DOA estimation for periodically modulated sources
© Chen; licensee Springer. 2013
Received: 23 May 2012
Accepted: 23 April 2013
Published: 24 May 2013
This paper considers the problem of direction-of-arrival estimation for periodically modulated signals using one uniform linear array of sensors. By means of modulating the sources with periodic modulation sequences, we can form a series of linear equations relating the autocorrelation matrices of the received data and the outer products of the scaled steering vectors. Solving these linear equations yields a group of Hermitian matrices formed from the outer products of the scaled steering vectors. Then taking the eigendecomposition of these Hermitian matrices, we can obtain all the scaled steering vectors. By utilizing a special structure of the scaled steering vectors, we can find the directions of signals impinging on the array. We also examine the relation of the modulation sequences and the estimation performance, and a design of the modulation sequences to resist the effect of spatial noise is proposed. One merit of the proposed method is that it can be used in the scenarios of more sources than sensors. The simulation result also shows that it has the capacity to distinguish the closely spaced sources.
The subject of array signal processing is concerned with the extraction of information from signals collected using an array (or arrays) of sensors [1, 2]. One important information is the direction of arrival (DOA) of the incident signals. Take wireless communications for example, the information of DOA can be used for mobile localization for directional transmission in the downlink . Hence, the DOA estimation of sources is one important research topic, and various algorithms in this field over the past decades have been proposed [1–7]. One of the famous algorithms is the multiple signal classification (MUSIC) algorithm proposed in . The merit of the algorithm is that the accuracy of estimation can be obtained for large data samples or at high signal-to-noise ratio (SNR) scenarios. Another famous algorithm is the deterministic method developed by Van Der Veen . Instead of requiring the statistical data and the search procedure in the angular spectrum inherently in the MUSIC algorithm, the deterministic method estimates the DOA directly in terms of eigenvalues of a certain matrix obtained from the received data. Due to the limitation of space, we cannot introduce all algorithms of DOA estimation, and we refer the interested readers to [1, 2, 7] and the references therein for a detailed review.
In digital communications, although the actual transmitted symbol stream is unknown to the receiver, some a priori information of the transmitted signals, for example, the modulation scheme, is available to the receiver. The receiver can take advantage of this extra information with the received data to carry out some tasks including source separation and channel estimation [8–11]. Particularly in [9, 10], the signal sources transmitted are first multiplied at a symbol rate by known amplitude-variation sequences, called modulation sequences, to aid symbol recovery or channel estimation at the receiver. However, there is little research for DOA estimation using modulation sequences. This motivates our research interest in developing a new DOA estimation method for one uniform linear array (ULA) based on modulation sequences.
Our idea and method are shown as follows: By means of modulating the sources with periodic modulation sequences, we can form a set of autocorrelation matrices of the received data. Then the set of autocorrelation matrices allows us to formulate a series of linear equations relating the outer products of the scaled steering vectors. Solving the set of linear equations produces a group of Hermitian matrices, which are the outer products of the scaled steering vectors. Then taking the eigendecomposition of these Hermitian matrices, we can obtain all the scaled steering vectors. By utilizing a special structure of the scaled steering vectors, we can find the directions of signals impinging on the array. We also examine the relation of the modulation sequences and the estimation performance, and a design of the modulation sequences to resist the effect of spatial noise is proposed. The merit of the proposed method is that it can be used in the case of more sources than sensors. In addition, the method has the capacity to distinguish the closely spaced sources.
It is worth to mention that since the periodically modulated signals are artificial, the proposed method is suitable for communication signals. One possible application of the proposed method is mobile localization for directional transmission in the downlink since the modulation formats of the mobile units are available to the base station in the uplink [3, 11].
This paper is organized as follows: Section 2 briefly reviews the system model and provides basic assumptions. In Section 3, we derive the estimation method and discuss some properties of the proposed algorithm. Simulation results are given in Section 4. Section 5 concludes this paper.
(·)∗, (·) T , and (·) H denote the complex conjugate, transpose, and conjugate transpose operations, respectively. The notation ∥ · ∥2 is the 2-norm. The symbols and represent the set of real numbers and the set of complex numbers, respectively. I M is the identity matrix of dimension M × M. A ∘ B is the Hadamard product of matrices and (, p. 190), and A(:, k) and A(l, :) are the k th column vector and the l th row vector of A, respectively. For a vector , b(r 1: r 2) is the subvector formed from the r 1th element to the r 2th element of b. In addition, for any M × M matrix G = [g k, l ]0 ≤ k, l ≤ m − 1, we define the operation Γ j (G) = [g 0, j g 1, j + 1⋯g m − 1 − j, m − 1] T , for 0 ≤ j ≤ m − 1, i.e., Γ j (G) is the column vector formed from the j th superdiagonal of G.
2 Problem statement
where and .
The source vector s(k) is a zero-mean, temporally and spatially uncorrelated, and wide-sense stationary vector with . The noise vector w(k) is zero-mean, wide-sense stationary, and , where δ(·) is the Kronecker delta function. In addition, the source signal is uncorrelated with the noise, i.e., E[s(m)w(n) H ] = 0, ∀ m, n .
The DOA , i = 1, 2, ⋯, N.
Each of the modulation sequences c i (k), i = 1, 2, ⋯, N, is periodic with period P ≥ N + 1, i.e., c i (k) = c i (k + P).
3 DOA estimation
In this section, we first derive the estimation method when noise is absent. The design of the modulation sequences when noise is present is given in Section 3.2. Some further discussions about the proposed method are given in Section 3.3.
3.1 The proposed method
Here we let with h i = d i a(θ i ) be the scaled steering vector for i = 1, 2, ⋯, N.
Since C(k) is a diagonal matrix formed from the periodic modulation sequences c 1(k), c 2(k), ⋯, c N (k) with period P by assumption (iii), we know that C(k)2 is also periodic with period P, i.e., C(k)2 = C(k + P)2, which implies that R k+P = R k , for example, R 1 + P = R 1. In addition, for the purpose of DOA estimation, we need the following proposition to aid our derivation of the proposed method.
where and is a vector for i = 0, 1, ⋯, M − 1.
Please see Appendix. □
Then we divide each scaled steering vector into two subvectors, namely and . It is clear that and can be obtained from the least squares solution. Then the DOA θ N is thus obtained from the angle of .
If the array is only composed of two sensors, i.e., M = 2, then h n is a 2 × 1 vector, and it is obvious that . In this case, we can divide the first entry of by the second one to obtain the DOA for n = 1, 2, ⋯, N. In other words, the proposed method can carry out the DOA estimation using only two sensors in theory.
3.2 The design of the modulation sequences
We have derived the estimation method in Section 3.1. We now discuss how to design the modulation sequences to combat the effect of noise on DOA estimation.
where q = (W T W) − 1 W T 1 P . From (3.18), we know that is the actual X 0(:, k) plus a perturbation term σ w 2q due to noise. Since q is formed from the modulation sequences, we need to design the modulation sequences to minimize ∥q∥2 and the effect of the resulting perturbation term. However, the high nonlinearity of the modulation sequences contained in q makes it difficult to design. Hence, we adopt another reasonable design criterion which is also used in  to tackle this problem.
for i = 1, 2, ⋯, N. In the study of blind channel estimation using periodic modulation , the two-level sequence in (3.23) is also shown to be optimal for mitigating the channel noise effect. In addition, with the optimal solution in (3.23), the corresponding γ i is , ∀i = 1, 2, ⋯, N. Note that γ opt decreases as τ decreases, and thus, the noise effect imposed on V 0 is reduced and hence estimation performance improves. We will give a simulation example to illustrate this property.
From (3.23), we know that each of the N modulation sequences is a two-level sequence with a single peak in one period. However, to make the matrix W be full column rank such that the least squares solutions (3.7) can be computed, the peak locations of the N modulation sequences in one period need to be distinct with one another, i.e., m k ≠ m l for all k ≠ l. Without loss of generality, we can let m i = i for i = 1, 2, ⋯, N.
From (3.26), it is obvious that if , then the effect of noise can be eliminated. From here, the discussion and derivation are the same as the content from (3.19) to (3.23). Hence we know that the optimal sequences given in (3.23) can work well for the colored noise case. We will give a simulation in Section 4 to demonstrate this feature.
We now discuss some notable features of the proposed method. First, from the result at the end of Section 3.1, we know that after is estimated, the DOA angle θ N can be obtained from the linear equation . Since is an M × 1 vector, the DOA angle θ N can be acquired provided that M ≥ 2, where M is the number of sensors for the ULA. Hence, the proposed method can carry out DOA etimation not only for the case of more sensors (M ≥ N), but also for the case of less sensors (M < N), as long as the number of sensors M ≥ 2. Second, the DOA , n = 1, 2, ⋯, N, may not be distinct with each other since from Section 3.1, we know that the estimates of θ 1, θ 2, ⋯, and θ N are independently obtained from the corresponding scaled steering vectors , , ⋯, and , respectively. Hence, the proposed method possesses the capacity to distinguish the closely spaced sources. We will give a simulation to demonstrate this feature. The third feature of the proposed method is that it provides a design of the modulation sequences to minimize the effect of noise on DOA estimation and thus improves the accuracy of the solution.
Collect the received data , where S divides P, the period of the modulation sequences.
- 2.Compute the autocorrelation matrices , , ⋯, via the following time average:(3.27)
Use the autocorrelation matrices in (3.27) to form V 0, V 1, ⋯, V M−1, and use the designed modulation sequences (3.23) to form W.
Obtain the matrices X 0, X 1, ⋯, X M−1 using the least squares solutions in (3.7) with the aid of the matrices V 0, V 1, ⋯, V M−1, and W obtained from the previous step.
Form N rank-one Hermitian matrices Q 1, Q 2, ⋯, Q n from , i = 0, 1, ⋯, M − 1, with the aid of (3.10).
For each , compute the estimate as the unit-norm eigenvector associated with the maximal eigenvalue of Q n , ∀n = 1, 2, ⋯, N.
Divide each scaled steering vector into two subvectors, namely and . Then obtain the DOA from the angle of the least squares solution of the linear equation .
In this section, we use several simulations to demonstrate the performance of the proposed method. For all simulation examples, the SNR is defined as . We use the root-mean-square error (RMSE) of angles as the performance measure, which is defined as , where is the estimate of θ N . The number of Monte Carlo trials is 500. The source symbols are independent and identically distributed binary phase-shift keying signals. The noise is zero-mean and white Gaussian (except for simulation 3).
4.1 Simulation 1 - underdetermined DOA estimation
4.2 Simulation 2 - comparison with existing methods
4.3 Simulation 3 - underdetermined DOA estimation in the presence of colored noise
This paper has proposed a new DOA estimation algorithm for one ULA based on periodic modulation. The proposed method has three notable features. First, the proposed algorithm can handle more sources than sensors, which may be few as two. In addition, the great capacity to distinguish the closely spaced sources is the second feature of the proposed method. The final feature of the proposed method is that the performance of the estimation algorithm depends on the choice of the modulation sequences to resist the noise effects. Hence, we can properly choose the modulation sequences to improve the performance of estimation. Simulation results are used to demonstrate the performance of the proposed method and to compare it with some existing methods.
Proof of Proposition 1
Let , , and be three row vectors. Then it is easy to verify that (a ∘ b)f H = (b ∘ f ∗)a T .
asserts the result given in Proposition 1. □
This research was sponsored by the National Science Council of Taiwan under grant NSC-99-2221-E035-056-.
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