Semi-Blind Separation of Multiple Asynchronous Wideband Frequency Hopping Signals Based on MWC and Spectral Entropy Method

محورهای موضوعی : Majlesi Journal of Telecommunication Devices

Mohsen Rezaee ¹ (Assistant Professor ICT Research Institute, Tehran, Iran)
Morteza Babaei ² (Assistant Professor Faculty of Electrical Engineering Department, Imam Hossein University, Tehran, Iran)
Mohammadreza Motedayen ³ (Phd Candidate، Imam Hossein Comprehensive University)

تاریخ دریافت : 1402/07/10 تاریخ پذیرش : 1403/02/10 تاریخ انتشار : 1403/06/07

کلید واژه: Semi-Blind Separation, Modulated Wideband Converter, Time-Frequency Distribution, Spectral Entropy, Wideband Frequency-Hopping Signal.,

چکیده مقاله :

Wideband Frequency Hopping Spread Spectrum (FHSS) communications are widely used in both military and commercial applications. In military applications, it is very important to investigate these communications, especially when frequency hopping signals are received simultaneously by a single antenna. This paper investigates the problem of estimating interfering wideband asynchronous frequency hopping (FH) signals parameters with the same hop rate, and using narrow-band receivers. Due to minimal knowledge about the transmitted signals, the problem is analyzed in semi-blind mode. For this purpose, time-frequency (TF) processing has been applied to the modulated wideband converter (MWC) output. The proposed method consists of two stages; In the first stage, frequency-hopping signals with different amplitudes are received by a single antenna. By passing through baseband receivers, the TF distribution of the converter's output signal is obtained. In the next stage, by computing instantaneous spectral entropy (SE), and finding the local maxima in the spectrum, the hop time of each signal is obtained. We use MWC for sub-Nyquist sampling and simultaneous extraction of time and frequency information from signals while eliminating irrelevant signals. The results obtained from estimating hop time parameters demonstrate improved performance compared to other traditional source separation methods such as sparse linear regression (SLR). Based on evaluation metrics such as root mean squared error (RMSE), in lower signal-to-noise ratio (SNR) values, the amount of error has been substantially reduced.

چکیده انگلیسی:

Wideband Frequency Hopping Spread Spectrum (FHSS) communications are widely used in both military and commercial applications. In military applications, it is very important to investigate these communications, especially when frequency hopping signals are received simultaneously by a single antenna. This paper investigates the problem of estimating interfering wideband asynchronous frequency hopping (FH) signals parameters with the same hop rate, and using narrow-band receivers. Due to minimal knowledge about the transmitted signals, the problem is analyzed in semi-blind mode. For this purpose, time-frequency (TF) processing has been applied to the modulated wideband converter (MWC) output. The proposed method consists of two stages; In the first stage, frequency-hopping signals with different amplitudes are received by a single antenna. By passing through baseband receivers, the TF distribution of the converter's output signal is obtained. In the next stage, by computing instantaneous spectral entropy (SE), and finding the local maxima in the spectrum, the hop time of each signal is obtained. We use MWC for sub-Nyquist sampling and simultaneous extraction of time and frequency information from signals while eliminating irrelevant signals. The results obtained from estimating hop time parameters demonstrate improved performance compared to other traditional source separation methods such as sparse linear regression (SLR). Based on evaluation metrics such as root mean squared error (RMSE), in lower signal-to-noise ratio (SNR) values, the amount of error has been substantially reduced.

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متن کامل:



Majlesi Journal of Telecommunication Devices                                       Vol. 13, No. 2, June 2024

 

 

 

Semi-Blind Separation of Multiple Asynchronous Wideband Frequency Hopping Signals Based on MWC and Spectral Entropy Method

 

Mohsen Rezaee1, Morteza Babaei 2, Mohammadreza Motedayen 3 

1-       ICT Research Institute, Tehran, Iran.

Email: [e-mail] (Corresponding Author)

2-       Imam Hossein Comprehensive University, Tehran, Iran.

Email: mbabaei @ihu.ac.ir 

3-       Faculty of Electrical Engineering Department, Imam Hossein University, Tehran, Iran.

Email: p9519894887@ ihu.ac.ir

 

ABSTRACT: Wideband Frequency Hopping Spread Spectrum (FHSS) communications are widely used in both military and commercial applications. In military applications, it is very important to investigate these communications, especially when frequency hopping signals are received simultaneously by a single antenna. This paper investigates the problem of estimating interfering wideband asynchronous frequency hopping (FH) signals parameters with the same hop rate, and using narrow-band receivers. Due to minimal knowledge about the transmitted signals, the problem is analyzed in semi-blind mode. For this purpose, time-frequency (TF) processing has been applied to the modulated wideband converter (MWC) output. The proposed method consists of two stages; In the first stage, frequency-hopping signals with different amplitudes are received by a single antenna. By passing through baseband receivers, the TF distribution of the converter's output signal is obtained. In the next stage, by computing instantaneous spectral entropy (SE), and finding the local maxima in the spectrum, the hop time of each signal is obtained. We use MWC for sub-Nyquist sampling and simultaneous extraction of time and frequency information from signals while eliminating irrelevant signals. The results obtained from estimating hop time parameters demonstrate improved performance compared to other traditional source separation methods such as sparse linear regression (SLR). Based on evaluation metrics such as root mean squared error (RMSE), in lower signal-to-noise ratio (SNR) values, the amount of error has been substantially reduced.   KEYWORDS: Semi-Blind Separation, Modulated Wideband Converter, Time-Frequency Distribution, Spectral Entropy, Wideband Frequency-Hopping Signal.

 

 

1.        Introduction

In recent years, spread spectrum telecommunications has been expanded due to its appropriate performance in different applications [1]. In the meantime, Frequency Hopping (FH) communication is one of the most essential types of spread spectrum communications in which the signal is controlled by a pseudo-random sequence and is spread across the entire spectrum [2].

Frequency-hopping communication is widely used in military and civilian communications due to its low probability of detection (and intercept) [3-5], high reliability, anti-jamming capability, and low sensitivity. Considering these features, estimating the parameters of FH signals is difficult, especially when there exist multiple interfering signals with no prior information about hop patterns [6-7]. In practical communications, the electromagnetic environment is very complex, especially in the short wave band, where all types of environmental signals are mixed and make it challenging to detect FH signals [8]. The identification of FH signals typically occurs in two stages: parameter estimation and signal sorting [9-11]. Estimating parameters such as hop time [12], frequency [13], and angle of arrival (DOA) [14-16], is crucial for sorting multiple users without prior information in practical applications.

On the other hand, MWC structure is widely used in cognitive radar [18], estimating the direction of arrival [19], and detection of FH signals of spread spectrum systems [20], due to its features such as flexibility, sub-Nyquist sampling [17], easy implementation, etc.

In many articles, it is used to implement compressed sensing (CS) methods. However, the CS-based MWC structure has several disadvantages, such as imposing sparse condition constraints on input signals, analyzing the problem only in an underdetermined state, poor performance with reduced SNR, etc. In [21], the structure of MWC is used based on channeling and energy detection, but it is focused on a single signal without time information. In [20], MWC-based frequency analysis with dynamic programming (DP) is considered track and sort wideband FH signals in two steps. Also, the stable performance of the DP section is only for signal-to-noise ratios above 9 dB, indicating poor performance in low SNR.

FH signal analysis requires a suitable method to express time-varying signals that can simultaneously represent time and frequency variations of the signal [22]. In this regard, TF analysis is widely used to estimate FH signal parameters [3], [23-24]. In a classification for non-stationary signals, the analysis of FH signals is divided into two aspects: estimation methods based on time-frequency analysis and estimation methods without it [25]. Generally, TF analysis is divided into two parts: linear and quadratic-order analysis. Linear TF analysis mainly includes Short Time Fourier Transform (STFT), Gabor Transform, and Wavelet Transform (WD) [22]. The window used in STFT distribution is rectangular, while the window used in the Gabor transform is a Gaussian function. The rectangular function in STFT has a fixed width, so STFT has poor time-frequency resolution [26]. In addition, the time-frequency resolution of the Gabor transform is better than STFT. Also, among these transformations, wavelet transformation is one of the most effective TF analysis tools, providing a suitable approximation for signals that exhibit sharp discontinuities. However, it is a method that requires prior knowledge of a completely known signal. Generally, the main TF analysis tools [3], [27], include the STFT analysis, which has less computational complexity compared to other TF distributions. Still, its estimation error is slightly higher than other distributions [28]. The spectrogram (SPEC) distribution is considered one of the simplest quadratic-order TF analysis methods, where the time-frequency representation of the non-stationary signal energy is shown [29]. An analysis based on the Wigner-Ville distribution (WVD) provides better resolution than the SPEC distribution. Still, it introduces cross-term interference components [3]. An analysis based on the Smoothed Pseudo Wigner-Ville Distribution (SPWVD) detects cross-term in a non-cooperative environment. However, it leads to increased computational complexity and reduced performance at low SNR [24], [30-33]. An analysis based on the Re-Smoothed Pseudo Wigner-Ville Distribution (RSPWVD), which overcomes the effects of cross-term in the SPWVD distribution, and leads to increasing computational load [23]. Meanwhile, the STFT distribution analysis is linear, and the rest of the distributions are of second order. Also, in [34-35], the combined STFT-SPWVD method has been used to estimate FH signal parameters. However, the performance of this method is significantly affected by noise, and it has a high computational load [36]. Moreover, in [37] which aimed at direction finding of FH signals, a method based on relative entropy has been employed to extract a hop-free dataset, and focused on estimating the direction of arrival of the signal.

In this article, semi-blind separation of multiple asynchronous wideband FH signals using MWC and spectral entropy in the TF domain is addressed. For this purpose, in Section 2, the signal model for receiving N signals by a single antenna is investigated. In Section 3, the mathematical relationships for the proposed method, including the analysis of the MWC structure and instantaneous SE method in estimating hop times are presented. In Section 4, the results obtained from simulating the proposed method will be evaluated. Finally, in Section 5, the conclusions are presented.

 

2.        Signal model

     Suppose a set of N frequency-hopping signals       , which are time-varying, asynchronous, non-stationary signals generated by fixed sources and received through a single antenna in the far field. For asynchronous  signals, we assume the dwell time is the same. The analytical representation of the n-th received signal, at time  is:

 

(1)                                                                                          

  

      Where  ,  and  are the amplitude, instantaneous frequency, and phase of the n-th FH signal in the k-th hop instant, respectively, and  is the number of frequency hops of . The k-th hop timing is defined as , and  is the observation time.

Considering linear combining model for a single antenna, the received signal  reads

 

 

 

     Where  is the attenuation coefficient of the n-th signal's amplitude, and is the added white Gaussian noise with zero mean and variance .

 

3.        The proposed method

     In non-cooperative communications, the receiver lacks a pseudo-random sequence generator similar to the transmitter and cannot receive the frequency-hopping pattern of the signals. These receivers consist of three main parts: signal detection, parameter estimation, and separation of combined signals. Non-cooperative receivers usually receive a combination of FH signals simultaneously. Therefore, to separate the signals, the parameters of each signal are estimated [21]. 

    We propose a semi-blind method for separating multiple asynchronous FH signals, using the MWC, TF analysis and applying an instantaneous SE-based method. The MWC structure is utilized due to its sub-Nyquist sampling [17], TF analysis is used because of its simplicity and fast computations [40] in analyzing time-varying signals [41] and also for evaluating the parameters of FH signals in blind mode [42]. Additionally, SE method is used for estimation of signal parameters such as hop times [37]. Some features of the proposed  method for the Blind Source Separation (BSS) process are as follows:

 

   1- Signal processing capability in the baseband [38] and sub-Nyquist sampling [17], [39], employing a single antenna.

   2- Analysis of MWC structure in TF domain and simultaneous extraction of time and frequency information of signals.

   3- Separation of signals based on the  method.

   4- Removal of other interfering signals.

 

     In summary, the proposed method consists of two stages for solving the BSS problem:

 

   (1) Passing signals through MWC, and transfer to the TF domain.

   (2) Instantaneous SE method in the estimation of signal parameters.

 

     In the following, the proposed method is presented in detail.

 

3.1.        MWC structure

      In the context of the BSS process using a narrow-band receiver, two flexible and popular structures of random demodulator (RD) and MWC can be mentioned. These structures can be employed in spectrum sensing to alleviate the limitations caused by the bandwidth shortage of receivers for wideband signals. In the meantime, the RD is a well-known single-branch structure for recovering the input signal, with  limited frequency components, such as . It is suitable for implementing analytical methods like CS. However, it leads to complex integral computations. Therefore, it is possible to use an alternative wideband converter, such as MWC, which is a generalization of the RD structure Some of the features of multi-branch compared to single-branch MWC are:

 

1. Parallel Processing and Data Diversity: Multiple branches allow parallel processing of the input signal, which can lead to the extraction of diverse features from different branches simultaneously. This can enhance the system's ability to capture various characteristics of the input signal.

2. Improved Signal Separation: In cases where the input signal contains different components or sources, multiple branches can help separate and process these components independently, potentially leading to more accurate signal decomposition.

3. Enhanced Noise Robustness: By processing the input signal through multiple branches and then combining the results, noise and interference from one branch may be mitigated by the information gathered in other branches, resulting in a more robust output.

4. Feature Combination: Having multiple branches allows for the combination of features extracted from each branch, potentially yielding more comprehensive and informative representations of the input signal.

 

     Fig. 1 illustrates the standard structure of MWC. Consider the frequency band range  for the signal , where  is the upper limit of the frequency band .  

         According to the figure (1), the signal  is multiplied by a set of periodic sequences . These periodic sequences have a period of  and a frequency range of , as defined by

 

(3 

 

   Then, MWC passes the resulting product through the low-pass filter  in the frequency range , and it samples the filter output uniformly at a frequency of , while satisfying the conditions of  [17]. In this case, the output signal of the i-th branch is equal to

 

 

 

 

     Where  is the input signal,  is the periodic sequence of the i-th branch,  is the baseband filter, ,  is the length of the , and  is the sampling time interval. Considering  branches of , the set of vectors  is obtained at the output of .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig1. MWC structure consisting of M parallel channels [17]

 

    By simultaneously processing each of the signals in the set , a set of signal features will be obtained. By processing these features, the desired parameter can be extracted. But the increase in the number of MWC branches leads to an increase in the volume of calculations. Therefore, it is possible to select one of the signals of the set  and perform further processing on it. In this case, the effects of noise and interference in the selected signal are less observed compared to other members. In this article, the second mode is chosen.

 

3.1.1. TFD of MWC output data 

TF distribution is an analytical tool that provides time and frequency information of signals simultaneously. In the BSS process using TF distribution leads to recovery and estimating signal parameters.

According to the Eq. (a-5) in Appendix (A), the discrete TF distribution of the observed signal at the output of the i-th MWC branch is equal to 

 

              (5)

 

where  is the conjugate of ,  is the time-lag kernel, and . By considering  branches of , the set of distributions is obtained where  is the discrete TF distribution of the observed vector.

 

    In the next stage of the signal separation process, by applying the instantaneous SE method on the distribution , we will take a significant step toward recovering the signals.

 

3.2.        Spectral entropy

     The spectral entropy of a signal is a measure of its spectral power distribution. The concept is based on the Shannon entropy, or information entropy. The SE treats the signal's normalized power distribution in the frequency domain as a probability distribution, and calculates its Shannon entropy. This property is useful for feature extraction [43-44], and is also widely used in areas such as signal processing [37], and biomedical signal processing [45].

      signal structure has a unique feature compared to other signals. When the frequency changes, a discontinuity will appear in the frequency spectrum of the signal. Methods based on entropy, variance, etc, are sensitive to discontinuous points in the signal, and can be used to extract information from these points. In this article, by applying the instantaneous SE method to the  matrix, such as the STFT matrix, the hopping times of  signals are determined.

 

3.2.1.  Instantaneous spectral entropy estimation

    To estimate SE, we obtain the TFD matrix in the output of one of the MWC branches, such as the  distribution. According to Eq. (a-7) of Appendix (A), the STFT distribution for the i-th MWC branch is equal to

 

 

where  is the window function, and  is the number of samples of . In the  distribution, an analysis window, such as Hamming window, slides over the signal and then the Discrete Fourier Transform () is computed for the windowed data. The number of rows in the STFT matrix equals the number of DFT points, and the number of columns is given by  ، , where  is the length of , and  represents the hop distance between consecutive DFT,  is the length of .

   Each column of the STFT matrix is a spectral segment of the signal and provides a short-term estimate of the time-frequency content of the signal [43]. The STFT matrix in terms of spectral segments can be expressed as . The element corresponding to m-th row and h-th column of the matrix  is equal to

 

 

 

  For , where  is the number of  DFT points, and the  vector  is obtained [43].

 

 

    The probability function for each column of the STFT matrix is used for detecting hop times. Columns, with frequency hops have high entropy due to the presence of instantaneous frequency variations. Entropy can be regarded as a measure of uncertainty between pairwise elements of a discrete set. Any possibility of displacement between the elements of this set, which have close probabilities, increases the entropy of the collection. In a time window with frequency hopping from one moment to another, it causes the signal to show its spectrum dispersion with a high entropy value.

Generally, the spectral entropy in the domain  is defined in terms of TF distributions as follows [22]:

 

 

 (8)                                                                       

 

where  ,  are the total time and frequency points, respectively. The above relation is only valid for the TFDs that do not assume negative values, e.g., the . Considering the  distribution, the instantaneous  of the i-th MWC branch is equal to

 

 

(9)                                                                      

 

 

where . In every time window, as long as the frequency is constant, the elements of  remain constant, as soon as the frequency changes, the elements of  show significant change and is considered a hop time.

    Therefore, by applying the  method to the  matrix of the  distribution, a set of peaks is created in the spectrum. In the next step, by detecting and processing these peaks, signal parameters are separated and estimated.

 

3.2.2. Estimating hop times of signals

    In practice, the presence of frequency hop in the signal leads to an increase in the SE spectrum. Therefore, detecting peak points is essential for extracting time and frequency measurement information from the signal.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 2. (a) shows the sample distribution of  for two asynchronous FH signals with four frequency hops, (b) the  of Figure (a) in the ideal state, and the creation of impulse functions at hop times.

 

    Fig. (a-2) shows the  distribution of two asynchronous  signals with four frequency hops. The frequency dwell time, , is the same for all frequency hops in both signals. Figure (b-2) shows the performance of the  method in an ideal state, where a set of impulse functions is created at the time instants of signal hops.

    According to Eq. (10), extraction of signal peaks based on instantaneous  and obtain a set of frequency-hopping times  of the signals.

 

 

                    (10)

 

where  represents the number of frequency hops per signal, and  is the number of signals. Increasing  in the  method leads to a reduction in false peaks and improvement in the estimating the set of time instants . Considering that all equations are related to a single branch, in order to simplify the equations, the  variable has been omitted in the following equations.

 

3.3.        Separation of hop times

 

     By Applying the  method, a set of hop times, , is obtained. In this stage, to separate the hop times of each signal, we first sort the set of times  in ascending order. Then, by measuring the difference between each successive pair of time values ,

 

 

 

Fig. 3. The representation of three asynchronous  signals, each with four hopping frequencies and the same dwell time.

a set of rounded numbers is obtained as follows:

(11)                                                 

 

 

 

where  and  for .

 

   It is assumed that all three signals have the same dwell time as shown in Figure (3). Some of the features of the  vector are as follows:

·                      In the  set, the highest frequency (with a value of ) corresponds to the time delays between the signals. that is,

 

 

 

 

where , , ,         ,… and that,

   

 

 

 

      where  represents the delay time between the first and second signals, and  represents the delay time between the second and third signals.

·                      In the set , after , the highest frequency (with a value of ) corresponds to the set , In which, , … and that,

 

    So, for three signals,  represents the time delay between the signals, and  will dwell time .

·                      In the  set, the repetition of all three consecutive components indicates the accuracy of

the  method, in estimating the two parameters  and . According to Figure (3), and Eq.(11), the equality , which means that in the m-th hop, the m-th component of all three signals is correctly estimated.

·                      Previous properties of the  set are discussed in the ideal state, but in practice, factors such as environmental noise, and  reduction lead to a decrease in the frequency of  and .

·                      So, Increasing the frequencies  and  in the  set, leads to an increase the accuracy of the method.

 

     Generally, for  asynchronous  signals, assuming the dwell time is the same, the set of hop times for each signal can be recovered and separated using two parameters  and , as follows:

 

 

 

 

 

 

}  

 

 

    By extracting the hop times from the STFT matrix of the TF distribution, other information, such as hopping frequencies, energy density of T-F plane points, period, and frequency hop of each signal based on matrix STFT, can be determined and separated.

    One of the advantages of the proposed method is that in a real environment, FH signals have time interference with other environmental signals, such as burst, constant, periodic, and other signals. However, if the two parameters  and  are accurately estimated, the interfering signals will not impact the estimation of signal parameters, and are easily eliminated. 

Because, the basis of recovery of signal hopping times is based on the difference frequency of both consecutive components of the time vector , and disturbing signals lack frequency due to scattering. Therefore, they have little effect on the estimation process of  and , and can be easily removed.

    Also, in the proposed method, according to the level of signal contamination with noise, the accuracy of estimating time parameters decreases, and leads to an increase in estimation error. Therefore, improving processing variables, such as increasing SNR, increasing input samples, increasing experiment repetitions, etc, results in a reduction of estimation errors for time parameters  and , consequently leading to increased accuracy in separating the time intervals of each signal hop.

 

4.        Evaluation of the proposed method

 

   This section is intended to validate the proposed method in the article, and it aims to evaluate, and calculate errors in the estimated parameters. We consider three asynchronous FH signals, each of these signals consists of six frequency hops (). The signals have different magnitudes [1.4 1.6 1.7], zero phases, different hop times, and the same hop length.

 

    The signals are placed in the frequency band range of , without frequency interference. Table (1) shows the normalized frequency-hopping pattern of the three input sources.

 

Table 1.  Frequency-hopping pattern of input signals.

 

     Figure (4) shows the  distribution of asynchronous  signal over a period. In the asynchronous mode, the hopping times of  signals are different. According to the figure (4), the total hopping time for all signals is equal to  seconds (hopping rate ), but the signals have time delays relative to each other. To simulate MWC input, for a sampling rate of  , the number of sampling points is . An increase in  or the duration of  leads to an increase in the value of . Increasing the number of  points for each frequency- hopping leads to improved estimation of hopping parameters with less error and better accuracy. However, increasing  leads to increased computational complexity and hardware limitations.

 

 

 

Fig. 4. SPEC, distribution of three asynchronous input signals.

 

   Therefore, for designing a suitable system, a trade-off between variables should be considered. Table (2) shows the numerical values of MWC parameters.

 

 

Table 2. MWC parameters.

Normalized hopping frequencies	Inputs
	Signal (1)
	Signal (2)
	Signal (3)

Value	Parameter	Value	Parameter
5MHz	Sampling frequency	10	Number of branches (M)
2.5MHz	Filter bandwidth (B)	30	Sequence compression (k)

 

    According to Table (2), a low-pass filter with a limited bandwidth of 2.5MHz has been used to pass  signals with a frequency-hopping range of 500MHz.

    According to Eq. (10), Figure (5-a) shows the amplitude of  in terms of time, to display the hop times of  input signals. Each of these signal consists of six frequency hops. The duration of each hop for every signal is the same, and equal to  seconds. According to Figure (5-a), all hopping times are equivalent to:

 

 

 

 

    The signals have a time delay relative to each other of approximately  seconds.

 

     Fig. (5-b) shows the results obtained from estimating the hop times of the signals in the  output. The estimation error is reduced by repeating the experiment, and increasing the .

 

    As shown in Figure (5-b), in Experiment 400, for over 100 iterations, the hop times of the signals with an error less than , sorted from small to large, have been estimated as follows:

 

 

 

 

 

Fig. 5. (a) plotting the amplitude of   in terms of time, and showing the hop times of three input asynchronous FH signals (b) Estimated hop times at  output.

 

    In addition, In a noisy environment, decreasing the  leads to an increase in false peaks in the spectrum, and applying the  method significantly increases the estimation error. 

    By measuring the difference of both consecutive components of the time vector , A set of rounded numbers is obtained as follows:

 

 

 

 

   In this set, the number 17.5 is observed with the highest frequency value, which indicates the time delay between signals. Because, it is the same frequency dwell time for all FH signals in each frequency hop. The higher the frequency value, the higher the accuracy of the  method. Additionally, increasing the  leads to an increase in the frequency value.

    In the next step, the position of each pair of consecutive and repeating numbers, 17.5, in the vector  represents the time delay of three signals at hop times. For example, in the  vector, the third and fourth components have a common value of 17.5, indicating that in the second hop, the second component of all three signals has been correctly registered. Similarly, the sixth and seventh components of the  vector have the same value. it means, The SE method has correctly estimated the third time component of all three signals in the third hop. Also, the total value of , along with two delays of , indicates the approximate duration of each signal at a constant frequency.

    Therefore, using the  vector, it is possible to obtain the time delay between signals (i.e., ) , and the approximate time  (i.e., ).

 

   Considering the  vector, the second, third, and sixth components of all three signals can be estimated using two-time parameters, , and . For example, the third component of all three signals is equal to the following:

 

 

 

 

 

    Similarly, the sixth component of the signals is:

 

 

 

 

 

   Thus, using the estimated time parameters  and , the set of hop times for each signal can be recovered and separated as follows:

 

 

 

 

   As previously mentioned, improving processing variables, such as increasing , increasing , and increasing the repetition of experiments result in reducing the estimation errors of time variables  and  to less than  (and even zero). As a result, the accuracy is increased in separating the hop times of each signal.

 

 

4.1.         Calculating the RMS/RF error

     To evaluate the results of the analysis, the estimated hop time errors have been measured using  and  methods. The estimation error of hop times is defined using Eq. (12) and Eq. (13).

 

      (12)

 

 

     In Eq. (12),  represents the estimated hop time,  represents the actual value of the hop time,  indicates the n-th signal, and  for the r-th frequency of the signal hop.

 

                                           (13)

 

 

     In Eq. (13),  represents the estimated time,  represents the actual value of the hop time,  for the sampling frequency of the , and , represents the total number of experiments. It should be noted that for all calculations, the number of experiments is greater than . Naturally, increasing  leads to improvements in the results.

   Figure (6), displays the curve of the  error value compared to the  error for estimating hop times of three asynchronous  signals at different  values. The results are obtained by registering the error for  processing iterations. 

     According to the Figure (6), the RMS error has decreased compared to the RF error and recorded a smaller amount of error, so that in the value of , the RMS error is approximately 0.25 of the RF error and recorded a value of . In addition, in low and negative SNR values, the RMS error is more reduced. Also, in the value of , the RMS error amount has almost reached zero, and in this case, the hopping times are estimated with better accuracy. It should be noted that the amount of recorded error has a direct relationship with the amount of rounding of the estimated hopping times during processing.

 

 

 

Fig 6. Compares, the RF error value with RMS in estimating the hopping times of three asynchronous FH signals.

 

      Figure (7), shows the curve of the RMS error value compared to the RF error for estimating the hopping times of three asynchronous FH signals. The repetition of processing is . According to the figure(7), increasing the repetition by 10 times has led to an error reduction of approximately 0.1 while maintaining the previous features compared to figure (6).

 

    In another investigation, the increase in the number of experiments and its impact on the  error for three signals with six hop frequencies has been examined. As shown in Figure (8), increasing  results in a reduction in the  error of the system. Moreover, at low  values, the error slope becomes slow and nearly linear.

 

 

Fig. 7. Compares, the  error with the  error in estimating the hopping times of three asynchronous  signals with increased processing repetition.

 

 

Fig. 8. Shows, the  error values in estimating the hopping times of three asynchronous  signals for different numbers of processing repetitions.

4.2.        Comparison with other methods

    In this section, to validate the accuracy of the proposed method compared to other methods, the estimation accuracy of parameters has been evaluated in terms of  for three asynchronous signals. Figure (9), For , compares the  error curve of the   method within the range of  -4dB to 10dB, simultaneously with two other methods, the  distribution based on array processing [8], and the parameter estimation method using sparse linear regression () [46]. The results represent the recorded error rate for a processing repetition of  times.

    According to the figure (9), the proposed method has achieved better results than the  method using  over the entire  range. Moreover, for , the  error of the proposed method is significantly reduced. So, in , the recorded error is approximately half of the other two techniques.

 

 

 

Fig. 9.  Compares, the  error of different methods in estimating the hopping times of  signals.

 

    Additionally, in the entire  range, the slope of the error curve for the proposed method is slow, indicating its robustness compared to the other two techniques. However, for , the error of the research method converges towards a constant value of . This is due to the rounding errors in the parameter extraction process of the signal. Another point to note is that increasing variables, such as the number of sampling points, , and the number of iterations significantly contribute to improving results and reducing errors.

 

5.        Conclusion

 

      This paper proposes a novel Semi-BSS and parameters estimation method of FH signals. The proposed method exploits the structure of MWC and  method to estimating the parameters of  signals. In the proposed method, initially, three  signals in the frequency range of  were received by a single antenna and passed through .   

     The  structure consists of ten branches with a constant sampling rate of , and a receiver with a cut-off frequency of . Then, the instantaneous  method was applied to the  distribution of the output data from . The accurate estimation of the time delay variables, , and the duration of stay, , ensures the precision of the proposed method in signal separation and parameter estimation. Some features of the proposed  method for the BSS process are as follows:

·                      Signal processing capability in the baseband and sub-Nyquist sampling condition by a single antenna,

·                      Ability to remove interfering arrival of the signals,

·                      Improved performance of the method with reduced , compared to similar methods for  values,

·                      Significant reduction in the volume of data in the input  analysis, proportional to the  sampling rate, and increased program execution speed.

   The obtained evaluation results show that this algorithm has an acceptable performance compared to other traditional separation methods, is applicable at lower , and has higher accuracy. According to Figure (9), the error rate decreases significantly compared to other methods at . Specifically, around , the  error reaches an approximate value of , representing a 50% reduction compared to other methods in parameter estimation.

 

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Appendix (A)

 

 

     For the analytical signal , the discrete-time instantaneous autocorrelation function (IAF) is expressed by [22]:

 

                          (a-1)

 

where ,  is the length of ,, and  is the conjugate of . Also, the smoothed IAF (SIAF) of signal  is equal to

 

 

                            (a-2)

 

 

 

where  is the time-lag kernel, and  represents the sign of discrete time-convolution. In this case, the discrete time TF distribution of the signal  is expressed by

 

 

 

                                 (a-3)

 

 

where , and  represents the discrete-time Fourier transform of variable  in matrix .

 

Substituting Eq. (a-2) into Eq. (a-3), writing out the transform, and substituting from Eq. (a-1), the distribution time frequency  is obtained.

 

                                                              (a-4)

 

 

 

which can be expanded as

 

                                                              (a-5)

 

 

 

In Eq. (a-4) and Eq. (a-5), P and  are a positive integer,  is the time-lag, and  [22]. For example, the time-lag kernel of some common distributions is equal to

·       WV distribution

 

     

 

·       SPEC distribution

 

                 

 

      WV and SPEC distributions are bilinear transformations. By placing the time-lag kernel  of the SPEC distribution in the equation (a-5), the frequency-time distribution  of the signal y[n] can be calculated. 

   The squared magnitude of the STFT, is called the spectrogram and is expressed as follows [22]:

 

                        (a-6)

 

 

where w[m] is the window,   is of length M. According to the Eq. (a-6), the TF distribution of the discrete STFT is equal to

 

 

 

 

Paper type: Research paper

https://doi.org/10.30486/MJTD.2024.901781

Received: 2 November 2023; revised: 18 March 2024; accepted: 29 April 2024; published: 1 June 2024

How to cite this paper: M. Rezaee, M. Babaei, M. R. Motedayen, “Semi-Blind Separation of Multiple Asynchronous Wideband Frequency Hopping Signals Based on MWC and Spectral Entropy Method”, Majlesi Journal of Telecommunication Devices, Vol. 13, No. 2, pp. 97-111, 2024.

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