Distinguishing CPFSK from QAM and PSK Modulations

Abstract

Digital modulation classification is important for many civilian as well as military applications. In this paper, we propose a simple and robust feature to distinguish continuous-phase FSK from QAM and PSK modulations. The feature is based on product of two consecutive signal values and on time averaging of imaginary part of the product. Conditional probability density functions of the feature given modulation type are determined. In order to overcome the complexity of calculating probability density functions, central limit theorem for strictly stationary m-dependent sequences is used to obtain Gaussian approximations. After calculating probability density functions, thresholds are determined based on minimization of total probability of misclassification. Since threshold-based results are valid for special cases requiring knowledge of some parameters, we resort to usage of support vector machines for classification, which require little training and no a priori information except for carrier frequency. Following that joint effects on the performance of carrier offset, fast fading, and non-synchronized sampling are studied in the presence of additive white Gaussian noise. For comparison purposes, rectangular pulse shape is used. To prove practical usefulness, not only the performance is analyzed for root-raised cosine pulses but also for quite less oversampling of symbols than what is found in other approaches. In the course of doing that, the performance is compared with wavelet-based feature that uses support vector machines for modulation separation.

Appendix

Here, we determine the mean and C of F when $\varDelta '\ne 0$. Assuming $\varDelta '\ne 0$, the means for 16-QAM, BPSK and 4-PSK signals for $N_s>1$ are

$$\begin{aligned} E[F]=\biggl (1-\frac{1}{N_s}\biggr )\sin (\varDelta '). \end{aligned}$$

(35)

For BFSK the mean value is given by (18). For $\tau \ge 0$, $C[\tau ]$ for BFSK, 16-QAM, BPSK and 4-PSK, respectively, are

$$\begin{aligned} C[\tau ]= & {} \left\{ \begin{array}{l} \frac{N_s-\tau }{4N_s}\Bigl (\sin (\beta _1'+\varDelta ')-\sin (\beta _2'+\varDelta ')\Bigr )^2 \\ -\,\frac{\sigma ^2}{2}\biggl \{\frac{1}{4N_s}\Big [\cos ((\beta _1'-\beta _2')N_s+\beta _1'+\beta _2'+2\varDelta ') \\ +\,\cos ((\beta _2'-\beta _1')N_s+\beta _1'+\beta _2'+2\varDelta ') \\ +\,\cos ((\beta _1'-\beta _2')(N_s-1)+\beta _1'+\beta _2'+2\varDelta ') \\ +\,\cos ((\beta _2'-\beta _1')(N_s-1)+\beta _1'+\beta _2'+2\varDelta ')\Bigr ] \\ +\,\Bigl (\frac{1}{2}-\frac{1}{2N_s}\Bigr )\Bigl (\cos (2\beta _1'+2\varDelta ')+\cos (2\beta _2'+2\varDelta ')\Bigr )\biggr \}\delta _{\tau ,1} \\ +\,\Big (\sigma ^2+\frac{\sigma ^4}{2}\Bigr )\delta _{\tau ,0},\quad 0\le \tau \le N_s-1 \\ 0, \quad \tau \ge N_s \end{array} \right. \end{aligned}$$

(36)

$$\begin{aligned} C[\tau ]= & {} \left\{ \begin{array}{l} \frac{1}{2N_s}+\frac{8}{25}\Bigl (1-\frac{1}{N_s}\Bigr )\sin ^2(\varDelta ')+\sigma ^2+\frac{\sigma ^4}{2},\quad \tau =0 \\ \frac{8}{25}\Bigl (1-\frac{\tau +1}{N_s}\Bigr )\sin ^2(\varDelta ')\\ -\,\biggl [\sigma ^2\Big (\frac{1}{2}-\frac{1}{N_s}\Bigr )\cos (2\varDelta ')\biggr ]\delta _{\tau ,1},\quad 1\le \tau \le N_s-1 \\ 0,\quad \tau \ge N_s \end{array} \right. \end{aligned}$$

(37)

$$\begin{aligned} C[\tau ]= & {} \left\{ \begin{array}{l} \frac{1}{N_s}\sin ^2(\varDelta ')+\sigma ^2+\frac{\sigma ^4}{2},\quad \tau =0 \\ ~ -\biggl [\sigma ^2\Big (\frac{1}{2}-\frac{1}{N_s}\Bigr )\cos (2\varDelta ')\biggr ]\delta _{\tau ,1},\quad \tau \ne 0 \end{array} \right. \end{aligned}$$

(38)

$$\begin{aligned} C[\tau ]= & {} \left\{ \begin{array}{l} \frac{1}{2N_s}+\sigma ^2+\frac{\sigma ^4}{2},\quad \tau =0 \\ -\,\biggl [\sigma ^2\Big (\frac{1}{2}-\frac{1}{N_s}\Bigr )\cos (2\varDelta ')\biggr ]\delta _{\tau ,1},\quad \tau \ne 0. \end{array} \right. \end{aligned}$$

(39)

Note that $C[-\tau ]=C[\tau ]$. The closed-form expressions of C for BFSK, 16-QAM, BPSK and 4-PSK signals, respectively, are

$$\begin{aligned} C_{\text{ BFSK }}= & {} \frac{N_s}{4}\Bigl (\sin (\beta _1'+\varDelta ')-\sin (\beta _2'+\varDelta ')\Bigr )^2+\frac{\sigma ^4}{2}\nonumber \\&+\,\sigma ^2\Biggl \{1-\biggl \{\Bigl (\frac{1}{2}-\frac{1}{2N_s}\Bigr )\Bigl (\cos (2\beta _1'+2\varDelta ')+\cos (2\beta _2'+2\varDelta ')\Bigr )\nonumber \\&+\, \frac{1}{4N_s}\Big [\cos ((\beta _1'-\beta _2')N_s+\beta _1'+\beta _2'+2\varDelta ')\nonumber \\&+\, \cos ((\beta _2'-\beta _1')N_s+\beta _1'+\beta _2'+2\varDelta ')\nonumber \\&+\, \cos ((\beta _1'-\beta _2')(N_s-1)+\beta _1'+\beta _2'+2\varDelta ')\nonumber \\&+\, \cos ((\beta _2'-\beta _1')(N_s-1)+\beta _1'+\beta _2'+2\varDelta ')\Bigr ]\biggr \}\Biggr \} \end{aligned}$$

(40)

$$\begin{aligned} C_{\text{16-QAM }}= & {} \frac{1}{2N_s}+\frac{8}{25}\Bigl (N_s+\frac{1}{N_s}-2\Bigr )\sin ^2(\varDelta ')+\frac{\sigma ^4}{2}\nonumber \\&+\, \sigma ^2\biggl [1-\Big (1-\frac{2}{N_s}\Bigr )\cos (2\varDelta ')\biggr ] \end{aligned}$$

(41)

$$\begin{aligned} C_{\text{ BPSK }}= & {} \frac{1}{N_s}\biggl \{2\sigma ^2+\frac{N_s\sigma ^4}{2}+\Big [2\sigma ^2(N_s-2)+1\Big ]\sin ^2(\varDelta ')\biggr \} \end{aligned}$$

(42)

$$\begin{aligned} C_{\text{4-PSK }}= & {} \frac{1}{N_s}\Bigl [2\sigma ^2+\frac{N_s\sigma ^4}{2}+\frac{1}{2}+2\sigma ^2(N_s-2)\sin ^2(\varDelta ')\Bigr ]. \end{aligned}$$

(43)