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This paper addresses the problem of modulation classification of unknown signals. Recognition of the modulation type of an unknown signal provides valuable insight into its structure, origin and properties. Automatic modulation classification is used for spectrum surveillance and management, interference identification, military threat evaluation, electronic counter measures, source identification and many others. An empirical distribution function based-guassianity test is proposed to distinguish OFDM from single carrier. The model is simulated in MATLAB and numerical results are presented to verify the efficiency of the proposed scheme.
Keywords: OFDM, modulation classification, distribution identification
Recently, orthogonal frequency-division multiplexing (OFDM) - has received considerable attention. Orthogonal Frequency Division Multiplexing (OFDM) is an alternative wireless modulation technology to CDMA. OFDM has the potential to surpass the capacity of CDMA systems and provide the wireless access method for 4G systems. OFDM is a modulation scheme that allows digital data to be efficiently and reliably transmitted over a radio channel, even in multipath environments. OFDM transmits data by using a large number of narrow bandwidth carriers. These carriers are regularly spaced in frequency, forming a block of spectrum. The frequency spacing and time synchronization of the carriers is chosen in such a way that the carriers are orthogonal, meaning that they do not cause interference to each other. It has been adopted or proposed for a number of applications, such as satellite and terrestrial digital audio broadcasting (DAB), digital terrestrial TV broadcasting (DVB), broadband indoor wireless systems, asymmetric digital subscriber line (ADSL) for high bit-rate digital subscriber services on twisted-pair channels, and fixed broad-band wireless access. Important features of OFDM systems include immunity to multipath fading and impulsive noise .
The automatic recognition of the modulation format of a detected signal, the intermediate step between signal detection and demodulation, is a major task of an intelligent receiver, with various civilian and military applications. Obviously, with no knowledge of the transmitted data and many unknown parameters at the receiver, such as the signal power, carrier frequency and phase offsets, timing information, etc., blind identification of the modulation is a difficult task. This becomes even more challenging in real-world scenarios with multipath fading, frequency-selective and time-varying channels.
The need for distinguishing the OFDM signal from single carrier has become obvious for general military applications and software defined radios. Modulation classification methods have been studied from decades, but much less research is on multicarrier modulation systems.
We know the fact that OFDM is asymptotically Gaussian. To distinguish OFDM from single carrier, an empirical distribution function based guassianity test classifier is devised. This classifier is based on statistical test. Actually it is a hypothetical problem. An hypothesis H1 (non Gaussian process) is restricted to single carrier modulation, whereas hypothesis H0 (Gaussian process) is assigned to OFDM signals. The channel is assumed as AWGN channel. One important point to be noted is AWGN is also having Gaussian distribution, so to distinguish OFDM from AWGN cyclostationarity test is conducted.
The layout of this paper is as follows: system model is discussed in section II, section III describes the empirical distribution function based guassianity test, numerical results are presented in section IV to show the efficiency of the system, and concluding remarks are given in section V.
II SYSTEM MODEL:
Figure 1 shows the system model diagram for proposed work. The incoming signal is first down converted and sampled by the block of pre-processing. Then guassianity test based modulation classification test is applied to the received signal. Here the channel assumed is the Additive White Guassian Noise (AWGN) channel. If the guassianity test is failed then one may say that single carrier modulation technique is present with the received signal, and one may proceed with single carrier modulation. If guassianity test is passed then we can say that multicarrier modulation is present with the received signal. But one point is to be noted that this positive guassianity may be because of AWGN signal. AWGN signal may also have the guassian distribution.
So to check whether this positive guassianity because of AWGN or OFDM, cyclostationarity test is conducted.
Figure 1: System Model Diagram
We know that OFDM is cyclic stationary with period Ts  . Where Ts denotes period of one OFDM symbol.
Where Tb and Tcp, are data and cyclic prefix duration respectively.
If this cyclostationarity test is failed then one may conclude that positive guassianity was because of only AWGN signal, not because of OFDM.
III Guassianity test based OFDM classification:
In OFDM, all orthogonal subcarriers are transmitted, simultaneously. In other words, the entire allocated channel is occupied with the aggregated sum of the narrow orthogonal sub bands. The OFDM signal is therefore treated as composed of a large number of independent, identically distributed (i.i.d.) random variables. Hence, according to the central limit theorem (CLT), the amplitude distribution of the sampled signal can be approximated, with Gaussian. On the other hand, the amplitude distribution of a single carrier modulated signal cannot be approximated with a Gaussian distribution. Therefore, the identification task of OFDM from single carrier becomes a Gaussianity test (or normality test).
A Empirical Distribution Function-Based Gaussianity Test
The empirical distribution function (EDF) is a stair-wise function which is calculated from the signal samples. The population distribution function can be estimated by the EDF. Assume a given random sample of size n is Ω1 , Ω2 ,Ω3,... Ωn and arrange the sample in ascending order Ω(1) < Ω(2) < Ω(3),... Ω(n) , further that the cumulative distribution function of Ω is F(w), then the definition of the EDF is given by
Therefore, as w increases, the EDF Fn(w) takes a step up of height 1/n as each sample observation is arrived. We can expect Fn(w) to estimate F(w), and actually Fn(w) is a consistent estimator of F(w). As , decreases to zero with probability one.
Since in our case, a Gaussianity test is conducted, we the assume the random samples belong to a Gaussian distribution
With mean µ and variance σ², and suppose it is null hypothesis H0. Furthermore, the hypothesized distribution has an incomplete specification, i.e., with mean and variance unknown. Then H0 becomes a composite hypothesis and we estimate parameters from the sample.
In order to measure the difference between EDF and CDF quantitatively, the so-called EDF statistics are introduced. They are based on the vertical differences between Fn(w) and F(w). The closer two curves, the smaller EDF test statistics. We resort to the Cramer-von Mises (CV) statistic, which is defined by
(4) Thus, CV statistic is nothing but the integrated square error between the estimated cumulative distribution function and the measured empirical distribution function of the sample.
The computation of W2 is carried out via the Probability Integral Transformation (PIT), . When F(w) is the true distribution of Ω, the new random variable - is uniformly distributed between 0 and 1.Hence - has distribution function
, and let be the EDF of values. Thanks to the fact that
EDF statistic calculated from with the uniform distribution Will take the same value as if it were calculated from the. This yields following formula to calculate the CV test statistics (6) From the definition and derivation given above, the procedure of CV Gaussianity test is summarized as follows
Sample the incoming signal, take real or imaginary part of samples to obtain ;
Arrange the samples in ascending order ;
Estimate the sample mean µ and standard deviation σ
(7) Apply PIT, calculate the standardized value for k= 1,...., n-1, from
, and further where φ(x) indicates a cumulative probability of a standard normal distribution;
5) Calculate the CV statistics via formula 6
6) When the CDF is not completely specified and the parameters are estimated, the CV test statistics should be modified to obey asymptotic theory. so use the percentage points given in table 1, and calculate modified statistics.
7) If the modified CV statistics exceed the appropriate percentage points at level α, is rejected with significance level α. In other words no Gaussianity is present in the incoming signal.
Note that the significance level is a statistics expression, which corresponds to probability of false alarm in engineering. Both belong to so called type one error.
Modified form T*
Significance level α
Upper tail percentage points
Lower tail percentage points
Table 1 Modifications and percentage points for a test for normality with μ and σ² unknown
IV:Simulation and discussion on Guassianity test:
The raised cosine pulse shaping is implemented with roll off factor set to 0.35 without loss of the generality, CV statistics(Cramer von mises statistics) of OFDM signal compared with that of single carrier modulation, say, M-ary QAM. In simulations, normalized constellations are generated to assure fair comparison. A 512 subcarrier OFDM signal has been generated, with 16 QAM modulations on each subcarrier.
These CV statistics are sketched for different modulation techniques for 100 trials. The dashed line indicates the decision threshold which is set to 0.2 for 0.005 significance level. (i.e. 0.5 %). If the statistics exceeds this threshold, the H0 (Gaussian hypothesis) is rejected with 0.5% probability that actually H0 is true. As we can observe from figure the CV statistics for OFDM signal is below threshold, except for couple of trials. similarly the CV statistics for single carrier signal(M-ary QAM) is above the threshold. This module have been checked for two different environments, Point-to-point link, or wired link (without fading), and mobile(wireless) channel(with fading). Figure 1 shows CV statistics for each trail in the fading environment, whereas figure 7.2 shows for no-fading channel. It can be seen that, classifier has noise margin of 0.3 for the classification of single and multicarrier signal in fading and non-fading environment.
Figure2, CV Statistics with fading
Figure3, CV Statistics without fading
Here we are not testing every modulation type, rather, classification will be correct only if signal is Gaussian. With gaussianity test, even single carrier modulation, because of AWGN it shows Gaussianity some times. But this performance characteristic indicates that Single carrier modulations are affected at low SNRs. Even, as we can see in figure 7.3, higher order QAM performance is affected more than lower order QAM. But for the OFDM signal it is above 0.95 for low SNRs as well as for high SNRs.
Figure 4 Classification performances without fading
Figure 5 classification performances with fading.
From figure 7.3 we can observe that because of 0.5% significance level the performance curve for an OFDM signal fluctuates from .95 to 1. From figure 7.4 we can observe that the performance is improved more with fading environment. Here the performance curve for OFDM signal fluctuates above 0.99. Means we can say that the probability of correct classification is above 98%.
The comprehensive modulation classification method is proposed to identify the OFDM signal from single carrier signal. This is a statistical based test based on Cramor Von mises statistics. The test shows that for single carrier modulation the CV statistics exceeds threshold value, whereas for multicarrier signal(OFDM), these statistics are below threshold.
The performance of this Classifier is also tested. This performance is tested in two different environments, i.e. in fading environment as well as in without fading environment. The performance of such classifier is above 95% in non-fading environment, whereas it is around 98% in fading environment.