DOA estimation for wideband signals based on weighted Squared TOPS
- Hirotaka Hayashi^{1}Email authorView ORCID ID profile and
- Tomoaki Ohtsuki^{2}
https://doi.org/10.1186/s13638-016-0743-9
© The Author(s) 2016
Received: 8 December 2015
Accepted: 1 October 2016
Published: 12 October 2016
Abstract
This paper introduces a new direction-of-arrival (DOA) estimation method for wideband signal sources. The new method estimates the DOA of wideband signal sources based on squared test of orthogonality of projected subspaces (Squared TOPS) which is an improved method of TOPS. TOPS and Squared TOPS use the signal and noise subspaces of multiple frequency components of wideband signal sources. Although coherent wideband method, such as coherent signal subspace method (CSSM), performs high DOA estimation accuracy, it requires the initial estimate of signal source directions. On the other hand, TOPS and Squared TOPS can provide good performance of DOA estimation without the initial value of signal sources; however, some false peaks appear in spatial spectrum based on these methods. The proposed method, called weighted Squared TOPS (WS-TOPS), uses the modified squared matrix and selective weighted averaging process to improve DOA estimation performance. The performance of WS-TOPS is compared with those of TOPS, Squared TOPS, incoherent MUSIC, and test of orthogonality of frequency spaces (TOFS) through computer simulations. The simulation results show that WS-TOPS can suppress all false peaks in spatial spectrum and improve DOA estimation accuracy and also keep the same resolution performance as Squared TOPS.
Keywords
Array signal processing Direction-of-arrival estimation Wideband signals1 Introduction
Direction-of-arrival (DOA) estimation for wideband signals has been attracting much attention for decades because wideband signals are commonly used in real world for such as signal source localization in wireless communication and radar systems. To improve degree-of-freedom (DOF) and accuracy of DOA estimation, many researches on wideband DOA estimation have also been introduced over several decades [1]. DOA estimation methods for narrowband signals cannot be applied directly to wideband signals because the phase difference between array antennas depends on not only the DOA of the signals but also the temporal frequency. Thus, the common pre-processing for wideband signal estimator decomposes a wideband signal into some narrowband signals using filter banks or a discrete Fourier transform (DFT). Based on the method, many algorithms have been introduced and they are categorized into two groups: incoherent signal subspace method (ISSM) [2, 3] and coherent signal subspace method (CSSM) [4].
ISSM is one of the simplest wideband DOA estimation methods. ISSM uses several narrowband signals decomposed from wideband signal incoherently [3]. In particular, the method applies narrowband DOA estimation techniques, such as MUSIC [5], independently to the narrowband signals. Then, these results are averaged to estimate the DOA of incoming wideband signal sources. Although ISSM provides better estimation accuracy in high signal-to-noise ratio (SNR) regions, the performance deteriorates when the SNR of some frequency bands are low. In other words, the poor estimates from some frequency bands will degrade the final estimation accuracy.
To overcome these disadvantages and to improve DOA estimation performance, CSSM was proposed [4]. In CSSM processing, the correlation matrix of each frequency band is focused by transformation matrices and the focused matrices are averaged to generate a new correlation matrix. Then, CSSM estimates the DOA of incoming wideband signal sources by applying a DOA estimation method for narrowband signals. The key point of CSSM algorithm is how to focus correlation matrices. Many techniques have been proposed to obtain a proper focusing matrix [6, 7]. However, each focusing technique requires the initial values, which means the pre-estimated direction of incoming signal sources, and the performance of CSSM is sensitive to the initial values [8]. The weighted average of signal subspaces (WAVES) [9] is also a well-known DOA estimation method for wideband signal sources. However, WAVES also needs the initial DOA estimates and its performance greatly depends on the accuracy of the initial values.
A novel wideband DOA estimation method, which is named test of orthogonality of projected subspaces (TOPS), was proposed [10]. TOPS uses the signal and noise subspaces of several frequency bands and provides good DOA estimation performance without requiring the initial values. However, the method has a drawback that the spatial spectrum calculated by TOPS algorithm has some false peaks and they make it difficult to estimate the true DOA of signal sources.
Squared TOPS was proposed as an improvement method of TOPS [11]. Squared TOPS improves DOA estimation performance by using the squared matrix for orthogonality test instead of the matrix to be tested in the signal processing of TOPS. The method provides higher DOA estimation accuracy and better resolution performance than those of TOPS. However, the undesirable false peaks in spatial spectrum remain.
The test of orthogonality of frequency subspaces (TOFS) was proposed as a new wideband DOA estimation method [12]. TOFS uses the noise subspaces of multiple frequency bands with the steering vector and shows high estimation accuracy when SNR is high. However, TOFS cannot resolve closely spaced signal sources when SNR is low.
Recently, Khatri-Rao (KR) subspace approach was proposed as the method to expand the array structure and to increase DOF [13]. Applying KR subspace approach to CSSM algorithm, some DOA estimation methods of wideband signal sources were proposed [14]. The methods achieve higher DOA estimation accuracy and resolution performance than the conventional CSSM even if there are fewer sensors or antennas than the incoming signal sources. However, it also requires the initial DOA estimate of each signal source.
Furthermore, sparse signal representation algorithms have also been received much attention, which can provide new approaches for wideband DOA estimation [15–17]. These DOA estimation methods based on the sparse signal representation perform higher resolution than the conventional methods without requiring the number of sources. However, there are some difficulties in selecting properly parameters to calculate optimal solutions.
In this paper, we propose a new DOA estimation method for wideband signals called weighted Squared TOPS (WS-TOPS) based on Squared TOPS. WS-TOPS also uses signal subspace and noise subspace of each frequency like Squared TOPS and does not require any initial values. Using modified squared matrix and selective weighted averaging process, WS-TOPS can suppress all false peaks in spatial spectrum and improve DOA estimation accuracy of wideband signal sources and also keep the same resolution performance as Squared TOPS.
This paper is organized as follows. In Section 2, the signal model is described, and the conventional DOA estimation algorithms are explained in Section 3. In Section 4, WS-TOPS is proposed. In Section 5, simulation results are presented, and conclusions are provided in Section 6.
Notation: We denote vectors and matrices by boldface lowercase and uppercase letters, respectively. The superscripts T and H are transpose and complex conjugate transpose, respectively. E[·] denotes the expectation operator.
2 Signal model
3 Conventional wideband DOA estimation methods
In this section, we explain some conventional DOA estimation methods, which can estimate the DOA of incoming wideband signal sources without any initial value. In these methods, each wideband signal is decomposed into K narrowband signals by DFT as mentioned in the previous section.
3.1 Incoherent MUSIC (IMUSIC)
Since the DOAs estimated by Eq. (11) are averages of the result of each frequency band, the poor estimates from a single frequency band even degrades the final estimation accuracy.
3.2 Test of orthogonality of frequency subspaces (TOFS)
TOFS shows good DOA estimation accuracy in high SNR region by using the noise subspaces obtained from the correlation matrix of received signals. However, TOFS cannot resolve closely spaced signal sources when SNR is low.
3.3 Test of orthogonality of projected subspaces (TOPS)
TOPS uses both of the signal and noise subspaces of each frequency band to estimate the DOA of incoming wideband signal sources [10].
First of all, we obtain the signal subspace F _{ i } and the noise subspace W _{ i } from EVD of the correlation matrix of each frequency band, as described in the previous section.
Then, one frequency band ω _{ i } should be selected and the signal subspace F _{ i } of the selected frequency band is transformed into other frequencies. One of the advantages of TOPS over CSSM is that TOPS does not require the initial DOA estimates for the frequency transform process as follows.
where \(\hat {\boldsymbol \theta }\) is the transformed θ by using the frequency transform matrix Φ(ω _{ i },θ),G _{ i } is a full-rank square matrix that satisfies F _{ i }=A _{ i }(θ)G _{ i }. The transformation process just transforms an array manifold at any frequency and DOA into another array manifold corresponding to another frequency. Therefore, the transformed matrix is a full rank matrix and could be used for the following test of orthogonality between transformed matrix and noise subspaces as discussed in detail in [10].
where \(\sigma ^{\prime }_{\text {min}}(\theta)\) is the minimum singular value of the matrix D ^{′}(θ).
where \(\sigma ^{\prime \prime }_{\text {min}}(\theta)\) is the minimum singular value of the matrix D ^{′′}(θ).
The output signal of DFT or bandpass filter is not always a perfect narrowband signal. The filtered signals could degrade the DOA estimation accuracy. TOPS can reduce those degradations by using the certain signal subspace obtained from the estimated correlation matrix instead of the steering vector of the frequency band. It indicates that the method to select the frequency band, of which the signal subspace will be transformed to other frequency bands by Eq. (16), also affects the resulting DOA estimation. In contrast, the residual error of the subspaces causes some undesirable rank reductions of the matrix D ^{′′}(θ). Therefore, TOPS has the serious disadvantage that several false peaks appear in the spatial spectrum obtained by Eq. (24).
3.4 Squared TOPS
Squared TOPS applies two techniques to TOPS to improve the performance of DOA estimation [11]. One is the technique to select the frequency band of which the signal subspace will be used. The other is the technique to improve the sensitivity of rank decrease of the matrix D ^{′′}(θ) when θ is the one DOA of incoming wideband signal sources.
The reference frequency, which is defined as the frequency band of which the signal subspace will be used, should be the frequency band with the highest SNR. Squared TOPS uses the frequency band where the difference between the smallest signal eigenvalue λ _{ i,L } and the largest noise eigenvalue λ _{ i,L+1} is maximum as the reference frequency [11].
Both of the row and the column elements of the matrix Z _{ i }(θ) obtained by the squared operation should be close to zero when θ is the DOA of incoming wideband signal sources. It means that the operation improves the sensitivity to detect the rank reduction of the orthogonality evaluation matrix. Eventually, it provides improvement of the estimation performance of Squared TOPS. However, the undesirable false peaks in spatial spectrum remain.
4 Proposed method
In this section, we explain our proposed method named weighted Squared TOPS (WS-TOPS). WS-TOPS applies the following two approaches to Squared TOPS to improve DOA estimation performance. One is the modified squared matrix method, which is the algorithm to suppress the false peaks in the spatial spectrum of Squared TOPS. The other is the selective weighted averaging method, which is the algorithm to improve the DOA estimation accuracy by using the signal subspaces of multiple frequency bands. The details of these algorithms are shown in the following subsections.
4.1 Modified squared matrix (algorithm 1)
Although Squared TOPS and TOPS use the projection matrix P _{ i }(θ) to reduce the signal subspace component leakage in the estimated noise subspace, some false peaks in the spatial spectrum remain. This is the serious disadvantage of TOPS and Squared TOPS. The transformed signal subspace matrix \(\boldsymbol {U}^{\prime }_{ij}(\theta)\) has residual error, and it causes the undesirable rank decrease of the matrix Z _{ i }(θ). Thus, we propose the algorithm to suppress these false peaks by modifying the component of the matrix Z _{ i }(θ).
The steering vector a _{ i }(θ) is orthogonal to the noise subspaces only when θ is the DOA of the incoming wideband signal sources. Here, we define \(b_{j}(\theta)=\boldsymbol {a}_{j}^{H}(\theta)\boldsymbol {W}_{j}\boldsymbol {W}_{j}^{H}\boldsymbol {a}_{j}(\theta)\). Then, we can avoid the undesirable rank decrease of the matrix Z _{ i }(θ) by adding the square matrix of which the diagonal elements are b _{ j }(θ) of the frequency band ω _{ j }. We, however, need to consider how we add b _{ j }(θ) to the components of the matrix Z _{ i }(θ) because b _{ j }(θ) is similar to the components of TOFS, and it would cause the degradation of the resolution performance of closely spaced signal sources.
b _{ j }(θ) is calculated by using a steering vector and noise subspaces. As we can see from Eq. (5), \(\boldsymbol {a}_{j}^{H}(\theta)\boldsymbol {a}_{j}(\theta)\) is M that is the number of antennas. Therefore, b _{ j }(θ) changes between 0 and M. If the steering vector is orthogonal to all of noise subspaces, b _{ j }(θ) is 0. If the steering vector is not orthogonal to noise subspaces, b _{ j }(θ) comes close to M. On the other hand, the elements of \(\boldsymbol {U}_{ij}^{'H}(\theta)\boldsymbol {W}_{j}\boldsymbol {W}_{j}^{H}\boldsymbol {U}_{ij}^{\prime }(\theta)\) are calculated by using the transformed signal subspaces and noise subspaces. Here, we define the lth column of \(\boldsymbol {U}_{ij}^{\prime }(\theta)\) as \(\boldsymbol {u}_{ijl}^{\prime }(\theta)\), which is a transformed signal subspace. As we can also see from Eqs. (16) and (22), \(\boldsymbol {u}_{ijl}^{'H}(\theta)\boldsymbol {u}_{ijl}^{\prime }(\theta)\) is 1, thus each element of \(\boldsymbol {U}_{ij}^{'H}(\theta)\boldsymbol {W}_{j}\boldsymbol {W}_{j}^{H}\boldsymbol {U}_{ij}^{\prime }(\theta)\) changes between 0 and 1. Therefore, we divide b _{ j }(θ) by M to deal with the elements of \(\boldsymbol {U}_{ij}^{'H}(\theta)\boldsymbol {W}_{j}\boldsymbol {W}_{j}^{H}\boldsymbol {U}_{ij}^{\prime }(\theta)\) and b _{ j }(θ) as the same range. Based on the discussion, we modify the component of Z _{ i }(θ) as follows.
where \(\sigma ^{\prime }_{{zi}_{\text {min}}}(\theta)\) is the minimum singular value of the matrix \(\boldsymbol {Z}^{\prime }_{i}(\theta)\).
4.2 Selective weighted averaging (algorithm 2)
The spatial spectrum obtained from Eq. (32) is averaged with the weights α _{ i }. Therefore, the algorithm can improve the DOA estimation accuracy of signal sources in high SNR region. However, the algorithm causes deteriorations of sharpness of spectrum peaks when each frequency band shows different peaks to each other, e.g., in low SNR region. Eventually, the algorithm degrades the resolution performance of closely spaced signal sources in low SNR region.
To prevent the deterioration caused by the process, we propose a selective averaging approach. By using only the frequency bands ω _{ i } with weight α _{ i } larger than a certain threshold α _{th}, which are the reliable frequency bands, we can improve DOA estimation performance and also reduce computational cost.
where K ^{′} is the number of frequency bands with weight α _{ i } larger than the threshold α _{th}. If there is no frequency band with weight α _{ i } larger than the threshold α _{th}, we use the signal subspace of the frequency band with the largest weight α _{ i }. For example, there are signal sources with low power and every α _{ i } is smaller than α _{th}.
If we use only single frequency band, the proposed method keeps the same performance as Squared TOPS. In other words, the algorithm can provide better performance than that of Squared TOPS even in the case of all α _{ i } that are smaller than the threshold α _{th}. In what follows, we set α _{th}=9, which implies that the signal power of the frequency band is larger than (3σ _{ n })^{2} based on Eq. (31). α _{th} in the algorithm 2 determines the frequency bands of which spatial spectra are averaged based on SNR; therefore, the performance of the algorithm 2 depends on SNR of each frequency band.
4.3 Weighted Squared TOPS (WS-TOPS)
4.4 Computational complexity
Computational complexity
Complexity | ||
---|---|---|
Algorithm | Calculation for D,Z, and Z ^{′} | Calculation for SVD |
TOPS | O(L M(M−L)(K−1)) | +O(L ^{2}(M−L)(K−1)) |
Squared TOPS | O({2L M(M−L)+L ^{2}(M−L)}(K−1)) | +O(L ^{3}(K−1)) |
WS-TOPS | O({2L M(M−L)+L ^{2}(M−L)}(K−1)K ^{′}) | +O(L ^{3}(K−1)K ^{′}) |
5 Numerical results
This section shows numerical simulation results to demonstrate the performance of WS-TOPS with respect to those of the conventional methods which do not require the initial value of DOA: IMUSIC, TOFS, TOPS, and Squared TOPS.
5.1 Simulation parameters
Then, we calculate the signal subspace matrix F _{ i } and the noise subspace matrix W _{ i } from EVD of the correlation matrix \(\hat {\boldsymbol {R}}(\omega _{i})\) and estimate the DOA of incoming wideband signal sources by using WS-TOPS and each conventional method described in Section 4.
Simulation parameters
Item | Symbol | Quantity | Remarks |
---|---|---|---|
Antenna spacing | d | λ/2 | Uniform linear array |
Frequency ω _{ L } | ω _{ L } | π/3 | The lowest frequency of signal sources (ω domain) |
Frequency ω _{ H } | ω _{ H } | 2π/3 | The highest frequency of signal sources (ω domain) |
Parameter α _{ th } | α _{ th } | 9 | 3σ _{ n } |
5.2 Spatial spectrum
As shown in Fig. 2, WS-TOPS can hold the capability to suppress false peaks for all scenarios. The results prove that the WS-TOPS is robust to the system parameters, which are the number of antennas (M), the number of sources (L), and the number of frequency bins (K).
Regarding the computational complexity of WS-TOPS- and TOPS-based methods, we calculate the computational costs to obtain an inverse of the minimum singular value of each direction by using MATLAB. In the case of M=10,L=3, and K=7, the averaged computation time of WS-TOPS (K ^{′}=7) is 2.1 ms, that of WS-TOPS (K ^{′}=2) is 0.59 ms, that of Squared TOPS is 0.21 ms, and that of TOPS is 0.17 ms. In the case of M=10,L=3, and K=15, the averaged computation time of WS-TOPS (K ^{′}=7) is 5.1 ms, that of WS-TOPS (K ^{′}=2) is 1.47 ms, that of Squared TOPS is 0.44 ms, and that of TOPS is 0.42 ms. Although the actual computational times depend on the calculation system, the results show that the effective costs coincide with the computational complexity described in Fig. 1.
5.3 Probability of resolution
5.4 Root mean square error (RMSE) of estimated DOA
Figure 5 shows DOA estimation accuracy of the signal source from 33° where there is the closely spaced wideband signal source. From Fig. 5, it is found that WS-TOPS yields the best performance of DOA estimation accuracy for closely spaced wideband signal sources in full range of SNR. The results prove that the DOA estimation accuracy of WS-TOPS is better than the conventional methods and also show that the performance of WS-TOPS is robust to the system parameters.
6 Conclusions
In this paper, we propose a new DOA estimation method for wideband signals called WS-TOPS-based on Squared TOPS. WS-TOPS uses the selective weighted averaging method and the modified squared matrix method to improve DOA estimation performance. The simulation results show that WS-TOPS can suppress all false peaks in the spatial spectrum, while TOPS and Squared TOPS cannot. It is also shown that the DOA estimation accuracy and the resolution performance of WS-TOPS are better than those of the conventional methods. WS-TOPS can achieve the performance without requiring initial estimates. These results prove that WS-TOPS is effective in estimating the DOA of wideband signal sources.
Declarations
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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