Estimating the Bearings of Near-Field
Cyclostationary Signals
Ju-Hong Lee and Chia-Hsin Tung
Abstract—By exploiting favorable characteristics of a uniform
linear array (ULA), the far-field approximation (FFA) method re-duces the deterioration of most high-resolution bearing estimation techniques due to the invalidity of the planar wavefront assumption when near-field sources exist. In this paper, we first present a new technique that provides better performance than existing bearing estimation techniques for cyclostationary signals. Then, the pro-posed technique incorporating with the FFA method is propro-posed to estimate signal bearings in the presence of near-field sources without revising the planar wavefront assumption. Several simu-lation examples confirm the theoretical work and show the effec-tiveness of the proposed technique.
Index Terms—Bearing estimation, cyclostationary signals,
near-field sources.
I. INTRODUCTION
I
N general, high-resolution techniques for estimating bear-ings of multiple sources radiating narrowband signals assume that all the signals arrived at array sensors have planar wavefront, i.e., in the far field [1]–[9]. However, when the ranges of some sources to the array are not sufficiently large compared with the diameter of the array system, the wave-fronts emitted from these sources are spherical rather than planar at the site of sensor array. Therefore, bearing estima-tion techniques with planar wavefront assumpestima-tion generally show unsatisfactory performance. A method referred to as a far-field approximation (FFA) method has been proposed in [10] to effectively deal with this situation. The FFA method first constructs an FFA covariance matrix that is Toeplitz and approximate to the data covariance matrix associated with the environment that all sources are in the far field. Based on the FFA covariance matrix, high-resolution techniques are then applied to estimate the bearings of all signal sources without revising the planar wavefront assumption.Cyclostationarity [11], which is a statistical property pos-sessed by most man-made communication signals, corresponds to the underlying periodicity arising from cycle frequencies or baud rates. Cyclic MUSIC and conjugate cyclic MUSIC pre-sented in [11], [12] accommodate multiple signals having the same cycle frequency by using the same type of subspace-fit-ting as MUSIC [1]. Moreover, in certain cases as shown in [13], the mean-square error (MSE) of the direction estimates obtained by cyclic MUSIC is less than the Cramér–Rao lower bound
Manuscript received August 23, 2000; revised September 28, 2001. This work was supported by the National Science Council of Taiwan, R.O.C., under Grant NSC89-2213-E002-084. The associate editor coordinating the review of this paper and approving it for publication was Prof Michail K. Tsatsanis.
The authors are with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, R.O.C. (e-mail: juhong@cc.ee.ntu.edu.tw).
Publisher Item Identifier S 1053-587X(02)00415-4.
for stationary signals. Recently, the Gelli–Izzo bearing estima-tion (GIBE) method based on a cyclic beamforming approach was presented in [14] to provide lower complexity than cyclic MUSIC but lower resolution. However, a direct application of the mentioned techniques [11]–[14] in the presence of near-field signals of interest (SOIs) also suffers the performance degrada-tion as the convendegrada-tional high-resoludegrada-tion techniques.
In this paper, we present a new technique for estimating the bearings of cyclostationary signals. Then, the proposed tech-nique is incorporated with the FFA method to deal with bearing estimation of cyclostationary signals when near-field sources exist. Due to the fact that the effect caused by the mismatch between the actual spherical wavefront phase vector and the as-sumed planar wavefront phase vector is alleviated, the proposed technique achieves very satisfactory performance in estimating the bearings of near-field signal sources.
II. BEARINGESTIMATIONUSINGSIGNALCYCLOSTATIONARITY A. Signal Cyclostationarity
For a signal , its cyclic autocorrelation function and con-jugate cyclic autocorrelation function are defined as the fol-lowing infinite-time averages [12]
(1) and
(2) respectively, where the superscript “ ” denotes the complex conjugate. is then said to be cyclostationary if or does not equal zero at cycle frequency for some . Many man-made communication signals exhibit cyclostation-arity with cycle frequency equal to twice the carrier frequency or multiples of the baud rate or combinations of these [11].
Let the data vector received by an array be designated as . Its cyclic autocorrelation matrix (CAM) and conjugate cyclic autocorrelation matrix (CCAM) are given by
(3) and
(4) where the superscript “ ” denotes the conjugate transpose and “ ” the transpose.
B. GIBE Method [14]
Consider an -element antenna array excited by narrow-band far-field SOIs with a common cycle frequency and noise. The received data vector can be expressed as
(5)
where denotes the complex waveform of the th SOI, includes all signals not of interest (SNOIs) and spatially white noise whose cycle frequencies are different from , and represents the direction vector of the impinging on the array from off broadside. Moreover, is the
matrix of the direction vectors , . Let be uncorrelated with and all signals be zero-mean. The corresponding CAM is given by
(6) where represents the CAM of the SOIs. The part of corresponding to is eliminated in going from (5) to (6) because of the property of signal cyclostationarity and the definition given by (3) when using infinite data samples. Based on (6), the average power at the array output for a direction angle
, which is given by
(7) is used to estimate the signal bearings by running on the an-gular region of interest and finding the resulting peaks. These peaks are then taken as the estimates of .
III. PROBLEM FORMULATION AND A REVIEW OF THEFFA METHOD
A. Near-Field Source Problem
Let the -element antenna array be a uniform linear array (ULA) with interelement spacing and aligned on the axis, as shown by Fig. 1, where the array center is designated as the phase reference point and the origin of the coordinate system. Then, the signal received by the th sensor can be expressed as
(8) where and denote the signal amplitude and range of the th source observed at the array center, respectively, is the wavenumber corresponding to the frequency , and indicate the relative gain coefficient due to the related spatial decay and range of the th source referring to the th sensor, respectively, and is all SNOIs plus noise received by the
th sensor. The and are given by [15] and
(9) where is the bearing of the th source from the array broad-side measured at the array center, and is the
Fig. 1. Sensor-source configuration for the near-field problem.
location of the th sensor, where . Letting , (9) can be approximated as [16]
(10) which is essentially the so-called Fresnel zone [15] approxima-tion. From (8)–(10), it follows that the vector form of the re-ceived signal is given by
(11)
where
and
(12) We note from (11) and (12) that the received near-field sig-nals can be equivalently viewed as far-field sigsig-nals received by nonideal sensors with denoting the complex gain of the th sensor to the th source. Since depends on the sensor lo-cation, the source bearing, and range, the gain of each sensor is thus unknown, complex, nonomnidirectional, and distinct for different sensors. Following (11), we have the data correlation matrix of given by
where denotes the signal power of the th source received at the array center and the autocor-relation matrix of . Next, substituting (12) into (13) yields
(14) where
diag
and diag represents a diagonal matrix. We note that and are the corresponding far-field unit-powered signal co-variance matrix and phase vector, respectively. represents the finite range effect of the th source. Moreover, it is noted that the signal subspace of is spanned by ,
, which are the actual phase vectors of the near-field sources. Therefore, the performances of most bearing es-timation techniques with the far-field assumption deteriorate because the assumed far-field source model is and not
.
Next, the near-field source problem in the case of cyclosta-tionary signals is considered. Using (3) and letting , for simplicity, we get the CAM corresponding to of (11) as
(15) with the th entry given by
(16) Substituting (8) and (10) into (16) yields
(17) where is given by
(18) Similar to the case without using signal cyclostationarity, (15) can be expressed as
(19) On the other hand, following the same process described above, we can easily obtain
(20) with the th entry given by
(21) and
(22)
instead of of (15), of (17), and of (18), re-spectively, for the case of using CCAM. From these results, we again observe that and are the corresponding far-field unit-powered signal covariance matrix and phase vector, re-spectively. represents the near-field effect of the th source. Moreover, the signal subspace of is spanned
by , , which are the actual phase
vectors of the near-field sources. As a result, the performances of cyclic-based bearing estimation techniques with the far-field assumption deteriorate because the assumed far-field source
model is and not .
B. FFA Method [10]
The FFA method finds the FFA correlation matrix from (13) as follows:
FFA (23)
with
(24) where denotes the largest integer , and is the th entry of . In [10, Fig. 2], there is an example to illustrate the operation given by (24). From (14) and (23), we have
FFA
FFA (25)
Thus, the signal subspace of is approximately spanned by , . Using instead of will improve the performance of existing bearing estimation techniques. C. Incorporating the GIBE Method With the FFA Method
The performance of directly incorporating the GIBE method with the FFA method is investigated. The resulting method is termed the GIBE-FFA method. First, consider the term using the CAM for far-field sources. From (8), (15), and (16), we can show that becomes
(26) Equation (26) reveals that is Toeplitz. From (17) and (26), we can further show that
where represents the remaining terms that are at least inversely proportional to the cube of . and are the th terms of and , respectively. Equa-tion (27) reveals that the first and second terms in the bracket are zero for equal to an integer, whereas
they become and ,
re-spectively, for or when
is not an integer. Hence, is the term closest to when is an integer, and
oth-erwise, and
are the two terms closest to . As a result, applying the FFA method according to (23) and (24) to obtain the FFA cor-relation matrix can effectively alleviate the effect of near-field sources.
From (8) and (20), we can show the using the CCAM for far-field sources given by
(28) Equation (28) reveals that is Hankel. We can further show from (21) and (28) that
(29) We note from (27) and (29) that both have the same first term in the bracket . However, the second term of (29) equals , which is not zero in general even for equal to an integer, whereas it equals which is larger than that of
using CAM for or .
Hence, we expect that applying the FFA method would not effectively cure the deterioration caused by near-field sources for the case of using CCAM.
IV. PROPOSEDTECHNIQUE A. New Cyclic-Based Technique
To cope with the difficulty of the original cyclic-based tech-nique in dealing with the near-field source problem as presented in Section III, we propose a new cyclic-based technique that uses the following function when using CAM:
(30) or the following function when using CCAM:
(31) In fact, the function given by (30) or (31) is proportional to the squared average power at the array output for a direction angle to estimate the signal bearings by running on the an-gular region of interest and finding the resulting peaks. These peaks are then taken as the estimates of the direction angles ,
.
B. New Cyclic-Based Technique Incorporating With the FFA Method
Let for using CAM
and for using CCAM, and
let for simplicity. Then, we have
(32) Substituting (17) into (32) and performing some algebraic ma-nipulations yields
(33) for . For the case of using CCAM, we have
(34) instead. In contrast, the corresponding for far-field sources are given by
(35) and
(36) instead of (33) and (34), respectively. Equations (35) and (36) reveal that the resulting always possesses the Toeplitz property, regardless of whether we adopt CAM or CCAM. Hence, we would expect that the FFA method can be performed on to successfully cure the near-field problem.
Comparing (33) and (35) and (34) and (36), it is noted that
the terms represent the
near-field source effect on the matrix . To alleviate this effect, we present a technique that incorporates the new cyclic-based technique with the FFA method as follows:
(37) with
(38) Then, we estimate the signal bearings of the SOIs by finding the peaks of
(39) The analysis regarding the properties of the proposed technique is presented in the Appendix.
Fig. 2. Bearing spectra of PAM signals with SOI 5 and SNOIs 30 , 40 using (a) the GIBE method and (b) the proposed method for Example 1.0!: far-field. -.-: near-field source ranges= [20 30 40]. 0 2 0: using FFA.
V. COMPUTERSIMULATIONRESULTS
For all examples, a ULA with number of elements
and interelement spacing is used, where is the wave-length of the narrowband SOIs. Three signals are impinging on the array from 5 , 30 , and 40 with signal-to-noise power ratio (SNR) equal to 10, 10, and 20 dB, respectively. The noise is spa-tially white Gaussian with zero mean and unit variance. More-over, the proposed technique exploits the cyclic and conjugate cyclic autocorrelation at twice the carrier frequency regarding the SOIs only and the sampling interval and for simplicity in all simulations. The presented results are ob-tained by using 1024 data snapshots to compute the observed data covariance matrix and averaging the results of 50 indepen-dent runs with indepenindepen-dent noise samples for each run.
Example 1: The SOI is a real pulse amplitude modulated (PAM) signal with direction angle and the carrier frequency 3.5. The other two signals are real PAM SNOIs with carrier frequencies equal to 6 and 7, respectively. It is well known that the cyclic autocorrelation function and the conjugate cyclic autocorrelation function of any real PAM signal are identical. Fig. 2 shows the resulting bearing spectra of using the GIBE method, the GIBE-FFA method, and the proposed technique for the near-field situation where the source ranges are , , and , respectively, and the far-field situation. From Fig. 2, we observe that the proposed technique provides more satisfactory performance than the GIBE method. Table I lists the corresponding bias, root mean squared error (RMSE), variance, and the number of failures for each SOI estimate of using the proposed technique and the GIBE-FFA
TABLE I
RESULTINGPARAMETERS OF THESOI ESTIMATE FOREXAMPLE1
Fig. 3. Bearing spectra of BPSK signals with SOI 5 and SNOI’s 30 , 40 using (a) the GIBE method and (b) the proposed method for Example 2.0!: far-field. -.-: near-field source ranges= [20 30 40]. 0 2 0: using FFA.
method. Trials are designated as “failure” when the square of its estimation error is greater than the variance of using the proposed technique with FFA method.
Example 2: Here, the SOI is a binary phase-shift-keyed (BPSK) signal with direction angle , the carrier frequency 1, and a rectangular pulse shape. The other two signals are BPSK SNOIs with carrier frequencies equal to 3 and 5, respec-tively, and rectangular pulse shape. Fig. 3 depicts the resulting bearing spectra of using the GIBE method, GIBE-FFA method, and the proposed technique for the near-field situation where the three source ranges are , , and , respectively, and the far-field situation. Again, we observe that the proposed technique outperforms the GIBE method. Moreover, the pro-posed technique has the advantage of effectively alleviating the effect of near-field sources over the GIBE-FFA method for
TABLE II
RESULTINGPARAMETERS OF THESOI ESTIMATE FOREXAMPLE2
Fig. 4. Bearing spectra of (a) PAM signals and (b) BPSK signals with SOIs 5 , 30 , and SNOI 40 using the proposed method for Example 3.0!: far-field. -.-: near-field source ranges= [20 240 50]. 0 2 0: using FFA.
the cyclostationary signals like BPSK, whose cyclic autocor-relation function and conjugate cyclic autocorautocor-relation function are not identical when twice carrier phase is not equal to the multiples of . Table II lists the corresponding parameters like Table I for comparison.
Example 3: Finally, we consider the bearing estimation problem for the case of a near-field SOI accompanied by a far-field SOI to further demonstrate the effectiveness of the proposed technique. First, we have the same simulations as those in Example 2, except that the first two PAM signals are the SOIs with source ranges equal to and , respectively. Hence, the second SOI is essentially a far-field source. The other PAM signal is SNOI with source range equal to . Then, the same simulation is performed again, except that BPSK signals are used instead of PAM signals. Fig. 4 depicts
the resulting bearing spectra. We note that the effectiveness of the proposed technique is not affected by the existence of far-field sources.
APPENDIX
The modeling errors for the near-field problem with and without using the proposed technique are evaluated. For simplicity, consider the signal component of the case where . From (33)–(36), we note that the only difference in between using the CAM and CCAM is to replace with . In the following, we thus present the results for the case of using the CAM only. The results for using the CCAM can be obtained by simply using to replace in the results of using the CAM. From (19), we have
(40) for using the CAM. In contrast, for far field sources is given by
(41) Let the Frobenius norms of and
, namely, and be the
far-field modeling errors (FFME) without and with applying the proposed technique, respectively. Based on the above results, we have
(42) where and can be easily derived from (33)–(36) and are given by
respectively, where and represent the proportional con-stants. Equation (43) reveals that the FFME without applying the proposed technique is inversely proportional to the source range.
Next, we investigate the FFME when applying the proposed technique. To obtain better approximation results for this case, the approximation for of (10) is refined as follows [10]:
(44) As a result, (8) becomes
(45) where
(46) Accordingly, (33) and (34) become
(47)
for and
(48) for , respectively. Let for simplicity. Equation (47) has the simplified form
(49) for . Therefore, we obtain
(50)
for , where
(51) Substituting (10) into (51) and performing some algebraic ma-nipulations yields
(52) Moreover, substituting into (52) and
, we have
(53) and
(54)
From the above results, we investigate the effect of the FFA method on . For the case where is an integer, substituting (50), (53), and (54) into (38) provides
(55)
For the case where is not an integer, it can be shown that (55) becomes
(56)
Since for the near-field case
with , we have
and
. Equation (56) can be further approximated as
for odd
for even . (62)
Therefore, for the case where is an integer, we can show from (55) and (35) that
(58)
Since
for
due to , (58) can be further approximated as
(59)
Similarly, for the case where is not an integer, we can show from (56) and (35) that
(60)
Finally, we compute the resulting FFME due to using the FFA method as follows. Similar to (42), the FFME with applying the FFA method is given by
(61)
Substituting (59) and (60) into (61) and performing some alge-braic manipulations yields (62), shown at the top of the page. From (59), (60), and (62), we note that the FFME with applying the FFA method is inversely proportional to the square of the source range. Comparing the results of (42) and (62), we can
ensure that , regardless of
the source range. Hence, the near-field source problem can be alleviated by using instead of in (39).
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Ju-Hong Lee was born in I-Lan, Taiwan, R.O.C.,
on December 7, 1952. He received the B.S. degree from the National Cheng-Kung University, Tainan, Taiwan, in 1975, the M.S. degree from the National Taiwan University (NTU), Taipei, in 1977, and the Ph.D. degree from Rensselaer Polytechnic Institute (RPI), Troy, NY, in 1984, all in electrical engineering.
From September 1980 to July 1984, he was a Re-search Assistant involved in reRe-search on multidimen-sional recursive digital filtering with the Department of Electrical, Computer, and Systems Engineering at RPI. From August 1984 to July 1986, he was a Visiting Associate Professor and later, in August 1986, an Associate Professor with the Department of Electrical Engineering, NTU. Since August 1989, he has been a Professor at NTU. He was appointed Vis-iting Professor with the Department of Computer Science and Electrical Engi-neering, University of Maryland, Baltimore County, during a sabbatical leave in 1996. His current research interests include multidimensional digital signal processing, image processing, detection and estimation theory, analysis and pro-cessing of joint vibration signals for the diagnosis of cartilage pathology, and adaptive signal processing and its applications in communications.
Dr. Lee received Outstanding Research Awards from the National Science Council (NSC) in 1988, 1989, and from 1991 to 1994 and Distinguished Re-search Awards from the NSC from 1998 to 2001.
Chia-Hsin Tung was born in Taipei, Taiwan,
R.O.C., on November 3, 1975. He received the B.S.E.E. degree from the National Tsing Hua University, Hsinchu, Taiwan, in 1997 and the M.S. degree in communication engineering from the National Taiwan University, Taipei, in 1999.
His current research interests include adaptive signal processing and signal bearing estimation.
