Estimation is generally carried out using different statistics of the output. The authors of this book consider the blind identification and source separation problem in the complex-domain, where the available statistical properties are richer and include non-circularity of the sources — underlying components. They define identifiability conditions and present state-of-the-art algorithms that are based on algebraic methods as well as iterative algorithms based on maximum likelihood theory.
Her research interests concern statistical and adaptive signal processing, with an emphasis on nonlinear and complex-valued signal processing, and applications in biomedical data analysis and communications. Blind identification consists of estimating a multidimensional system through the use of only its output.
Source separation is concerned with the blind estimation of the inverse of the system. The estimation is generally performed by using different statistics of the outputs. They also consider the case of direct estimation of the inverse system for the purpose of source separation.
They then describe the estimation theory associated with the identifiability conditions and dedicated algebraic algorithms. Here at Walmart. Your email address will never be sold or distributed to a third party for any reason. Due to the high volume of feedback, we are unable to respond to individual comments. Sorry, but we can't respond to individual comments.
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To sort synchronous FH signals in underdetermined situation, in which the number of sensors is less than that of signals, Fu and Sha used underdetermined blind source separation UBSS method [ 19 , 20 ] based on time frequency TF analysis. Sha ignored the frequency effect on mixing matrix, so the results were not useful in practice.
Fu modified time frequency ratio of mixtures algorithm to estimate frequencies and mixing matrix, and then DOA was obtained to sort synchronous FH signals. Many other alternative methods are introduced, such as [ 21 , 22 ]. Many single source point SSP detection methods have been proposed to improve the accuracy of mixing matrix estimation. The single source points refer to those TF vectors with only one source contribution. A time frequency ratio of mixtures TFROM method was proposed in [ 27 ] by calculating the ratio values to estimate mixing matrix at each TF point.
This method requires that only one source is active at TF points so that the imaginary parts of the STFT coefficients are equal to zero. If more sources are active, the coefficients have complex values. A new method to detect SSP was introduced in [ 28 ], which calculated absolute directions of real parts and imaginary parts of TF coefficients. Similar methods were proposed in [ 29 , 30 ]. In [ 29 ], the mixing matrix estimation method was improved by Kernel K-hyperline clustering technology.
Sun improved the absolute directions methods in [ 25 ] by ignoring low-energy points in TF domain. Xie modified the SSPs detection method in [ 31 ], which utilized the assumption that the trace of Wigner—Ville distribution WVD was equal to its unique nonzero eigenvalue. The methods based on detecting auto term TF points were adopted in [ 32 , 33 , 34 ] to estimate mixing matrix with the same assumption in [ 31 ]. However, these SSP detection methods [ 27 , 28 , 29 , 30 , 31 ] and the auto term TF point detection methods [ 32 , 33 , 34 ] are only suitable for real value mixing matrix.
Since the mixing matrix of FH signals is complex-valued, some methods that deal with complex-valued mixing matrix estimation are discussed. Li and Nie considered the complex-valued mixing matrix estimation problem in [ 35 , 36 ] based on SSPs detection. Li introduced that the normalizations of coefficients of the Vandermonde mixing matrix were equal to one and the corresponding mixing coefficients could be calculated by utilizing a matrix inverse operation, so the mixing matrix estimation could be estimated by k-means clustering algorithm [ 35 ].
A similar theorem was introduced based on the inherent property of the Vandermonde mixing matrix in [ 36 ]. Although complex-valued mixing matrix estimation methods are developed, the performance is not satisfactory. An improved SSPs detection method is proposed in this paper. To recover sources, the methods in [ 37 , 38 , 39 ] based on subspace projection assumed that the number of active sources at any TF point was no more than the number of sensors, thus sources could be calculated by inverse STFT.
Practically, in the subspace projection method, since these methods do not consider the effect of the residual sources, a threshold value is chosen to calculate the residual power in this paper. Firstly, a new criterion is proposed to identify SSPs to improve mixing matrix estimation performance. Secondly, another improved source recovery method is introduced to estimate frequencies according to subspace projection. With the greatly improved estimations of mixing matrix and frequencies, the DOA of each piece of FH signals can be calculated.
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Finally, synchronous orthogonal FH signals can be sorted in underdetermined condition. This paper is organized as follows. In Section 2 , the FH signal model is introduced. The proposed methods are introduced in Section 3. In Section 4 , the analysis and the performance of the proposed algorithms are provided and conclusions are given in Section 5. The model is shown in Figure 1. T is the observation time. To avoid interference of irrelevant aspects, the amplitude and the phase can be ignored in this paper.
The delay time cannot be ignored when signals impinge on different antennas. The propagation time-delay from the m th antenna to the first antenna of the n th source can be formulated as follow according to Figure 1 ,. Therefore, the m th received signals can be written as. The vector formulation of 3 is. Assume that the FH signals are not hopping in delay time, thus, according to 1 and 2 , the transmission coefficient matrix A t is written as. Obviously, according to 2 , A t is a Vandermonde matrix. The data model in 4 is the same with blind source separation, so A t is the mixing matrix.
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The mixing matrix of FH signals changes with hopping frequencies and it is not easy to be estimated. In this paper, firstly the hop timings are obtained by other methods [ 40 , 41 ]. Then, the mixing matrix is constant in hop duration. The data model can be written in the k th hop duration.
To identify FH signal slices, more parameters need to be estimated, such as DOAs and frequencies, which are included in the mixing matrix. Thus, the major task is to estimate the mixing matrix. Under UBSS condition, the number of sensors is less than the number of sources. Similar to other UBSS methods [ 19 , 20 ], the following two assumptions should be satisfied. This assumption is easily satisfied due to the Vandermonde structure of mixing matrix. Second, suppose that only one source is active at some points in TF domain.
FH signal is sparse in time-frequency domain. Single-source-points detection method is effective to deal with sparse signals. The method in [ 42 ] assumes that only one source is active at each time-frequency point. The algorithm in [ 27 ] requires that there exists at least one time-frequency point of single-source-occupancy for each source. The algorithms in References [ 35 , 36 , 43 ] assume that there exist some points in the TF domain where only a single source is active and such points must exist for each source.
Therefore, the second assumption is reasonable. If the frequency hopping signal does not collide, this assumption is valid.
In this section, firstly, the proposed SSPs detection algorithm is introduced to estimate mixing matrix. Secondly, the improved subspace projection algorithm is proposed to recover sources. For simplicity, STFT is applied on both side of 6 without noise term. The mixtures in TF domain can be written as. Since the methods in [ 35 , 36 ] treat a large number of multi-source points as single-source points, the estimation results of mixing matrix are not satisfactory.
Therefore, a new theorem is proposed to detect the single source points in TF domain. The proof is given in Appendix A. After detecting all SSPs, the mixing matrix can be estimated by clustering methods. If the second row of the mixing matrix can be obtained, the mixing matrix is obviously calculated based on the Vandermonde structure. According to 7 ,.
Because the first row elements of Vandermonde matrix are equal to one, the following result can be obtained.
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Although it may not be the best clustering algorithm, the clustering method will be studied in the future work. An improved subspace projection method is proposed which is inspired by the method in [ 38 ]. The method in [ 38 ] assumed that there were no more than M active sources at each TF point.
It calculated the minimum value of the product of the orthogonal projection matrix of mixing matrix and the TF domain mixtures to identify the sources.
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However, this method did not consider the effect of redundant signals. That is why the number of active sources at each TF point is always estimated as M. To improve the accuracy of the recovered FH signals, the proposed method in this paper sets a threshold to solve this problem.
The l FH signals can be identified by. For synchronous FH signals, hopping times of FH signals are the same. DOAs do not change in a short time. The decision criterion is. The value of r will be described in Section 4. Step 8: Compare the DOAs in different hop durations. The recovered FH signals can be spliced according to In this section, numerical results are given to demonstrate the effectiveness of the proposed algorithms. The synchronous FH signals parameters are shown in Table 1. The influence of thresholds on the estimation performance is discussed later.
Hop timings can be estimated by the methods in [ 40 , 41 ]. This parameter is ignored in Table 1. In the following simulations, the SNR is written as. Following [ 44 ], the performance of parameter estimation can be measured by Root mean square error RMSE which is defined as. The performance of the mixing matrix estimation is demonstrated by mean square error MSE which is defined as.
A small MSE indicates superior quality. In addition, the columns of the estimated mixing matrix are normalized and the uncertainty of the columns is removed. This experiment does Monte Carlo tests. It can be seen that the proposed method can obtain higher accuracy than other algorithms. Since the methods of Li and Nie treat a large number of multi-source points as single-source points, the estimation results of mixing matrix are not satisfactory.
Figure 3 a shows N clusters clearly. Because the proposed method detects fewer error points than the other two methods, the proposed method works very well. To analyze the complexity of these methods, CPU time is calculated. The cost time of the proposed method is increasing with the number of SSPs. It demonstrates that the performance of the proposed method is also better than the others when the source number increases to six. In other words, the performance becomes bad when the number of sources increases. Therefore, the number of FH signals cannot be much larger than that of antennas.
This will lead to wrong results. This will also lead to wrong results. The effect of sample size on the performance of mixing matrix is discussed in this part.
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Fourth, the influence of hop timing error on the MSE of mixing matrix is discussed. Figure 9 shows the impact of different hop timing errors on different sample sizes. Thus, the greater the radio of the hop timing error to sample size is, the worse the result is. According to the existing hop timing estimation algorithms in [ 40 , 41 ], the hop timing can be estimated correctly in a certain range of error.
Hop timing error is not very restricted to the proposed method. The recovery performance is quantified by source to interference ratio SIR , which is defined as. A large SIR indicates superior quality. First, Figure 10 compares the recovery performance between the improved subspace projection method and the method in [ 38 ]. These two methods utilize the mixing matrix that is estimated by the proposed SSPs detection method.
It can be seen that the improved subspace projection method performs better than the method in [ 38 ].
Second, according to estimated mixing matrices that are obtained by the three methods, the improved subspace projection method is used to recover signals, and the performance is shown in Figure It can be known that the higher the estimation accuracy of the mixing matrix is, the better the recovered performance is. This also proves that the proposed SSPs detection method is effective and can improve recovered signal performance.
Third, frequencies can be estimated by FFT operator when the signals are obtained by the proposed recovery method in time domain. Figure 12 shows the comparison between the true and the estimated frequencies with SNR increasing. The square represents the estimated frequency and the straight line represents the true frequency. Figure 13 shows the RMSE of frequency estimation. Even when SNR is equal to 0 dB, the proposed methods can estimate the frequencies accurately. Figure 14 and Figure 15 show that frequencies can be calculated by FFT even when using bad recovered waveforms.
The DOAs estimation can be obtained by 14 using the estimated mixing matrix and the frequency which are obtained by the proposed methods. In addition, the proposed algorithms can successfully sort synchronous orthogonal FH signals in the underdetermined case. The correlation of mixing matrix columns is related to the product of DOA and frequency.
This paper proposes a novel method to detect SSPs in TF domain and also an improved subspace projection method is proposed to recover signals. The proposed SSPs detection method improves the complex-valued mixing matrix estimation accuracy. In this paper, DOAs and frequencies are also estimated by the improved subspace projection method. The task of sorting FH signals has been completed by the proposed methods. Expand the real and the imaginary part of A2. Then, A5 is written as. If there is no noise, the A8 can be rewritten as. Then, A9 can be written as.