rank profile (SRP) to deduce the source coherency structure. Let {g h i = 1, • • • , L } denote the number of coherent groups of degree / and let L be the highest degree of source coherency present (i.e., maximum number of sources that are coherent with one another in any group). Note that {gh i = 1, • • • , L} is a characterization of SCS. The total number of coherent groups Q is then Q =

s,

Let Uj denote the (M — /) x (M — i) smoothed covariance matrix, obtained by averaging (/ + 1) subarray covariance matrices, corrsponding to a subarray of length (M — i). The smoothed rank profile is then defined as the set of ranks r ( t / , ) of the matrix sequence { t / , } , i.e., SRP (Uo) = { r ( I / , ) , i = 0, • • • , M - 1} where it can be shown that k = 0

The augmented smoothed rank profile of the set (U, V) is then defined as ASRP (U, V) = [ {T( U,), i = 0, 1, • • • , M } U {T(V,), / = 0, 1, • • • , M } ] . In order to see how the additional information in ASRP may help us, let us first look into the nature of the rank profiles of the matrix sequence { V,} in relation to that of {{/,}. It has been established in [6], [4], and [8] that, under favorable situations, the forward-backward averaging can restore the rank deficiency of R due to signal coherency, nearly twice as fast as conventional averaging. Therefore, the rank profile of the matrix sequence { V,} may be expected to rise faster than that of { U,}, with the rank becoming equal to the total number of sources present after the use of only L/2 subarrays. On the other hand, the rank profile of the sequence { {/,} will rise to this value only after the use of L subarrays. It follows, therefore, that the information about the total number of sources present is available sooner from the stationary segments of the rank profile of {V,}. This additional information present in the ASRP, therefore, can potentially help determine the coherency structure more efficiently (i.e., with fewer number of sensors). Consider now the situation where the required condition regarding the stationarity of the SRP is not satisfied. Although this does imply that a complete determination of the SCS is not possible from the SRP, it is clear from (2) that the "increasing" segments of the SRP will still yield a correct, although incomplete, determination of SCS. The failure will only occur at the qth stage where the (M - q) x (M - q) matrix Uq becomes full rank for the first time in the sequence. Meanwhile, the stationary segment of the rank profile of { V,} will give us the total number of sources D. This additional information, along with the value of T(U0), which tells us the total number of coherent groups, Q, can be used to determine the SCS via the following procedure. Step 1: Using the computed values of gt, • • • , gq, calculate the number of coherent groups Q' as

M - k, S ig, + (k + 1) k = 1,

, M - 1.

The SCS is determined as the set of negative second-order differences in the increasing and the stationary segments of SRP, i.e.,

g, = - { r ( i / , . _ 2 ) - 2 r ( t / , _ , ) + r ( t / , ) } where

r ( I / _ , ) = 0.

(2)

It has also been shown by Shan et al. [3] that in order to obtain the SCS, the minimum number of sensors required is equal to the total number of sources plus the highest degree of source coherency present. In the next section, we show that it is possible to reduce this number via a simultaneous examination of the rank structures of two sequences of matrices, viz. the smoothed matrix sequence { {/,} as discussed above, and a corresponding sequence { V,} of the forward-backward smoothed matrices to be defined in the sequel. III. AN AUGMENTED SMOOTHED RANK PROFILE PROCEDURE

(ASRP) Let us define the (M — /) x (M — /) matrices V, as follows: V, = U, + E,[U,]*En

i = 0, 1, • • • ,M

Step 2: Obtain the estimated number of sources D' from the obtained coherency structure as D' =

2 g, = Q and H ig, = D. i

E-, =

Li o •• • o oJ and [ • ]* denotes complex conjugate.

(5)

I

Step 4: If a unique solution of the above equations exists, the solution for SCS is obtained. Otherwise, the problem is declared unsolvable. Elaboration of Step 3: If D' < D or Q' < Q, solution of (5) gives q- I

q-\

L

S g, + S g, = 2 and / = 1

L

S ig, + S igi = D.

=q

I

i-\

(6)

i=q

Since { g t , • • • , gq^ , } are known to be correct, we can write /.

/.

S g, = Q" and

£ ig, = D".

I

0 0 ••• 1 0

(4)

'8i-

If Q' = Q and D' = D, the problem is solved. Step 3: If D' < D or Q' < Q, compute {g q , • • • , gL } using {£i> ' ' ' • gq~\ }, D, and Q as a solution of the following equations:

where E,is an (M — i) X (M — i) exchange matrix given by "0 0 • • • 0 1

(3)

Q' =

=

q

i

•••

+

=

(7)

q

From (7), we get 8q+i

+

2gq-2

+

(L

-

q)gL

=

D"

-

qQ".

(8)

Now D" - qQ" can be either positive or zero. If D" ~ qQ" is zero, it follows that gq+ , = gq_,, • • • g, = 0. This gives gq =

IEEE TRANSACTIONS ON ACOUSTICS. SPEECH, AND SIGNAL PROCESSING. VOL. 37. NO. 7. JULY 1989

1146 TABLE 1

TABLE 11

RESULTS FOR EXAMPLE 1

RESULTS FOR EXAMPLE 2

K

ViUk)

ft

nv A )

k

-1 0 1 2 3"

0 3 6 7 8

0 0 0 2 0

0 3 8 8 8

-1 0 1 2 3 4 5 6"

"Denotes the matrix becomes full rank; hence, the obtained gk may not be correct.

In general, D" - qQ" > 0 and solution of (8) along with (7) would give the source coherency structure. Therefore, augmenting the SRP procedure with the information obtained from "forward-backward" smoothing, it is possible to obtain the source coherency structure using a smaller number of sensors. In Section V, we shall illustrate the above algorithm via some numerical examples. IV.

T(Uk) 0 2 4 6 8 10 12 13

ft _ — 0 0 0 0 0 1

T(Vt) 0 2 8 12 14 14 14 13

'Denotes that the matrix becomes full rank; hence, the obtained gk may not be correct.

than the number that would be needed (i.e., M = 23) with the SRP procedure. Finally, we note that, as suggested in [7], the actual rank tests for the finite sample size case are best carried out via MDL or AIC tests.

SOLVABILITY

The conditions for solvability using the ASRP can be summarized as follows. 1) The problem is unsolvable by ASRP if the rank profile of the matrix sequence { V,} contains no stationary segment. 2) The problem is solvable if the rank profile of the matrix sequence { Vj} contains a stationary segment of two or more periods. 3) In the critical case, defined as the case when there is only one stationary segment in the rank profile of the matrix sequence { V,}, the problem may or may not be uniquely solvable via the ASRP procedure. Additional sensors are then required to resolve the ambiguity. Furthermore, in view of the fact that the minimum number of subarrays needed for the largest rank of { V,} to equal D is L/2, the ASRP holds the possibility of SCS determination via a minimum of L/2 subarrays. V. EXAMPLES AND IMPLEMENTATION

Example 1: The first example consists of eight planar wavefronts impinging on an 11-element array at angles such that no two DOA's are closer than half a beamwidth. The sources are divided into three coherent groups with g2 = 2 and g4 = 1. Table I summarizes the rank profile of { Uk }, the corresponding values of gk from (2), and the rank profile of { Vk }. It is assumed that no noise is present in the field. The problem is not solvable using only the SRP information. The rank profile of the matrix sequence { Vk } is stationary for three points, and hence the problem is solvable via the ASRP procedure. The ASRP gives us the total number of sources as eight. From the additional information obtained from the ASRP, we get g2 = 2 and g4 = 1, i.e., the SCS is determined correctly. Example 2: This example consists of 14 planar wavefronts impinging on a 19-element array. The sources are comprised of two coherent groups with g6 = 1 and g8 = 1. The results obtained are summarized in Table II. The problem is, once again, not solvable using only the SRP information. From the rank profile of the matrix sequence { Vt }, we get the total number of sources as 14. Using the ASRP procedure, we get two possible solutions, i.e., either g7 = 2 or g6 = 1 and gg = 1. Since a unique solution is not obtained, we say that the problem is not solvable by ASRP. However, it is easy to see that it becomes solvable by ASRP for M = 20, which is still less

REFERENCES

|1] R. O. Schmidt, "Multiple emitter location and signal parameter estimation," IEEE Trans. Antennas Propagat., vol. AP-34, pp. 276-280, Mar. 1986. [2] T. J. Shan, M. Wax, and T. Kailath. "On spatial smoothing fordirection-of-arrival estimation of coherent signals," IEEE Trans. Acousl., Speech, Signal Processing, vol. ASSP-34, pp. 806-811, Aug. 1985. [3] T. J. Shan, A. Paulraj, and T. Kailath, "On smoothed rank profile tests in eigenstructure approach to direction-of-arrival estimation," in Proc. IEEE ICASSP 1986, Tokyo, Japan, pp. 1905-1908. [4] R. Williams. S. Prasad, A. K. Mahalanabis. and L. H. Sibul, "An improved spatial smoothing technique for bearing estimation in a multipath environment," IEEE Trans. Acoust., Speech, Signal Processing, vol. 36, pp. 425-432, Apr. 1988. [5] T. J. Shan and T. Kailath, "Adaptive beamforming for coherent signals and interference," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-33, pp. 527-536. June 1985. |6] J. E. Evans, J. R. Johnson, and D. F. Sun, "Applications of advanced signal processing techniques to angle of arrival estimation in ATC navigation and surveillance systems." Tech. Rep.. Lincoln Lab.. M.I.T., 1982. [7] M. Wax and T. Kailath, "Determining the number of signals by information theoretic criteria." presented at the 2nd ASSP Workshop Spectral Est., Nov. 1983. |8] Y. Bresler and A. Macovski, "On the number of signals resolvable by a uniform linear array," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 1361-1375, Dec. 1986.

A Note on the Computational Complexity of the Arithmetic Fourier Transform NAZIF TEPEDELENL1OGLU

Abstract—It is shown that the number of data points the arithmetic Fourier transform (AFT) needs for an N-point Fourier transform is proportional to A'2. Manuscript received April 11, 1988; revised November 7, 1988. The author is with the Department of Electrical and Computer Engineering, Florida Institute of Technology, Melbourne. FL 32901. IEEE Log Number 8928126.