The Cramer-Rao lower bound for directions of arrival of Gaussian cyclostationary signals

1418 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 4, JULY 1992

where sl = 1 (mod n) . It follows that

(Note that s is odd. Thus, s(n /4) = n /4 (mod n ) or s( n /4) = n /4 + n/2 (mod n ) and applying (17) if necessary) Hence, by using (18) and (19), we can see that conditions from (13) to (16) are satisfied when every b, is replaced by Pi (or every bi by 1 + b,), which means that y(x)(or y(x) + P(x)) gives rise to a miracle configuration.

Example: n = 24: 68195E (First row in Table I)

01101OOOOOO1100101011110 go, g,;.., g,, 0011010OOOOO110010101111 1001 101OOOOOO11001010111 shifts 110011010000001100101011 etc. 100101 11 11 1001 10101oooO1 01 11 10101001 lOOOOOOlO110 reverse 11oooO101011001111110100 “even” complement 001 11 10101001 lOOOOOO1011 “odd” complement

complement

go, g,, g 2 , ; ” , giS; . . , gZ3*, with s = 5 , 7 , 1 1 , 13, 17, 19 ,2

(see the last above remark).

V. CONCLUSION AND OPEN PROBLEMS

We obtain many almost perfect autocorrelation sequences by using the two miracle configurations of Section IV. This gives rise to several open questions.

Is there an algebraic explanation of the special form A A* for the vector E introduced in Corollary lo? Furthermore, is it possible to specify the convenient vectors A? Our results are obtained in the case where the parameter 0 of Corollary 6 is equal to 1. What about other values for 0? (For example we know a few sequences with 0 = 2.) Why does not our construction give results if n = 32,44,68,72,80,92? We arbitrarily stopped the computer search at n = 100. What happens for larger values? Is it possible to use the g( x‘) transforms to get new properties or constructions?

ACKNOWLEDGMENT

The author would like to express his gratitude to S. Harari for his assistance in computing the numerical results of Table I.

REFERENCES [l] R . Alexis, “Search for sequences with zero auto-correlation,” Lec-

ture Notes in Computer Science, vol. 311, New York: Springer- Verlag, 1988, pp. 159-172. L. D. Baumert, “Cyclic difference sets,” Lecture Notes in Muthe- motics, vol. 182, T. Hqholdt and J. Justesen, “Ternary sequences with perfect periodic auto-correlation,’’ IEEE Trans. Inform. Theory, vol. IT-29, pp. 597-599, 1983.

[2]

[3] New York: Springer-Verlag, 1971.

The Cramer-Rao Lower Bound for Directions of Arrival of Gaussian Cyclostationary Signals

Stephan V . Schell, Member, IEEE, and William A . Gardner, Fellow, IEEE

Abstract-The Cramer-Rao lower bound on the variance of estimates of directions of arrival, signal strengths, and carrier frequencies and phases of cyclostationary signals arriving at an array is considered. Numerical evaluations of the bound for the estimates of directions of arrival of cyclostationary signals show that it can he orders of magnitude lower than the corresponding bound for stationary signals, despite the greater number of unknown parameters in the former case. Compar- isons with the mean squared error of direction estimates obtained by the Cyclic Least Squares algorithm show that the hound i s not overly optimistic.

Index Terms-Cramer-Rao lower bound, parameter estimation, cyclostationarity.

I . INTRODUCTION

The CramCr-Rao lower bound (CRLB) for the directions of arrival (DOA’s) at a sensor array, signal powers, and noise power has been derived and evaluated for stationary Gaussian signals in stationary white Gaussian noise [l], [lo] and for unknown signals in stationary white Gaussian noise [ l I]. However, no results seem to be available for cyclostationary random signals. Recent work [7] on direction-finding methods that exploit cyclostationarity shows that the mean-squared errors (mse’s) of the estimated DOA’s for these signals can be much less than the CRLB for stationary signals. Thus, it is of interest to evaluate the corresponding CRLB.

The statistics of cyclostationary signals contain periodically time- varying (PTV) components [3] which reflect the periodicities of the underlying physical phenomena that generate the signals. That is, the time-variant mean, autocorrelation, and/or higher order mo- ments contain finite-strength additive sine waves, the frequencies of which are called cycle frequencies. These periodicities can result from the application of PTV transformations to otherwise stationary data, as in the keying, gating, sampling, and frequency-shifting operations applied in a modulator to a stationary message signal. Thus, most communication and telemetry signals, including double- sideband amplitude modulation (DSB-AM), digital pulse AM (PAM), digital quadrature AM (M-ary QAM), frequency-shift keying (FSK), and other modulation formats exhibit cyclostationarity, and the cycle frequencies of these signals are typically equal to the doubled carrier frequency, integer multiples of the symbol rate, and sums and differences of these [3] .

The particular application of interest here is that of estimating the directions of arrival (DOA’s) of signals impinging on an array of sensors and, more specifically, of resolving the DOA’s of two closely spaced signals. This application arises frequently in intelli- gence, surveillance, reconnaissance, and in other applications in which little or nothing is known about the signal environment. The setting to be considered is a physically stationary environment in which the signal sources are fixed in space and the modulation parameters (signal power and carrier frequency and phase) are also

Manuscript received January 21, 1991. This work was supported in part by the National Science Foundation under Grant No. MIP-88-12902 and in part by the Army Research Ofice under contract DAAL03-89-C-0035 sponsored by the U.S. Army Communications Electronics Command Center for Signals Warfare.

The authors are with the Department of Electrical Engineering and Computer Science, University of California. Davis, Davis, CA 95616.

IEEE Log Number 9107515.

0018-9448/92$03.00 0 1992 IEEE

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 4, JULY 1992 1419

fixed. From this environment the data collection system (i.e., sensor array) collects data records that are then processed to obtain estimates of the parameters of interest (DOA's, signal powers, carrier frequencies and phases, and noise power).

In this paper, where all signals are assumed to be either stationary or cyclostationary, expectation can be interpreted within either the stochastic process framework or the nonstochastic time-average framework (e.g., see 131, [4]). That is, expected values can be obtained from either a time-indexed average over an ensemble or an average over time using sinusoidal weighting functions. In the latter case, expectation is the additive-sinewave extraction operation [3] , P I .

The remainder of the paper is organized as follows. In Section 11, the mathematical description of the Gaussian cyclostationary signal is given, and the time-variant autocorrelation is evaluated. In Sec- tion 111, the CRLB for unknown parameters (DOA, signal power, carrier frequency and phase, and noise power) of this type of signal is compared to the CRLB for stationary signals and to the mean- squared error (mse) of an existing direction-finding method for cyclostationary signals. Finally, conclusions are drawn in Section IV .

11. SIGNAL MODEL

Most cyclostationary communication signals, especially those having digital modulation formats, are not Gaussian. However, to determine the effects of cyclostationarity , the comparison of the CRLB for the DOA estimates of cyclostationary signals with existing results, essentially all of which consider only stationary Gauss- ian signals, should first be carried out by considering Gaussian cyclostationary signals. This first step is further justified by the tractability of the CRLB for Gaussian signals. The Gaussian cyclostationary signals of interest in this paper are DSB-AM signals having flat spectra throughout the reception band, although varia- tions such as stacked carriers and nonflat spectra can be accommo- dated in the signal model. The simplification of considering only flat spectra is made to allow direct comparison with the majority of existing results for the stationary case. In some direction-finding applications (e.g., surveillance), the carrier frequencies and bandwidths of the signals are unknown and the signals can be completely spectrally overlapping. This is allowed for in the model adopted here. It is also assumed that the response of the sensor array does not vary significantly over the receiver bandwidth, allowing the use of the narrowband approximation. Note, however, that this does not imply that the signals are sinewaves, nor does it imply that signals in the same band have the same carrier frequency: loosely speaking, it implies only that the band-pass bandwidth is much less than the center frequency (see [2] for a more rigorous investigation of the conditions under which signals can be approximately modeled as being narrow-band). This narrow-band assumption is exactly the same as that used in obtaining existing results [ 13, [ 1 1 1 for stationary Gaussian signals.

Consider L independent zero-mean real stationary white Gaussian baseband messages b ( t ) = [ b , ( t ) , . . . , bL(f)lT. For the Ith signal, denote the angle of arrival by 8 ) . the spectral height by P,, the carrier frequency offset relative to the center frequency of the receiver by f , , and the carrier phase by +/. Under these assumptions, the complex envelope of the vector-valued output signal x( t ) from an M-element sensor array receiving the signals in noise n( t ) is modeled by

x ( t ) = H ( t , u ) b ( u ) du + n(f), -00

where the Ith column h, ( t , U) of H ( t , U) is given by

for which a(8,) is the response of the array to a signal arriving from angle 8, and filter g , ( t ) has Fourier transform G,(f) given by

G , ( f ) = { f/"" s f S f F ' , otherwise.

( 3 )

The noise n( t ) is zero-mean stationary complex white Gaussian noise having autocorrelations R,, = 0 2 Z and R,,, = 0. The spectrum of a typical signal, having carrier offset f , , before and after reception at a receiver having one-sided bandwidth B is shown in Fig. 1 ; the output at each of the sensors in the array is a linear combination of noise and L such signals which can have differing carrier frequencies and phases, spectral heights, bandwidths, and DOA's. Since the frequencies and bandwidths are unknown, no signal will necessarily have zero carrier offset or lie completely within the receiver band.

To allow direct comparison between the numerical evaluations presented in Section 111 and the majority of existing results, the bandwidth parameters f/"" and f?' of all signals are taken to be known and have values -0.5 and 0.5, respectively, corresponding to the full bandwidth of the receiver. That is, in all cases considered here, the spectral support of each signal includes the receiver bandwidth, although the mathematical model of the signal presented next does not require this assumption and can be easily generalized to stacked carriers and/or nonflat spectra.

Using the input-output relations for linear periodically time-varying systems [3] , it can be shown that that the autocorrelation and conjugate correlation of x ( t ) are given by

I = I

and

L

= PIe j2+/s , ( tm , t , )a (e , )aT(e , ) , I = I

respectively, where

with


A P r

f:" = -B 0 f i fp'= B (b)

Fig. 1 . Spectrum of the complex envelope of a typical signal having carrier offset f, (a) before and (b) after reception at a receiver having one-sided bandwidth B.

In this correspondence, (.)' denotes transposition, (. )* denotes conjugation, and ( * ) H denotes conjugate transposition. Since x ( t ) is a complex-valued Gaussian process (or Gaussian time-series in the time-average framework), the joint probability density function (or the joint composite fraction-of-time density in the time-average framework [3], [5]) of X = [ ~ ( t , ) ~ ; . . , ~(t,)']', evaluated at Y = [ y r , y ? ; . . , y;, y,"IT, is given by

J,(Y) = ( ( ~ T ) ~ ~ R ~ ) - ~ ~ x ~ ( ~ Y ~ R - ' Y 1 , (11)

where the autocorrelation matrix R is given by

with

The factors of two present in the expression for the density are needed even though Y is complex-valued because the length of Y is twice that of X .

111. CRLB

Since x ( t ) is zero-mean and Gaussian, the CRLB for a vector c of unknown parameters is given by

cov {C} 2 J - J (14)

(i.e., wTE{(C - c)(C - c)'}w 2 w'JJ-'w for all possible choices of w and unbiased estimator C), where the i, j th element of the Fisher information matrix J is given by (e.g., see [l], [lo])

In this application, the unknown parameters include the noise power U', and, for each of the L signals, the DOA e / , spectral height Pr, carrier frequency offset f, and phase c$/. The CRLB for these parameters must generally be evaluated numerically for the signal environments of interest, because analytical inversion of R so far appears to be intractable, even under the simplifying assumptions stated above, except for the case where each signal has zero carrier offset or is stationary. In the stationary case, a closed-form expression of the CRLB for the directions of arrival is given in [ 121.

I I

1

0.9

0.8

0.7

0.6

0.5 0 0.1 0.2 0.3 0.4 0.5

Carrier Frequency Offset

Fig. 2. Ratio of CRLB for DOA of a cyclostationary signal to the CRLB for DOA of a stationary signal, evaluated as a function of carrier frequency offset and SNR.

A . Single Signal

When each signal present is either cyclostationary with zero carrier offset or stationary, R is block diagonal, and evaluation of the CRLB can proceed analogously to the case in which all signals are stationary. In particular, when only one signal is present and has zero carrier offset, it can be shown (e.g., by following the general outline of the proof in [lo]) that the CRLB for the DOA 0 is uncoupled from the CRLB for the other unknown parameters, and that it can be expressed as

where SNR is the ratio of the signal power to the noise power. In contrast, the CRLB for the DOA 0 of a stationary Gaussian signal is given by (e.g., see [lo]).

It is easily verified that CRLB,,,,,{O} for M sensors equals CRLB,,,{ 0 ] for 2 M sensors. This fact arises because the autocorrelation of the 2M-vector [ x ( t ) ' , ~ ( t ) ~ ] is needed to fully describe the received cyclostationary data, whereas only the M-vector x( t ) is needed in the stationary case. The ratio of (16) to (17) is given by

MSNR ) . (18)

That is, for all possible numbers M of sensor elements and signal- to-noise ratios SNR, the CRLB for the cyclostationary signal is less than or equal to the CRLB for the stationary signal. In particular, in the limit as M S N R approaches infinity, the two bounds are the same, whereas in the limit as M SNR approaches zero (i.e., the SNR is very low), the CRLB for the cyclostationary signal is equal to half of the CRLB for the stationary signal. Since the performance of estimation methods typically decreases with decreasing SNR, this result can be interpreted to mean that the cyclostationarity properties of the signal should be exploited if possible by estimation methods operating in low-SNR environments.

This analytical result is confirmed by the results of numerical evaluations. The ratio of the CRLB for the DOA of a single cyclostationary signal to the CRLB for the DOA of a single stationary signal is shown in Fig. 2 as a function of carrier frequency offset for several different values of SNR (ranging from -20 dB to 20 dB). A 4-element uniform linear array having sensor spacing equal

CRLB,,clo{~) = (1 +

CRLB,t,,{e) , ? 1 + M S N R

1YYL 14L1

Stationary CRLB - - - ~ - ~ Stat. CRLB (bias adj.) -+---

CLS (measured) +

i x*\ Cyclostatioiary CRLB ~ ’;: 1 ‘‘ ’-Cyclo. CRLB (bias adj.) t-

*. . % +>\\

z 1

-1 0.1 0.01

0.001 --*:I-- -.---___

0.1 1 10 100 Angular Separation (deg.)

Fig. 3. CRLB for DOA’s of two cyclostationary signals, mse of the Cyclic Least Squares method, and the CRLB for DOA’s of two stationary signals. CRLB’s for estimators having bias equal to that of CLS are also shown for comparison.

to half of the wavelength of the center frequency of the receiver is used. Note that the ratio is smallest when the offset is 0, in which case the receiver is perfectly tuned to the signal and the received data exhibits the highest degree of cyclostationary , and that the ratio increases monotonically as the offset (or the SNR) increases. For offset equal to unity, the bandpass-filtered DSB-AM signal is actu- ally stationary, and so the ratio of the two CRLB’s is exactly unity.

B. Two Signals

Unlike the results for the particular single-signal environment considered in Section 111-A, the results for a two-signal environment show that the CRLB for the DOA’s of cyclostationary signals can be many orders of magnitude less, rather than only 50% less, than the CRLB for the DOA’s of stationary signals, provided that the carrier frequencies are different. For example, consider two cyclostationary signals as specified in Section I1 each having SNR of 10 dB and arriving at a 4-element uniform linear array. One signal, having carrier offset - 1/16, arrives from 0 degrees, and the other signal, having carrier offset 3/32, arrives from a different angle (which ranges from 0.1 to 50 degrees). The CRLB is evaluated numerically under these assumptions and separately under the assumption that both signals are stationary. For purposes of comparison with an existing estimator, the Cyclic Least Squares (CLS) direction-finding method [9], using adaptively chosen estimates of the cycle frequencies [8], is also simulated (lo00 independent trials). The CRLB for the DOA’s of the two cyclostationary signals, the mean-squared error (mse) of the CLS method, and the CRLB for the DOA’s of two stationary signals are shown in Fig. 3 for N = 8192 data samples. Also shown are the CRLB’s adjusted according to the measured bias of CLS, so as to provide a more meaningful comparison between the CRLB’s and the mse of CLS. In particular, as the signals move closer together the Fisher information matrix in the stationary case becomes more ill-conditioned, which results in rapidly increasing mse, whereas it remains almost constant in the cyclostationary case. Also, the mse of the CLS method can be seen to be nearly identical to the CRLB for the cyclostationary signals, which indicates that 1) the CLS method is nearly efficient when the signals are close together in this environment, and 2 ) the CRLB for cyclostationary signals is not overly optimistic. However, the mse of the CLS method is greater than the CRLB for the stationary signals in this environment when the signals are more than 5 degrees apart and, in fact, exhibits a global maximum near 20 degrees; this is predicted by the analytical results in [6] and is due to residual contributions of the undesired signal (the one not having the chosen

cycle frequency a used by CLS) to the cyclic conjugate correlation matrix

used by CLS, where a is set equal to one or the other of the two estimated cycle frequencies (for which the exact values are twice the carrier frequency offsets of the signals). Also, although results are not shown for angular separations greater than 50 degrees, it is easily proven that as the direction of arrival of the second signal approaches 90 degrees, both CRLB’s grow without bound because the derivative of R with respect to the direction of arrival approaches zero. This may explain why the CRLB for cyclostationary signals decreases slightly as the angular separation decreases

Other results not shown here indicate that the difference between the CRLB for the cyclostationary signals and the CRLB for the stationary signals becomes even larger as the SNR decreases, as it does for a single signal. Also, similar results can be expected for non-Gaussian signals having different cycle frequencies (e.g., two BPSK signals that have either unequal carriers or unequal keying rates), because exploiting cyclostationarity allows signals to be separated according to their differing cycle frequencies in addition to their differing spatial characteristics.

IV. CONCLUSION

The Cram&-Rao lower bound on the covariance of unbiased estimates of parameters of Gaussian cyclostationary signals arriving at an array of sensors is considered. It is shown that the ratio of the CRLB for the direction of arrival of a single cyclostationary signal to the CRLB for the direction of arrival of a single stationary signal is between one half and unity. However, it is shown that the ratio for two signals arriving at the array can be many orders of magnitude smaller than unity, and decreases rapidly as the angular separation between the two signals decreases. Comparisons with the Cyclic Least Squares direction-finding method indicate that the CRLB for cyclostationary signals is nearly attainable with a practi- cal algorithm.

REFERENCES W. J. Bangs, “Array processing with generalized beamformers,” Ph.D. dissert., Yale Univ., New Haven, CT, 1971. K . M. Buckley, “Spatial/spectral filtering with linearly constrained minimum variance beamformers,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASP-35, pp. 249-266, Mar. 1987. W. A. Gardner, Statistical Spectral Analysis: A Nonprobabilistic Theory. - , “Two alternative philosophies for estimation of the parameters of time-series,” ZEEE Trans. Inform. Theory, vol. 37, pp. 216-218, Jan. 1991. W. A. Gardner and W. A. Brown, “Fraction-of-time probability for time series that exhibit cyclostationarity ,” Signal Processing J . EURASIP, vol. 23, no. 3, pp. 273-292, 1991. S. V. Schell, “Estimating the directions of arrival of cyclostationary signals-Part 11: Performance,” IEEE Trans. Signal Processing, in review. - , ‘ * Exploitation of spectral correlation for signal-selective direction finding,” Ph.D. dissert., Dept. of Elect. Eng. and Comput. Sci., Univ. of California, Davis, CA, 1990. S . V. Schell and W. A. Gardner, “Progress on signal-selective direction finding,” in Proc. Fifth ASSP Workshop on Spectrum Estimation and Modeling, Rochester, NY, Oct. 1990, pp. 144-148. - , “Signal-selective high-resolution direction finding in multipath,” in Proc. IEEE Int . Con f. Acoust . , Speech, Signal Processing, Albuquerque, NM, Apr. 1990, pp. 2667-2670. R. 0. Schmidt, “A signal subspace approach to multiple source location and spectral estimation,” Ph.D. dissert., Stanford Univ., Stanford, CA, 1981. P. Stoica and A. Nehorai, “MUSIC, maximum likelihood, and

Englewood Cliffs, NJ: Prentice-Hall, 1987.


Cramer-Rao bound,” IEEE Trans. Acoust., SPeech, Signal processing, vol. 37, pp. 720-741, May 1989.

[ 121 -, “Performance study of conditional and unconditional direction- of-arrival estimates,” IEEE Trans. Signal Processing, vol. 38, pp. Pr ( Y 5 y) = 1 - Q N ( S , y ) = J’ Z - ‘ h ( Z ) e Y ‘ G , 1783-1795, Oct. 1990. c+

N one computes instead the cumulative distribution function itself,

dz

1

2 z = x,++ -cy2 + iy, (5)

Computing the Generalized Marcum Q-Function

Carl W. Helstrom, FeNow, IEEE

Abstract-A modified version of the method of saddlepoint integration is shown to compute the generalized Marcum @function accurately over a broad range of parameters. Bounds on the truncation error incurred by stopping the numerical quadrature at a certain point are presented.

Index Terms-Q-function, saddlepoint method, numerical integration.

A recent paper by D. A. Shnidman [l] illustrates the complexities of computing the generalized Marcum Q-function by recursive methods: even for modest values of the parameters a computer program must cope with underflow and overflow, and when the parameters are large, the great number of summations required makes certain of these methods susceptible to round-off error. These problems are largely avoided in the method of saddlepoint integration [2]. This correspondence presents a modification of its formula- tion in [2] that extends its applicability to the broad range of parameters contemplated in [l], and slightly simpler bounds on the truncation error are established.

As in [l] we treat the Q-function in the form

which equals P,(S/N, N) in the notation of [l]; in the notation of [3, p. 2191, this is Q,(&%, m). As explained in [2], this function can be computed by numerical quadrature of the Laplace inversion integral

where

(3)

is the moment-generating function of the random variable y of which QN(S, Y ) = Pr(y > Y) is the complementary cumulative distribution function [2], [4, p. 1641. The contour C- is a parabola of suitable curvature c passing through the saddlepoint x i of the integrand of (2) lying in the segment -1 < Rez < 0; on C-

z = x;+ icy’ + jy. (4)

The form in (2) is used when Y 2 E( y ) = S + N. When Y < S + Manuscript received October 5 , 1990; revised June 17, 1991. The author is with the Department of Electrical and Computer Engineer-

IEEE Log Number 9106693. ing, University of California, San Diego, La Jolla, CA 92093-0407.

where C, is a similar parabola passing through the saddlepoint x: of the integrand of (5) lying on the positive Re z axis.

We write the integrands of (2) and (5) as

called the “phase”of the integrand. The saddlepoints x; and x: are roots of

Z = X , , X ; ; (8)

primes indicate differentiation. This reduces to the cubic equation,

Yz3 + (2Y - N - 1 ) z * + ( Y - N - s - 2 ) z - 1 = 0, (9)

of which the third root is always real and less than - 1 . If one’s computer has a routine for solving polynomial equations, it can be used to determine the appropriate saddlepoint directly. Otherwise, one can use Newton’s method [5]; an initial trial value of z is replaced at each stage by

(10) @‘(z> a‘‘( z ) ’ z t z - -

where

The starting value of z in this search is conveniently taken as

Z , = [ - D * (0’ + 4 0 ~ ) ~ ’ ~ ] / 2 o ’ ,

D = Y - S - N , u 2 = v a r y = N + 2 S . (12)

The - sign is used for D 2 0, the + sign for D < 0. This value represents the saddlepoint when the distribution of y is approxi- mated by a Gaussian with mean S + N and variance U * , where- upon the phase @( z) is

+ ( z ) = D z + i a 2 z * - I n ( + z ) . (13)

If zl 5 -1, one instead starts the search for the saddlepoint at

To the parabolic path of integration is assigned a curvature c that makes it closely approximate the path of steepest descent of the integrand; as in [2] we take

z = -0.99.

3 s N

0018-9448/92$03.00 O 1992 IEEE

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The Cramer-Rao lower bound for directions of arrival of Gaussian cyclostationary signals