IEEE SIGNAL PROCESSING LETTERS, VOL. 20, NO. 5, MAY 2013
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Efficient Multidimensional Harmonic Retrieval: A Hierarchical Signal Separation Framework Chun-Hung Lin and Wen-Hsien Fang
Abstract—This paper presents a low-complexity one-dimensional (1-D) Unitary Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT)-based algorithm for multidimensional harmonic retrieval (MHR) problems based on an HIerarchical Signal Separation (HISS) technique, which interleaves the parameter estimation and filtering processes. The filtering process not only progressively partitions the signals with close parameters into separate groups, but also reduces the power of the additive noise, both of which entail higher parameter estimation accuracy. The pairing of the estimated parameters is also automatically achieved. Simulations show that the new algorithm provides satisfactory performance compared with previous works but with drastically reduced computations. Index Terms—Filtering, low-complexity algorithm, M-D harmonic retrieval, parameter estimation, subspace algorithm.
I. INTRODUCTION
M
ULTIDIMENSIONAL harmonic retrieval (MHR) is of importance in various signal processing applications such as radar, sonar, and multiple-input multiple-output communication systems [1], [2]. Under certain conditions joint estimation of multipath parameters, such as azimuth, elevation, delay, and Doppler shift, leads to an MHR problem. Therefore, there has been a flurry of interests for the MHR problems. The widespread maximum likelihood approach is asymptotically efficient, but requires multidimensional search, which is in general computationally prohibitive in practice. To alleviate the computations, miscellaneous subspace algorithms such as MUltiple SIgnal Classification (MUSIC) or Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) [3]-based algorithms have been reported. The latter are in particular computationally attractive as they are free of a higher-dimensional search. For example, a higher-dimensional Unitary ESPRIT was proposed in [4], the accuracy of which is further enhanced by a high order SVD in [5]. Liu et al. [6] and Mokios et al. [2] proposed a multidimensional folding (MDF) method for the MHR problems, which used the 2-D (forward-backward) smoothing to enhance the accuracy of the parameter estimates. However, MDF is a direct data approach, which is in general inferior to the covariance matrix based algorithms [5]. Pesavento et al. [7] addressed a rank reduction estimator (RARE) method which decoupled the multidimensional MUSIC by searching over the specific RARE manifold
in each individual dimension instead of the multidimensional manifold. Despite the effectiveness of [4]–[7], they, however, still call for enormous computational overhead due to the higherdimensional data stacking and the related eigendecomposition. To mitigate the computational load, this paper proposes an efficient, yet accurate algorithm for the MHR problems via an HIerarchical Signal Separation (HISS) technique, an extension of the tree-structured estimation scheme in [8]. The essence of HISS lies in a succinct interleaving of the parameter estimation and the filtering processes, where signals with close parameters are progressively partitioned into separate groups and the power of the additive noise is reduced accordingly to entail higher estimation accuracy. The data in every stage are appropriately re-stacked so that only the 1-D Unitary ESPRIT are required. Furthermore, with such a hierarchical estimation scheme, the pairing of the estimated parameters is automatically achieved. Simulation results show that the new algorithm provides satisfactory performance but with substantially reduced computations compared with previous works. The remainder of this paper is organized as follows. Section II introduces the data model for the MHR problems and describes the proposed low-complexity algorithm. Some simulation results are furnished in Section III to justify the new approach. Section IV contains some concluding remarks. II. PROPOSED 1-D UNITARY ESPRIT-BASED ALGORITHM Consider the data which are comprised of a summation of multidimensional exponentials [4] (1) , , where is the data size for in the dimension, and denote respectively the signal in the amplitude and the harmonic component of the dimension, and denotes the additive zero-mean complex white Gaussian noise with variance . , The proposed algorithm begins with estimating , in the first dimension ( ). To achieve this, we first construct a matrix as
(2) Manuscript received December 02, 2012; accepted January 01, 2013. Date of publication January 09, 2013; date of current version March 12, 2013. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Michael Rabbat. The authors are with the Department of Electronic and Computer Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan (e-mail:
[email protected];
[email protected]). Digital Object Identifier 10.1109/LSP.2013.2238528
where the time index has been omitted for brevity. Based on (1), (2) can be re-expressed as
1070-9908/$31.00 © 2013 IEEE
(3)
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IEEE SIGNAL PROCESSING LETTERS, VOL. 20, NO. 5, MAY 2013
where , and , in which denotes is the Kronecker product [9], and the noise matrix constructed by in the same way as from in (2). By using (3), the covariance matrix of , , where denotes the Hermitian operator and , can be shown as
where which
denotes the column stacking operation [9], , in , , and we have used the fact that
(4) where
matrix
, , in which , and we have used the facts that and the covariance , as the noise
is white. and share the same column Note from (4) that space and thus the 1-D Unitary ESPRIT [10] can be employed to yield a set of coarse parameter estimates in the first dimension, say, , where is the number of resolvable parameters in the first dimension. However, ESPRIT can not well resolve the parameters when they are close to each other [10], as the data matrix tends to be rank deficient. To overcome this setback, we partition the signals into smaller groups before proceeding to estimating the parameters in the second dimension. More specifically, based on we can construct a set of given by [8] projection matrices (5) , in which where is obtained by replacing the true parameter by the estimated one in and denotes the matrix Pseudo inverse [9]. We can , which, then obtain the filtered data matrix based on , can be re-written as (6)
It is noteworthy that the above process – parameter estimation, signal separation, and restacking of filtered data can be readily extended to higher dimensions by mathematical induction. For simplicity, we use the shorthand notation for the subscript , so the data matrix for the subgroup in the stage, , is denoted as
.
To follow, we show that
can be expressed as
(8) for , where and denote respectively the number of resolvable parameters and signals in the subgroup, , , is the projection matrix given by (9) in
which , is obtained by replacing the true parameter
estimated one in
by the
. The noise matrix is (10)
where the new subscript denotes the signal in the group, and we have used the fact that , , denotes the Euclidean norm, in which and . Thereby, we partition the signals into groups, where the group has signals with close yet diverse parameters in the other dimensions. in (6) does not contain the parameter We can find that , so we have to re-stack the data before we can employ ESPRIT to estimate . For this, we partition into sub-block matrices, , , and then stack them into as .. .
.. .
where ,
, and
. By (7), it is obvious that (8) is true for . Next, we show that it is also true for the data matrix of the subgroup in the stage after the following three steps. Dimension
A. Estimation of the Parameters in the Based on (8), the covariance matrix of
,
can be expressed as (11) where
(7)
,
and
, .
in
which
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By using the facts that
where of
, , , and that , is the Kronecker delta, the noise covariance matrix in (10) can be shown as (12)
where , and we have used the facts that the projection matrices are Hermitian and idempotent, and [9]. Equation (12) implies that the noise components remain to be white. Therefore, the 1-D Unitary ESPRIT can be invoked to . estimate the parameters, resulting in estimates
Note that in the stage, the parameters are well separated due to the HISS scheme and can then be precisely estimated. In contrast, in the previous dimensions, the estimated parameters are rough because the parameters in these dimensions may be close to each other and are not easy to resolve. To overcome this setback, we resort to as comprises of only one signal, so the 1-D Unitary ESPRIT can be employed to precisely estimate the parameters. More specifically, for the refined parameter estimates in the first dimension, we partition into sub-block matrices, , , and stack them as .. .
.. .
B. Signal Separation Next, we further decompose the signals with close or the same parameters in each group into finer subgroups based on the parameters estimated above. To attain this, we construct a set of projection matrices, , as (9) (for ), and pre-multiby yields , which will plying annihilate the signals that do not belong to the subgroup. Based on (8), the filtered data matrix can be approximated by
(15)
is constructed from
where from
in the same way as
. ,
It follows from (15) that the covariance matrix of is given by
(16) ,
where (13) C. Re-Stacking of the Filtered Data Thereafter, we partition
into
sub-block matrices, ,
, and stack them into a matrix
as
,
have matrix by following the manipulations similar as those leading to (12). We can note that the noise is not white and does not possess the Vandermode structure, so the 1-D Unitary ESPRIT can not be invoked directly. To resolve the problem, we consider an auxiliary matrix , where
used
the
fact
that
the
noise
and we covariance
is the normalized eigenvector .. .
corresponding to the largest eigenvalue of
.. .
[8], and
is as defined in (5). Let , then by using the inverse of a border matrix [10, (Eq. A.69)], it can be readily . Since
shown that
possesses the Vandermonde structure again, we can carry (14)
out the 1-D Unitary ESPRIT with respect to precise estimate of
where we use the fact that . The proof of the mathematical induction is then complete by comparing (8) and (14). The above process will proceed from to , resulting in estimates , , .
to get a more
.
Along the same line as above, we can partition into ( ) sub-block matrices and then stack them to get as (14) to attain a , . Note that since every subgroup in this step only contains one signal, the pairing process is automatically achieved without extra computations. more precise estimate of
,
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IEEE SIGNAL PROCESSING LETTERS, VOL. 20, NO. 5, MAY 2013
The overall steps can be delineated as follows: For do 1) to 3); only do 1) and 2) for 1) (Rough Parameter Estimation) mate the covariance matrix
: Esti-
, where is the number of snapshots (available data matrices) is as defined in (8) with initialization and given in (2). Invoke the 1-D Unitary ESPRIT to yield a set of parameter estimates in the dimension, ; 2) (Signal Separation by Projection) Employ
Fig. 1. Comparison of RMSE versus SNR.
to construct a set of projection matrices by (9) and then obtain the filtered data matrix . 3) (Re-stacking of the Filtered Data) Partition into ( ) sub-block matrices and stack them based on (14) to form . : , do 4): For and 4) (Refined Parameter Estimation) Partition to get with given re-stack them as in (15). Determine the normalized eigenvector corresponding to the largest eigenvalue of ,
, and
in (9) to
. Invoke form the 1-D Unitary ESPRIT to obtain precise estimates , . Remarks: 1. The number of groups and the number of parameters resolvable in each stage, , are assumed to be perfectly estimated, say by the AIC or MDL criterion [10]. 2. As for complexity, since in general , the proposed algorithm requires about real multiplications, where for simplicity we assume that . In contrast, [4] requires roughly real multiplications, [6] needs about real multiplications, and [7] requires about real multiplications. Therefore, the proposed algorithm is generally computationally more parsimonious than [4], [6] and [7]. III. SIMULATIONS AND DISCUSSIONS Consider a 3-D harmonic retrieval problem, where , and the size of the noisy observation is , in which , , . For each specific signal to noise ratio (SNR), 100 Monte Carlo trials are carried out. Four algorithms, including the higher-order ESPRIT [4], MDF [6], RARE [7], and the proposed one, are conducted for comparison. The related Cramer-Rao lower bound (CRLB) is also provided for reference. The comparison of the root-mean-square-errors (RMSEs) for all the parameters versus SNR is shown in Fig. 1, where
and the . We can observe from Fig. 1 number of snapshots that MDF is in general inferior to the other algorithms based on the covariance approach with the same snapshot. Also, RARE can not yield satisfactory performance in the low SNR region because some parameters are close to each other in some dimensions. The performance of the proposed algorithm is close to the others. This is due to the fact that although the proposed algorithm only involves the 1-D Unitary ESPRIT, the degradation is compensated by the combination of the estimation and filtering processes to enhance the parameter estimation accuracy. IV. CONCLUSIONS This paper has developed an efficient, yet accurate algorithm for the MHR problems, which combines the parameter estimation and the filtering process. As such, the parameters are appropriately grouped and estimated, and the pairing of the estimated parameters is automatically achieved. The complexity, meanwhile, is substantially reduced. Conducted simulations demonstrate the efficacy of the new approach. REFERENCES [1] A. B. Gershman and N. D. Sidiropoulos, Space-Time Processing for MIMO Communications. Hoboken, NJ, USA: Wiley, 2005. [2] K. N. Mokios, N. D. Sidiropoulos, M. Pesavento, and C. F. Mecklenbräuker, “On 3-D Harmonic retrieval for wireless communicaiton channel sounding,” in Proc. IEEE Int. Conf. Acoust., Speech, and Signal Process., Montreal, QC, Canada, 2004, pp. II89–II92. [3] R. Roy and T. Kailath, “ESPRIT-Estimation of signal parameters via rotational invariance techniques,” IEEE Trans. Acoust., Speech, Signal Process., vol. 37, no. 7, pp. 984–995, Jul. 1989. [4] M. Haardt, C. Brunner, and J. H. Nossek, “Efficient high-resolution 3-D channel sounding,” in Proc. IEEE Int. Conf. Vehicular Technol., May 1998, vol. 1, pp. 164–168. [5] M. Haardt, F. Roemer, and G. Del Galdo, “Higher-order SVD-based subspace estimation to improve the parameter estimation accuracyin multidimensional harmonic retrieval problems,” IEEE Trans. Signal Process., vol. 56, no. 7, pp. 3198–3213, Jul. 2008. [6] X. Liu and N. D. Sidiropoulos, “On constant modulus multidimensional harmonic retrieval,” in Proc. IEEE Int. Conf. Acoust., Speech, and Signal Process., Orlando, FL, USA, 2002, vol. 3, pp. 2977–2980. [7] M. Pesavento, C. F. Mecklenbräuker, and J. F. Böhme, “Multidimensional rank reduction estimator for parametric MIMO channel models,” EURASIP J. Appl. Signal Process., no. 9, pp. 1354–1363, 2004. [8] J.-D. Lin, W.-H. Fang, Y.-Y. Wang, and J.-T. Chen, “FSF MUSIC for joint DOA and frequency estimation and its performance analysis,” IEEE Trans. Signal Process., vol. 54, no. 12, pp. 4529–4542, Dec. 2006. [9] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. Baltimore, MD, USA: Johns Hopkins Univ. Press, 1996. [10] H. L. Van Trees, Optimum Array Processing. Hoboken, NJ, USA: Wiley-Interscience, 2002.
