Higher Order Cummulants based Digital Modulation Recognition Scheme

In this study, we have presented that Higher Order Cummulants (HOC) based modulation recognition scheme for Pulse Amplitude Modulation (PAM), Quadrature Amplitude Modulation (QAM) and Phase Shift Keying (PSK) modulated signals having orders of 2 to 64. Modulation recognition is a process to recognize the signal modulation type which is received by the receiver in the presence of channel noise. The HOC based MR is accomplished in two modules. First is feature extraction using higher order cummulants. These features are distinct for different modulated signals. Second is recognition process which gives decision based upon the features extracted from higher order cummulants. The Probability of Correctness (POC) curves shows the recognition accuracy for sample size and number of iterations. The Additive White Gaussian Noise (AWGN) is considered throughout the simulations.


INTRODUCTION
Modulation Recognition (MR) is the intermediate step between detection of the information contained in the signal and demodulation of signal.Automatic Modulation Recognition (AMR) has various applications in Cognitive Radio (cooperative and noncooperative communication), civilian and military communication, electronic warfare and surveillance.AMR is to recognize the received signal modulation type, which has undergone through channel effects like fading, noise and interference etc., during transmission of the signal.AMR is basically a non-cooperative communication, (Panagiotou et al., 2000) which also includes some aspects of cooperative communication, such as tracking and identification of channel and estimation and detection of signal parameters.
The major techniques of automatic modulation recognition are decision theoretic methods and pattern recognition.The decision theoretic approach is based on likelihood function of the received signal.The modulation recognition in decision theoretic approach can be viewed as multiple hypothesis tests, or may be considered sequence of pair-wise multiple hypothesis test.Once the likelihood function of the received signal is constituted, Average Likelihood Ratio Test (ALRT) and Generalized Likelihood Ratio Test (GLRT) can be pragmatic to determine the modulation type.Likelihood based approach is theoretically optimal but computationally complex (Wang and Wang, 2010).Due to phase errors, frequency offset, channel effects and timing jitter, the decision theoretic methods are not robust to model mismatch (Wei and Mendel, 2000;Yucek and Arslan, 2004;Zhao and Tao, 2004).
The Feature Based (FB) pattern recognition method is the suboptimal solution (Swami and Sadler, 2000).In FB approach modulation recognition is carried out in two modules; the first module is feature extraction subsystem, in which features are extracted from the received signal with channel effects; the second module is pattern recognizer subsystem, in which features extracted from the received signal are compared with the theoretical values of the reference features and determines the modulation type of transmitted signal.Due to robustness with respect to model mismatches and low computational complexity, FB approach is used for modulation recognition.Few research contributions in this are Marchand et al. (1997), Dobre et al. (2003), Kadambe and Jiang, (2004), Guan et al. (2004) and Dobre et al. (2007).
In this study we have used the Higher Order Cummulants (HOC) to recognize the modulation type of received signal corrupted by additive white Gaussian noise.The features extracted based on moments and cummulants for recognition purpose considering PAM and QAM modulations for order 2 to 64.The theoretical values of higher order moments and higher order cummulants for considered modulations are

SYSTEM MODEL AND FEATURES USED
System model: Figure 1 shows the system model.The generalized expression for signal received is given by: where, ‫ݎ‬ሺ݊ሻ : Complex baseband envelop of received signal ‫ݕ‬ሺ݊ሻ : The additive white guassian noise ‫ݏ‬ሺ݊ሻ : Given by: where, ‫ݏ‬ሺ݈ሻ : Input symbol sequence which is drawn from set of M constellations of known symbols and it is not necessary that symbols are equi-probable K : Amplitude of signal ݂ : Frequency offset constant T : Symbol spacing ߠ : The phase jitter which varies from symbol to symbol ℎሺ… ሻ: Channel effects ߳ ் : The timing jitter Features used: As Cummulants are made up of moments, so various moments have been used as features.For the complex valued stationary random process ‫ݎ‬ሺ݊ሻ, Cummulants of 2 nd , 4 th , 6 th and 8 th order have the following definitions: C ସଵ = M ସ − 3M ଶ M ଶଵ = cumm{yሺnሻ, yሺnሻ, yሺnሻ, y * ሺnሻ} (6) ‫ܯ‬ stands for moments of received signal and it is given:

Feature values based on moments and cummulants:
The theoretical values of Moments and Cummulants which were used for various signal constellations of Fig. 1: The system model

SIMULATION RESULTS
The recognition of PAM and QAM modulated signals in the presence of additive white Gaussian noise is evaluated here.The modulated signals considered here is PAM 2, PAM 4, PAM 8, PAM 16, PAM 32, PAM64 and QAM 2, QAM 4, QAM8, QAM 16, QAM  32, QAM 64.Table 3 shows the simulation results of correct recognition of modulated signals using the higher order Cummulants features under different SNR.
The recognition of PAM and QAM modulated signals are acceptable above -5 dB.As SNR increases from -5 to 20 dB, the correct rate of recognition also increases, while at SNR = 0 dB the correct rate of recognition reaches 100%.
For example considering the modulated signal PAM2; the graphical representation of the probability of correctness curve for varying SNR is shown in Fig. 2. The probability of correctness gradually increases with the increase in SNR.The probability of correctness approaches 1 at SNR = -2 dB.
For example considering the modulated signal PAM4; the graphical representation of the probability included.The following modulated signals PAM 2 to PAM 64, QAM 2 to QAM 64 and PSK 2 to PSK 64 are used for recognition purpose.The Probability of Correctness (POC) curves are simulated, based on Signal to Noise Ratio (SNR), number of iterations and sample size.The simulation results using HOC for the considered modulated signals show that high recognition rate is achieved at low SNR.The cummulants based tree structure for recognition of PSK signals is briefly presented.

Fig. 4 :
Fig. 4: POC curves for the different number of iterations

Table 1 :
Theoretical values of moments of different modulation types

Table 2 :
Theoretical values of cummulants of different modulation types PSK , the computed values of moments of PSK, QAM and PAM with orders 2 to 64 are listed.The moments (row wise) M 21, M 42 and M 63 are same for all orders of PSK modulated signals.The moments M 20, M 41 and M 60 are also same for all orders of PSK modulated signals.The moment M 60 has same value for all orders of QAM modulated signals?Also all moments for PSK 2 have same constant value i.e., 1.In Table2, the computed values of cummulants of PSK, QAM and PAM with orders 2 to 64 are listed.The cummulants C 21and C 42 are same value for all orders of PSK and QAM modulated signals.

Table 3 :
The correct rate of recognition of PAM and QAM under different SNR