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An identification technique for the cofrequency mixed communication signals based on cumulants
EURASIP Journal on Advances in Signal Processing volume 2018, Article number: 50 (2018)
Abstract
Identification of the cofrequency interference is a common problem in wireless digital communication systems. The highorder statistics (HOS) feature based on cumulants is a widely adopted solution. However, prior knowledge of the timings and the symbol period is required to extract the cumulants, which is quite difficult in a blind environment. In order to solve this problem, this paper proposed a more general calculation for the cumulants based on the oversampled data with consideration of the intersymbol interference (ISI). The generalized theoretical value of the cumulants is deduced in this paper. Besides, the HOS feature based on the generalized cumulants for identification is found to be robust with different rolloff factor and sampling number per symbol. Computer simulations are performed to prove the validity of the proposed method.
1 Introduction
With the frequency spectrum becoming more and more intense in wireless communication, the receivers suffer from the intentional or unintentional cofrequency interference more and more frequently [1,2,3]. Specifically, paired carrier multiple access (PCMA) is a new technique adopted in satellite communication system, and the earth stations receive a mixed signal of two intentional cofrequency signals [4, 5]. The conventional demodulation performance is severely degraded by the cofrequency interference, which leads emerging need for the receivers to identify the type of the received signal, a single signal or a mixed signal, before adopting corresponding procedure.
The problem of the mixed signal identification has been explored in current literatures [6,7,8,9,10,11,12,13,14]. The commonly adopted methods can be divided into two types. One is the information theoretic based method [6, 7]; the other is featurebased method [8,9,10,11,12,13]. The former method adopts the correlation matrix of the received signal to estimate the number of the transmit signals. In [6], the classic Akaike information criterion (AIC) algorithm and minimum description length (MDL) are reviewed. However, these algorithms require complex computation for eigen decomposition of the correlation matrix. Moreover, the information theoretic methods require the number of the receive antennas larger than the number of the transmit signals, which is not always satisfied.
On the other hand, the featurebased methods extract features from the received signal, which means a priori knowledge of the received signals is required. In [8, 9], the features of secondorder cyclostationary and the spectral coherence function are adopted to detect the mixed signal. However, this method requires high computation complexity and performs poorly when the signals’ carrier frequencies are nearly the same. In [10, 11], the cyclic stationary characteristics of digital communication signals are adopted to detect the number of the cofrequency signals. However, it fails when the cyclic frequencies are overlapped. In [12], a novel singlechannel signals’ number detection algorithm based on wavelet transform is investigated. The signals’ symbol characteristics in Haar wavelet transform are extracted to achieve the blind estimation of the signals’ number. However, the symbol rates of the signals are required to be different. In [13], the highorder moments of the received signal are employed to detect the number of the cofrequency signals. However, this method requires a priori information of the noise variance. Inspired by the tolerance of white Gaussian noise, the cumulants are adopted to distinguish OFDM signals from single carrier signals in [14]. However, the cumulants are extracted from a timing synchronization sampled sequence which removes the intersymbol interference (ISI) from the oversampled data of the received signal. That means the signal bandwidth and the timing information should be estimated if prior knowledge is not available. However, the accurate estimation of these parameters is quite difficult under low signaltonoise ratio (SNR) condition [15,16,17,18].
In this paper, a more generalized theoretical value of the cumulants of the oversampled data with consideration of ISI is deduced. Instead of only using the optimal sample per symbol for cumulant calculation, all the samples per symbol are used. A highorder statistics (HOS) feature based on cumulants is adopted to identify the received signal’s type with prior knowledge of the modulation type of the two cofrequency signals. Moreover, the HOS feature based on the cumulants is found to be stable for arbitrary symbol period and rolloff factor of the transmitter pulseshaping filter. Thus, it releases the need for timing synchronization and estimation of the symbol rate of the two cofrequency signals when this HOS feature is adopted to identify the received signal’s type.
This paper is organized as follows. Section 1 introduces the research background. Section 2 presents the signal model and the definition of HOS. Section 3 derives the more generalized theoretical value of the cumulants of the oversampled data with consideration of intersymbol interference and introduces the HOS feature for signal identification. Section 4 introduces the proposed identification algorithm based on the HOS feature, and the algorithm’s performance simulations are also presented. The summation is concluded in the last section.
2 Signal model and the HOS
2.1 Signal model
In this paper, the received signal is supposed to be composed of two cofrequency signals with the same modulation type; its model can be expressed as
where h_{1} and h_{2} are the amplitude of the two signals, and the power ratio of these two signals is \( {P}_r={h}_1^2/{h}_2^2 \) (for simplicity, P_{r} is assumed to be larger than 1); ω_{1} and ω_{2} are the carrier frequency of the two signals and ω_{1} − ω_{2} ≪ π/T_{1} + π/T_{2}; T_{1} and T_{2} are the symbol period of the two signals, respectively; θ_{1} and θ_{2} are the initial phase and are supposed to be uniform random variables on [0, 2π]; v(t) is an additive white Gaussian noise; x_{1}(t) and x_{2}(t) are the baseband modulation signals which can be described as
where a_{i}(t) represents the modulated symbol of the ith transmitted signal at time t and is uniform distributed on the constellation of the modulation type; g_{i}(t) represents the pulse response of the equivalent filter including forming filter, channel filter, and matched filter and is usually substituted by the finite transmitter pulseshaping filter lasts from − L_{1}T_{i} to L_{2}T_{i} [19]; ∗ represents convolution operation.
Assuming the receiver is sampled at the sampling rate 1/T_{s},. the oversampled data can be written as
where r_{k}, x_{i,k}, and v_{k} are the sampling value of r(kT_{s}), x_{i}(kT_{s}), and v(kT_{s}), respectively.
2.2 Highorder statistics
The HOS of r_{k} can be described by the moments and the cumulants, which characterize the shape of the distribution of the signal constellation. The moments M_{pq} can be obtained by calculating the expectation of \( \left\{{\left({r}_k\right)}^{pq}{\left({r}_k^{\ast}\right)}^q\right\} \) [20, 21]
where (∙)^{*} represents taking the conjugate; E{∙} represents the expectation operation (in the following derivation, the expectation is replaced by ergodic average).
The cumulants can be obtained by the moments using the following formulas [20, 21]
3 Theoretical value of the cumulants and the HOS feature
The studies in [20,21,22,23,24] have derived the theoretical value of the cumulants of the phaseshift keying (PSK) signal and the quadrature amplitude modulation (QAM) signal. However, the derivation is based on the sequence where the timing synchronization is completed and ISI is removed. This is not easy in practice. In this paper, we derive a more generalized theoretical value of the cumulants of the oversampled data with ISI.
For simplicity, the additive noise v_{k} is omitted during the derivation of the theoretical value. The moments M_{pq} of the oversampled data of the received signal r(t) can be obtained as follows. The detail deduction is shown in Appendix 1. The deduction of the special case that the two signals are BPSK modulated and ω_{1} = ω_{2} is given in Appendix 2.
Substitute (6) into (5), the theoretical value of the cumulants C_{pq} can be obtained.
where the expectation of \( \left\{{\left({x}_{i,k}{x}_{i,k}^{\ast}\right)}^q\right\} \) can be referred to Appendix 1.
Seen from [21], in order to remove the influence of the signal power, a HOS feature F_{1} constructed by the cumulants of the received signal is introduced for signal identification, which can be regarded as a statistics feature characterizes the shape of the distribution of the mixed signal constellation with normalized signal power.
where \( {E}_{i,q}=E\left\{{\left({x}_{i,k}{x}_{i,k}^{\ast}\right)}^q\right\} \). The HOS feature F_{1} for a single signal, i.e., P_{r} approaching infinity can be obtained as follows.
In order to verify the above theoretical derivation, MonteCarlo method is adopted to obtain the simulation value of F_{1}. During the simulation, the pulseshaping filter g_{i}(t) is set to be square root raised cosine filter with L_{1}, L_{2} to be 2 (for the values of g_{i}(t) are quite small when t is larger than 2T_{i}, so we can neglect them). Figure 1 shows the theoretical value and simulation value of F_{1} of the mixed signal r(t) of two cofrequency signals with the same modulation type, symbol period, and rolloff factor. The theoretical values are obtained by the following process: first, calculate the expectations of \( \left\{{\left({x}_{i,k}{x}_{i,k}^{\ast}\right)}^q\right\} \), using (12), (13), or (14), and then substitute them into (8) to obtain F_{1}. The simulation values are obtained by calculating the moments and cumulants of a generated mixed signal with different power ratio using (4) and (5), and then substitute the cumulants into (8). Seen from Fig. 1, F_{1}increases with the increasing power ratio P_{r}. Moreover, it can be seen that the value of F_{1} when P_{r} → ∞ (that means the type of the received signal is a single signal) is nearly twice the value when P_{r} = 1 (in this case, the separation of the two cofrequency signals is quite difficult). Owing to the distinctive difference of F_{1} between single signal and mixed signal, F_{1} can be adopted to identify the type of received signals.
Moreover, MonteCarlo method is adopted to show the relationship between the value of F_{1} and the rolloff factor α_{i} and the symbol period T_{i} (which can be equivalent to the sampling number per symbol N_{i}). The maximum and minimum value of F_{1} under different α_{i} and N_{i} are obtained. The rolloff factor α_{i} is varied among 0~1 and the sampling number per symbol N_{i} is varied among 10~100. It is shown in Table 1 that the difference between the maximum and minimum value of the HOS feature F_{1} is quite small, which means the rolloff factor α_{i} and the symbol period T_{i} have little impact on the value of F_{1}. That releases the demand for estimation of the two signals’ bandwidths, the rolloff factors, and the timing information when adopting F_{1} to identify the received signal’s type. Thus, it makes the HOS feature F_{1} more robust in a blind environment (the special case that the two signals are BPSK modulated and ω_{1} = ω_{2} is not taken into consideration).
4 Results
4.1 A more generalized identification method based on the HOS feature
In this section, a variety of simulation experiments are presented to illustrate the performance of the proposed identification method. The total block diagram of the identification method is shown in Fig. 2. First, the HOS feature F_{1} is extracted from the oversampled data of received signal r(t) and then compared with a threshold η. If F_{1} ≥ η, the type of the received signal is identified as a single signal; else if F_{1} < η, the type of the received signal is identified as a mixed signal. For the value of F_{1} only vary with the power ratio P_{r}, the threshold’s value can be substituted by the theoretical value of F_{1} with the given modulation type of the two cofrequency signals and the power ratio P_{r0}. The mixed signal can be regarded as a single signal when the two signals’ power ratio P_{r} is larger than P_{r0} (in this paper, P_{r0} is set as 10 for the biterror rate (BER) performance degradation in the presence of a cofrequency interference is quite small when the power ratio P_{r} is larger than 10 [25]). This proposed identification method is also available for the cases that the mixed signal is composed of more than two cofrequency signals.
The SNR of the mixed signal is defined as.
where E_{si} is the energy per symbol of the ith signal. N_{0} is the noise power spectral density.
In experiment 1, we consider the identification performance under different SNR condition. To evaluate the performance, the probability of correct identification P_{CI} is counted by 500 times MonteCarlo simulations. The simulation parameters are set as follows: the carrier frequencies (ω_{1}/2π, ω_{2}/2π) are set as (100e6 + 8e3 Hz, 100e6 − 6e3 Hz); the baud rates (1/T_{1}, 1/T_{2}) are set as (10e6 bps, 10e6 bps) and (10e6 bps, 7e6 bps) for the two signals with same symbol period case and with different symbol period case, respectively; the sampling rate is set as 1400e6 Hz; the total symbol number used for simulation is set as 5e4. Figure 3 shows the identification performance when the received signal is a mixed signal with the power ratio P_{r} to be 5, and Fig. 4 shows the identification performance when the received signal is a single signal. Obviously, the P_{CI} increases with the increasing SNR and the identification method is robust for the two signals with different symbol period case. Seen from these figures, it is obvious that the proposed identification method performs better under a higher SNR condition.
In experiment 2, we consider the identification performance under different power ratio P_{r} of the two signals. The simulation parameters are set as experiment 1. The identification performance for the SNR = 0 dB is shown in Fig. 5. It is shown that the identification performance decreases with the increasing P_{r} for the value of the extracted HOS feature F_{1} gets closer to the threshold η.
In experiment 3, we consider the identification performance under different total symbol number used for simulation. Figure 6 shows the identification performance for the SNR = 6 dB and the received signal is a single signal; it is clear to see that the identification performance increases with the increasing total symbol number used for simulation.
In experiment 4, the identification performance comparison between the proposed HOS featurebased method and the cyclostationary spectrum coherence function (SOF)based method [8, 9] is made. For the two cofrequency, signals’ carrier frequencies are set to be too close in experiment 1~experiment 3 and the SOFbased method fails under that condition, the carrier frequencies (ω_{1}/2π, ω_{2}/2π) are reset as (100e6 + 8e3 Hz, 102e66e3 Hz) in this experiment. Moreover, for the SOFbased method that requires prior knowledge of the carrier frequencies, symbol period, we suppose those knowledge are known during our simulation for simplicity. The result is shown in Fig. 7. Seen from Fig. 7, it is obvious that our proposed method performs much better than the SOFbased method under low SNR condition and our proposed method is more robust in a blind environment.
5 Discussion
This paper derives the theoretical value of the cumulants of the whole sampled data with consideration of the intersymbol interference. The HOS feature is adopted to identify the cofrequency mixed signal. Since it releases the need for timing synchronization of the received signal, it makes the HOS feature more fitful in blind environment. However, this proposed method requires prior knowledge of the modulation type of the two cofrequency signals.
6 Conclusions
In this paper, a more generalized theoretical value of the cumulants and a HOS feature are derived based on the oversampled data. It is effective for the release of the demand for prior knowledge of timing information and symbol periods in current literatures [14, 20,21,22,23,24]. Based on the derived HOS feature, a novel identification technique for the cofrequency mixed signals with the same modulation type is proposed. This approach is robust for arbitrary cases of the cofrequency signals with different rolloff factor, sampling number per symbol, and power ratio.
Furthermore, the derived cumulants in this paper can be applied in blind modulation classification of a single signal for the needless of prior knowledge and estimation of the power ratio of the mixed signal for the power control in cellular systems [26,27,28]. Further discussions could be made in the future.
7 Methods
This paper studies the general theoretical value of the cumulants of the oversampled received data and adopts the HOS feature to identify the cofrequency mixed signal. The proposed identification method’s performance is obtained by MonteCarlo simulations using MATLAB software.
Abbreviations
 AIC:

Akaike information criterion
 BER:

Biterror rate
 HOS:

Highorder statistics
 ISI:

Intersymbol interference
 MDL:

Minimum description length
 PCMA:

Paired carrier multiple access
 PSK:

Phaseshift keying
 QAM:

Quadrature amplitude modulation
 SNR:

Signaltonoise ratio
 SOF:

Cyclostationary spectrum coherence function
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Funding
This work was supported by the National Natural Science Foundation of China (61271265 and 61671263) and Tsinghua University Independent Scientific Research Project (20161080057).
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CWD and YFZ proposed the original idea and designed the experiments. CWD and HL performed the experiments and analyzed the results. All authors read and approved the final manuscript.
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Appendices
Appendix 1
For simplification, the additive noise v_{k} is omitted during the derivation of the theoretical value of the moments. Seen from (3), the product of the oversampled data r_{k} and its conjugation can be written as follows.
Assume k = N_{i}(m − 1) + p (where N_{i} is the sampling number per symbol of the ith transmitted signal; m is the mth transmitted symbol of the ith transmitted signal at time kT_{s}). Then, the baseband modulation signal x_{i,k} can be extended as follows.
where a_{i}(k) and g_{i}(k) is assumed to be a_{i}(kT_{s}) and g_{i}((k − 1)T_{s} − L_{1}T_{i}), respectively; a_{i,m} is the mth transmitted symbol of the ith modulated baseband signal; the finite pulseshaping filter g_{i}(k) lasts from − L_{1}T_{i} to L_{2}T_{i}.
In the case of MPSKmodulated baseband signal, the expectation of \( \left\{{\left({x}_{i,k}{x}_{i,k}^{\ast}\right)}^q\right\} \)(q ≤ 3) can be written as (13), where \( {q}_1,\cdots, {q}_{L_1+{L}_2+1} \) are positive integers; the expectation of \( \left\{{a}_{i,m}{a}_{i,m}^{\ast}\right\} \) is assumed to be 1; \( {C}_q^{q_1} \) denotes binomial coefficient. Note that when a_{i,m} is BPSK modulated, \( {a}_{i,m}^{\ast }={a}_{i,m} \). Then, the expectation of \( \left\{{\left({x}_{i,k}{x}_{i,k}^{\ast}\right)}^q\right\} \) of BPSKmodulated signal is different from (13), which is deduced in (14) (where \( {q}_1,\cdots, {q}_{L_1+{L}_2+1} \) are positive even integers).
While in the case of QAMmodulated baseband signal, a_{i,m} = I_{i,m} + jQ_{i,m} (where I_{i,m} and Q_{i,m} are independent with each other and take ± 1, ± 3, etc. with equal probability). Then, the expectation of \( \left\{{\left({x}_{i,k}{x}_{i,k}^{\ast}\right)}^q\right\} \)(q ≤ 3) should be written in another form.
where the expectation of {(I_{i}(k) ∗ h_{i}(k))^{2q}} and {(Q_{i}(k) ∗ h_{i}(k))^{2q}} can be obtained as (16), where \( {q}_1,\cdots, {q}_{L_1+{L}_2+1} \) are positive even integers.
Given (11), (13), (14), (15), and (16), the moments M_{pq} of the oversampled data can be written as follows.
Note that the expectation of the term E{Re[∙]} is zero except when ω_{1} − ω_{2} = 0 and the two baseband signal x_{i,k} are BPSK modulated. The HOS of the oversampled data r_{k} when the two baseband signal x_{i,k} are BPSK modulated and ω_{1} − ω_{2} = 0 is discussed in Appendix 2.
Appendix 2
When the two baseband signal x_{i,k} are BPSK modulated, \( {x}_{i,k}^{\ast }={x}_{i,k} \). The expectation of the term E{Re[∙]} in (17) can be obtained as (18).
Because E{cos(2(ω_{1} − ω_{2})kT_{s} + 2(θ_{1} − θ_{2}))} is obtained by calculating the ergodic average of cos(2(ω_{1} − ω_{2})kT_{s} + 2(θ_{1} − θ_{2})) as shown in (19).
It is easy to see that E{cos(2(ω_{1} − ω_{2})kT_{s} + 2(θ_{1} − θ_{2}))} is not zero only when ω_{1} = ω_{2}.
Given (17), (18), and (20), the moments M_{pq} of the oversampled data when the two signals are BPSK modulated and ω_{1} = ω_{2} can be obtained as (21).
Substitute (21) into (5), the theoretical value of the cumulants C_{pq} when the two signals are BPSK modulated and ω_{1} = ω_{2} can be obtained as (22).
It is obvious that the value of F_{1} varies with different values of θ_{1} − θ_{2} when the two signals are BPSK modulated and ω_{1} = ω_{2}. The theoretical values and the simulation values of F_{1} under different values of θ_{1} − θ_{2} are shown in Fig. 8. It is easy to see that our proposed identification method fails because the value of F_{1} is changeable. However, this condition that the two signals’ carrier frequencies are exactly the same is quite hard to reach in practice. So our proposed identification method is still acceptable.
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Duan, C., Zhan, Y. & Liang, H. An identification technique for the cofrequency mixed communication signals based on cumulants. EURASIP J. Adv. Signal Process. 2018, 50 (2018). https://doi.org/10.1186/s1363401805735
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DOI: https://doi.org/10.1186/s1363401805735
Keywords
 Highorder statistics
 Cumulants
 Cofrequency mixed signal
 Signal identification