An algorithm for space–time block code classification using higher-order statistics (HOS)
- Wenjun Yan^{1}Email author,
- Limin Zhang^{1} and
- Qing Ling^{1}
Received: 13 October 2015
Accepted: 12 April 2016
Published: 26 April 2016
Abstract
This paper proposes a novel algorithm for space–time block code classification, when a single antenna is employed at the receiver. The algorithm exploits the discriminating features provided by the higher-order cumulants of the received signal. It does not require estimation of channel and information of the noise. Computer simulations are conducted to evaluate the performance of the proposed algorithm. The results show the performance of the algorithm is good.
Keywords
Background
Blind signal classification of communication signals plays a pivotal role in both civilian and military applications, such as electronic warfare, radio surveillance, civilian spectrum monitoring, and cognitive radio systems (Axell et al. 2012; Dobre 2015; Dobre et al. 2005, 2007). The research on signal classification for multiple input multiple output (MIMO) scenarios is at an incipient stage. Regarding space–time block code (STBC) classification algorithms, they can be divided into several general categories: likelihood-based (Choqueuse et al. 2010), subspace-based (Swindlehurst and Leus 2002; Zhao et al. 2014), second-order statistics based (Via and Santamaria 2008a, b), cyclostationarity based (DeYoung et al. 2008; Shi et al. 2007; Marey et al. 2012), higher-order based (Choqueuse et al. 2008a, b, 2011; Eldemerdash et al. 2013a, b) and correlation function based (Marey et al. 2014; Mohammadkarimi and Dobre 2014).
The likelihood-based algorithm evaluate the likelihood function of the received signal ,and employ the maximum likelihood criterion for decision making. However, the likelihood-based algorithm need channel estimation and signal imformation (Choqueuse et al. 2010). To avoid the drawbacks of likelihood-based algorithm, several authors have investigated the use of subspace (Swindlehurst and Leus 2002; Zhao et al. 2014) and second-order statistics (SOS) (Via and Santamaria 2008a, b) algorithm. However excluding some specific low-rate codes, these approaches fail to extract the channel in a full-blind contex (Swindlehurst and Leus 2002; Zhao et al. 2014; Via and Santamaria 2008a, b). These semi-blind methods cannot be employed in a non-cooperative scenario since they require modification of the transmitter. To avoid the drawbacks above, the cyclostationarity-based and higher-order based algorithms are proposed. Most of the paper formed with more than a single antenna (Choqueuse et al. 2008a, b; Eldemerdash et al. 2013b; Choqueuse et al. 2011). Some of the articles study the classification of spatial multiplexing (SM) and Alamouti STBC (DeYoung et al. 2008; Shi et al. 2007; Eldemerdash et al. 2013b), and others study a large pool (Marey et al. 2012; Choqueuse et al. 2008a, b, 2011). Literature (Marey et al. 2014) and literature (Mohammadkarimi and Dobre 2014) performe under frequency-selective channels and impulsive noise respectively . However, few articles illustrate the STBC classification when a single antenna is employed at the receiver (Eldemerdash et al. 2013a; Mohammadkarimi and Dobre 2014). Since in reality the requirement cannot always be met, blind classification for STBC are of interest when a single receive antenna is available.
This paper proposed an efficient algorithm based on Higher-order cumulants for classification of STBC. We use the properties of higher-order cumulants to avoid the effect of noise. We exploit features based on fourth-order cumulants, and divide the STBCs with an interval detector. The proposed algorithm performs well in the simulation and does not need channel estimation and signal information.
Signal model and assumption
Signal model
We consider a wireless communication system which employs linear space–time block coding with multiple transmit antennas. Each symbol is encoded to generate \(n_t\) parallel signal sequences of length L. The sequences are transmitted simultaneously with \(n_t\) antennas in L consecutive time periods. The kth \(n_t\times L\) matrix can be denoted by \(C(S_k)\), from a block of n symbols denoted \(s=[s_1,s_2,\ldots ,s_n]^T\).
The received signal is assumed to be encoded by one of the following STBCs^{1}:SM (Choqueuse et al. 2008a) with \(n_t=1\) and L = 1, Alamouti STBC (Al for short) code (Alamouti 1998) with \(n_t=2\) and L = 2 (orthogonal with rate 1), ST3 (Choqueuse et al. 2008a) with \(n_t=3\) and L = 8 (orthogonal with rate \(\frac{3}{4}\)), ST4 (Tarokh et al. 1999) with \(n_t=4\) and L = 8 (orthogonal with rate \(\frac{3}{4}\)).
Main assumptions
In this study, the following conditions are assumed to hold.
(AS1) The data symbols are assumed to belong to an M-PSK or M-QAM signal constellation , and consist of independent and identically distributed random variables with zero mean and \(E[|s|^2]=E[|s|^4]=1\), \(E[s^2]=E[(s^*)^2]=0\), and \(E[s^4]=E[(s^*)^4]=-1\) Eldemerdash et al. (2013b).
(AS2) The received signal is affected by a frenquency-flat Nakagami-m fading channel Beaulieu and Cheng (2005), with m = 3, and \(E[|h_i|^2]=E[|h_i|^4]=1\), \(E[h_i^2]=i\), and \(E[h_i^4]=-1\), where \(i=1,\ldots ,n_t\).
(AS3) The noise vector \(B_k\) is a complex stationary, and ergodic Gaussian vector process, independent of the signals, with zero mean and variance \(\sigma ^2\). It implies that: \(E[B_kB_k^H]=\sigma ^2L\). The SNR is defined as \(10\log _{10}{\left( \frac{n_t}{\sigma ^2}\right) }\) (Swami and Sadler 2000).
AS4) The received signal intercepts a whole number \(N_b\) of space–time blocks \(Y=[Y_1,\ldots ,Y_{N_b}]\), i.e., the first and last intercepted samples correspond to the start and the end of a space–time block, respectively.
Classification based on HOS
In this section, we exploit the feature by using Higher-order cumulants. We will first define the fourth-order cumulants which we propose to use, discuss how they can be estimated from the data, and then give the theoretical values for various STBCs.
Definitions
Sample estimates
Theoretical values
Theoretical cumulants statistics C _{40} and C _{42} for various STBCs, and variances of their sample estimates
STBC | C _{40} | C _{42} | \(Nvar(\hat{C}_{40})\) | \(Nvar(\hat{C}_{42})\) | ||||
---|---|---|---|---|---|---|---|---|
0 dB | 5 dB | 10 dB | 0 dB | 5 dB | 10 dB | |||
SM | 1 | −1 | 0.12 | 0.01 | 0.00 | 0.02 | 0.00 | 0.00 |
Al | 2 | −2 | 0.14 | 0.08 | 0.08 | 0.02 | 0.01 | 0.01 |
ST3 | 3 | −3 | 0.81 | 0.71 | 0.41 | 0.10 | 0.08 | 0.05 |
ST4 | 4 | −4 | 2.93 | 2.85 | 2.55 | 0.19 | 0.19 | 0.15 |
The theoretical values are described in Table 1. Column 2 shows C _{40} of STBCs, and column 3 shows C _{42}. We can see that, The theoretical values of cumulants are different in various STBCs. The specific algorithm to be used depends upon the difference.
Threshold analysis
In this section, we develop thresholds for the tests in the hierarchical classification scheme. In order to do this, we need to derive expressions for the variance of the sample estimates of the cumulants in Eq. (10). The variance expressions are estimated by 1000 Monte Carlo trail for each \(\xi \in \{SM, Al, ST3, ST4\}\).
Simulation results
In this section, a variety of simulation experiments are presented illustrating the performance of the proposed classification schemes. For each Monte Carlo trial, the appropriate normalized statistics \(\hat{C}_{42}\) is estimated via Eqs. (9) and (10), based on N data samples, and the additive noise is complex White Gaussian, via QPSK modulation. All results are based on 1000 Monte Carlo trials.
The performance of the optimal likelihood-based algorithm is best, but it require estimation of the channel, noise information of the transmitted signal. The proposed algorithm does not need these estimation and more suitable to reality system. The performance of the proposed algorithm greatly outperforms the algorithm in Choqueuse et al. (2008a), which achieves a \(P_c=0.5\) even for high SNR. This can be explained as the second-order correlation provides a discrimination feature for SM, ST3, and ST4 only. For Al, it equals zero, leading to the mis-classification. Algorithm (Eldemerdash et al. 2013a) has a little better performance than the proposed algorithm when SNR > 8 dB, but in low SNR, it is on the contrary. The probability of classification of proposed algorithm researches 0.97 at 0 dB, when algorithm (Eldemerdash et al. 2013a) at 8 dB.
Complexity comparison
Conclusion
This paper proposed an algorithm for blind classification of STBC using a single antenna based on high-order cumulants. We have shown that simple HOS are useful for classification of STBC. The algorithm was evaluated through simulations in terms of average probability of correct classification. The proposed algorithm, with the advantages that it does not require channel estimation and noise information, performed better than any other classification algorithms using a single antenna in low SNR. Moreover, it can benefit from spatially correlated fading.
The decision thresholds were on the conservative side because they were obtained by assuming that the sample estimates of the test statistics \(C_{40}\) and \(C_{42}\) have equal variances under different hypotheses, and ignored the effects of additive noise. The performance could be improved by taking these issues into account.
We choose Al and SM as they are the most commonly used in wireless standards, and ST3 and ST4, as being commonly referred codes.
Declarations
Authors’ contributions
WY and LZ conceived and designed the study. WY and QL performed the experiments. WY wrote the paper. WY, LZ and QL reviewd and edited the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The Institute of Information Fusion is acknowledged for supporting and funding this research.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis 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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