INTRODUCTION
The identification of digital modulated signals is one of the important tasks in the field of mobile communication in general, Software Defined Radio (SDR) in particular. This has motivated the researchers to develop various digital modulation identification algorithms in the recent past (Enrico Buracchini, 2000). As the adaptive receiver can communicate with different communication standards like TDMA, CDMA and GSM, the identification of digital modulation type of a signal has to be optimized for their effective usage. The signal identification process is an intermediate step between signal reception and demodulation for characterizing signals in various communications applications including spectrum management, surveillance, electronic warfare, military threat analysis. Several identification algorithms have been reported so far (Azzouz and Nandi, 1996; Beran, 1997; Druckmann et al., 1998; Yu et al., 2003). Generally, two basic approaches in the identification problem as decisiontheoretical and statistical pattern recognition (Nolan et al., 2002). The decisiontheoretical approach is based on hypothesis testing for sourced hypotheses conditioned to a finite set of known candidate signals. The pattern recognition approach usually consists feature extraction, reduction of the feature space and classification based on the lower dimension feature space.
One of the basic feature extraction methods for identifying the nonstationary signals is timefrequency analysis, particularly the Wavelet Transform (WT). Lin and Jay Kuo (2002) have reported that the phase changes can be examined using Morlet wavelet and the likelihood function based on the total number of detected phase changes can be used to classify Mary PSK signal. One of the most complex and important identifier was introduced by Hong and Ho (1999). They applied the Haar WT and statistical decision theory to identify the Mary Phase Shift Keying (MPSK), Mary Frequency Shift Keying (MFSK) and QAM signals containing Additive White Gaussian Noise (AWGN) without baseband filtering. Binary PSK/CPFSK and MSK identification was investigated by Radomir Pavlik (2005) and complex Shannon wavelet was applied to identify the modulation schemes. The investigation was focused primarily on the identification of binary modulation signals under constant envelope class and it fails to identify the nonconstant envelop modulation schemes. Automatic Modulation Identification (AMI) algorithm was developed to classify QPSK and GMSK signals with simulated Additive White Gaussian Noise (AWGN) channel by Prakasam and Madheswaran (2007). The extracted transient characteristics and histogram peaks were used to identify the modulation scheme.
This study proposes the modulation classification algorithm considering the
AWGN channel to identify most of the Mary Shift Keying modulation schemes.
The wavelet transform, normalized histogram peak were applied for identification.
Also, the various higher order statistical moments based decision was considered
to compare. The performance analysis of the proposed algorithm was carried out
and compared with reported algorithms. The confusion matrix, throughput analysis
and Receiver Operating Characteristics (ROC) were carried out to measure the
correct identification capability of the proposed algorithm.
MATHEMATICAL MODEL
Let the received waveform r(t), 0≤t≤T be described (Haykin, 2005) as:
Where, s(t) is transmitted signal and n(t) is the additive white Gaussian channel noise. The signal s(t) can be represented in complex form as:
Where, ω_{c} is the carrier frequency and θ_{c} is the carrier phase. The analysis technique is required for nonstationary signal, which will analyze the signal frequency with time instants of occurring. The Fourier transform approach gives either the frequency components or time components. The wavelet transform has the special feature of MultiResolution Analysis (MRA), which provides the information in both frequency and time instants. The Continuous Wavelet Transform of a signal s(t) is defined (Chan, 1995) as:
Where, a is the scaling factor and τ is the translation factor. The function ψ*_{a}(t) is the complex conjugate of mother wavelet.
Generally, the complex envelope of s(t) in Eq. 1 may be expressed for all modulation types as:
Where, φ(t;a) represents the timevarying phase of the carrier, a represents
all possible values of the information sequence {a_{k}}. In the case
of binary symbols a_{k} = ±1. From Eq. 2,
3 and 4, the resulting integral of C(a, τ) is obtained
as:
Where:
is
the exponential integral and y = jt(2πf2πf_{c}). 
Classification of GMSK and Mary FSK with Mary PSK and Mary QAM: The normalized histogram peak of Wavelet transformed coefficient is used to classify the Class I (GMSK and Mary FSK) with Class II (Mary PSK and Mary QAM) signals. If n_{k} is the number of occurrence in a particular value then the normalized histogram (probability of occurrence) of a process is given by:
Where, n is total number in the particular process. This factor is applied to measure the probability of occurrence of the frequency components present in the incoming signal. Class I has multifrequency component and multiple peaks in its normalized histogram. But the Class II signal has constant transient characteristics and a single peak in its normalized histogram. Based on the histogram peak, the either Class I or Class II modulation scheme has been identified.
Sub classification of class I: The classification of various modulation schemes may be formulated using the statistical parameters such as moments and median. The higher order statistical moment plays the major contribution in nonstationary signal. Thus it has been considered for the classification of nonstationary signals. The n^{th} order moment for p(x_{i}), where i = 0,1,2,…N1 is given by
Where:
is the mean of the statistical process. The second order moment (variance) of Wavelet Transform can be computed using:
Where, N is the length of analyzed signal. Similarly, the higher order moments
can be formulated in the same way and then the classification problem as a binary
tree hypothesistesting problem.
Let H_{i} be the i^{th} modulation format assigned to the received signal, where i is associated with {GMSK, j} and j is with Mary FSK. The statistical decision needs the probability density function (pdf) of the test statistics conditioned on the assigned digitally modulated signal. The random variables generated from linear combinations of both sinusoidal signal and Gaussian noise is considered as Gaussian probability density function. The two conditional Gaussian pdfs allow a threshold setting to decide the GMSK and Mary FSK, when a certain probability of false identification of both signals is given. The conditional pdf is:
Under the hypothesis H_{GMSK} is true, the probability of GMSK misclassification is simply the probability that μ_{2, GMSK } x < μ_{2, GMSK}T_{1}, ie, μ_{2}>T_{1}. The probability of misclassification error for GMSK is given by:
which is reduced to
Where, erc(f) is defined as
Similarly, if it is assumed that the hypothesis H_{Mary FSK} is true, the probability of Mary FSK signal misclassification is simply the probability that x  μ_{2, GMSK} < T_{1}μ_{2, GMSK} ie, μ_{2 }< T_{1}. The Probability of misclassification error for Mary FSK is given by:
which is reduced to:
Where, erc (f) is in the same form. It is obvious that when the Gaussian noise increases, the variance of GMSK and Mary FSK decreases until the point when both the probabilities of misclassification are equal. Thus, P(e/H_{GMSK}) = P(e/H_{Mary FSK}) = 0.01 and the condition for setting the optimal threshold value T_{1} can be obtained by equating Gaussian distribution to zero. Then the related threshold value is obtained as:
Based on the variance the classification of GMSK with Mary FSK can be done.
Classification of Mary FSK signals: Similarly the threshold T_{Fr }(where r = 1,2,3,4 ... to represent 2ary, 4ary, 8ary, 16ary…, respectively) and the further classification of subclass I can be classified based on second or higher order statistical moment. Decision making between 2FSK, 4FSK, 8FSK, 16FSK and so on can be carried out based on the comparison of higher order statistical moment with computed threshold value and for 2^{r }FSK and 2^{r+1 }FSK is given by:
Where, n is ≥ 2.
Sub classification of class II: Similarly, the threshold value for identifying Mary PSK and Mary QAM can be obtained by:
Based on the mean and variance (or higher order moment) the classification of QPSK with QAM can be performed.

Fig. 1: 
Flow graph for proposed modulation identification algorithm 
PROPOSED ALGORITHM
The identification of Mary modulation schemes has been done using a common feature. As the transmission of any signal mainly concentrates in the high frequency components, these components can be obtained by extracting the coefficients using wavelet transformation. Wavelets are to be selected such a way that it looks similar to the patterns to be localized in the signal. A good approach to find a solution to this problem can be done by searching a function suitable for approximating both the analyzed signal envelope and frequency content. The wavelet Transform has been computed and the coefficients are recorded. These extracted coefficients are used to generate the histogram peaks. Based on the number of peaks, the identifier identifies that the received signal is either Class I or Class II (Fig. 1).
Table 1: 
Subsystem classificationdecision rule 

After the major classification the Subclass I and II is classified based on the following decision rules as shown in Table 1.
RESULTS AND DISCUSSION
The algorithm including test signal generation, noise addition, reception, feature extraction and modulation identification was developed and tested using MATLAB. The developed algorithm is verified for 2FSK, 4FSK, 8FSK, 16FSK, GMSK, MPSK and MQAM modulation schemes. The above specified modulation schemes were simulated MATLAB with 200 symbols input message and AWGN noise is simulated and added with a transmitting signal as a channel noise. The Wavelet Transform has been applied to extract the transient characteristics of the received signal. The magnitude of Haar wavelet transform for Class II is a constant, but Class I has a multistep function since the frequency is variable. This common feature made to consider the Haar wavelet as mother wavelet which is given (Chan, 1995) hereafter:
After extracting the transient characteristics, the coefficients were used to generate the histogram peak.
The Fig. 2 and 3 show that the Class I signal has more than single peak because these signals have multiple frequency components. But the Class II signal has constant transient characteristics and single peak in its histogram.
Then each subsystem is further classified based on decision rules shown in Table 1. The threshold value for Class I and II classification is provided in Table 2.
After identification as Class I, the subclassification is done based on the threshold values for 2nd, 4th and 8th order statistical moments tabulated in Table 3. After identification of the scheme, the demodulation is performed by conventional methods.
Performance analysis: The performance of the proposed algorithm was examined based on the confusion matrix, throughput and Receiver Operating Characteristics (ROC). These parameters are used to analyze the identification capability of the proposed algorithm.
Confusion matrix analysis: For the analysis purpose, the identifier has been tested for 1000 experiments with 250 symbols per experiments. The testing was carried out for different SNR starting from 20 dB and the confusion matrix were tabulated at 3 dB for the proposed identifier as shown in Table 4.
The spread of identification results is caused by the similarity of the waveforms
and could not really assess as false identification. Most of the nonsuccessfully
identified signals were assigned to the reject class REJ.

Fig. 2: 
Histogram peak of class 1 signal 

Fig. 3: 
Histogram peak of class 2 signal 
Table 2: 
Threshold value for classification of class 1 and 2 

Table 3: 
Threshold value for subclassification of class 1 for various
statistical moments 

This is a desired result because a rejection is preferred to a false identification.
The identification of 1.7% of 8FSK as 16FSK based on 8th order moment is caused
by comparatively low SNR. Both the waveforms are similar in some aspects. The
false identification of 1.6% of MPSK as MQAM is also not surprising because
for the lower SNR these waveforms are similar in some nature.
Table 4: 
Confusion matrix at SNR = 3dB for various decision parameters 

Table 5: 
Throughput of the proposed algorithm (%) 

Table 6: 
Receiver operating characteristics (ROC) 

The false identification of 0.9% of GMSK as MFSK and vice versa is acceptable
because both have some similar nature in frequency component. The above analysis
show that the number of false identification is 1.6% for worst case and it is
0.16 for 1000 experiments testing.
Throughput analysis: The throughput of the proposed algorithm was computed for various SNR starting from 20 to 3 dB and is tabulated in Table 5.
When SNR is greater than or equal to 6 dB, the percentage of identification is 100% and the identifier identifies the correct modulation schemes when SNR is greater than 3 dB with 99.1%.

Fig. 4: 
Receiver Operating Characteristics (ROC) for the Proposed
Algorithm 
Receiver Operating Characteristics (ROC): ROC is a plot of probability of detection (P_{d}) as a function of the probability of false alarm (P_{f}). The probabilities of 200 symbols with 1000 experiments are calculated and tabulated in Table 6.
Figure 4 shows the ROC curves for the identifier when SNR is equal to 10, 5 and 3dB. The performance of the identifier is better if the curve rise faster.
When SNR is 10dB, P_{d1} is 100% independent of P_{f1}. When SNR is 5dB and the P_{f2} is 0.1, the P_{d2} between GMSK and QPSK is 0.96. When SNR is 3dB and the P_{f3} is smaller than 0.3 the P_{d3} drops rapidly. This is because the hypothesis of moderate SNR used to obtain the optimum threshold in the decision device will no longer be valid.
Table 7: 
Comparison of various algorithms 

Comparison of various classifiers: Comparison of the proposed algorithm with several existing algorithms for classifying various Mary digital modulation schemes is shown in Table 7. The ideal scenario, i.e., no unknown parameters, as well as the scenarios with unknown carrier phase and unknown carrier phase/timing offset, respectively has been considered. Of course, when higher order modulations are included in the modulation pool, higher SNRs and/or a larger number of symbols are needed to achieve the same performance. From the comparison it is clear that the proposed algorithm is capable of identifying the various modulation schemes with low SNR values.
CONCLUSION
An automatic Modulation Identification algorithm is presented and is found to be well suited for Mary digital modulation schemes used in SDR. The proposed algorithm was tested for 2FSK, 4FSK, 8FSK, 16FSK, GMSK, MPSK and MQAM modulation schemes with different SNR. The simulated results obtained using Wavelet transform technique, normalized histogram peak and 8th order statistical moment measurement show that the correct modulation scheme identification is possible even at low channel SNR of 3 dB. The ROC analysis shows that the percentage of correct modulation identification is higher than 98.4% for 1000 experiments with 200 symbols when SNR is not lower than 3 dB. The comparison with existing reported methods shows that the proposed algorithm is capable of identifying the Mary Shift Keying modulation scheme with low SNR.