# Radiometric Identification of Signals by Matched Whitening Transform
## 4 Results
### Class Confusion Matrices  
Training the classifier involves the computation of 5 matched whitening matrices, \(\mathbf{W}_{i},i=1,2,\ldots,5\). The data consists of 35 million symbols taken from QPSK modulated signals originating from five different radios. The training set consists of \(5\times 10^{5}\) symbols which is about 1.4% of the total data. The Majority Vote Classifier needs a voting scheme. Votes are generated by dividing the data into 72 blocks of \(5\times 10^{5}\) samples each. Each block generates one vote which is then tabulated over the entire signal length. The test blocks are drawn from an \"unknown\" source, corrupted by Gaussian noise and repeatedly projected on whitening matrices corresponding to each source. The Forstner-Moonen  
Figure 8: The signal constellation for EBEM signals suffers smearing due to varying amount of \(f_{d}T\), (**a,b**). Smearing is completely reversed after phase offset correction. (**a**) Limited phase offset, (**b**) Phase offset of multiples of \(2\pi\), (**c**) Constellation after phase and frequency offset correction. Figure 9: Six constellations from six different sources are corrected for phase and frequency offsets and used for training and testing. There is noise of unknown strength already in the data. Constellation similarities illustrate the difficulty of radiometric identification by feature extraction. distance is used to compute the mode function in (6) leading to the compilation of the confusion matrices. Before creating the confusion matrices, the behavior of the Forstner-Moonen distance measure needs to be studied. According to (3), as the process is increasingly whitened, the Forstner-Moonen distance between the whitened covariance matrix and the identity matrix is narrowed. The theoretical minimum distance is zero for white noise. To test for this behavior, two random variables with adjustable correlation coefficients are created and placed in a two-column matrix. The covariance of this matrix is calculated as a function of correlation values and the corresponding Forstner-Moonen distance is plotted. The results are plotted in Figure 3. As Figure 3a shows, the distance is an increasing function of correlation, reflecting that the covariance matrix is moving away from that of a white noise process for increasing correlation. This is expected.The second property of the Forstner-Moonen measure is that the unknown data is closer to a white noise process when whitened by its own whitening transformation than any other, hence matched whitening. To show this property, the data from Agilent is whitened by its own whitening matrix and then by the whitening matrix of Viasat EBEM. Distance calculations are performed over 40 blocks of data and plotted in Figure 3b. What stands out is that the Forstner-Moonen distance for the Agilent data is almost always less than that when the Viasat EBEM whitening matrix is used. This behavior is expected, meaning that a correct decision is made every time it happens. This count is essentially the basis for populating the confusion matrices over all sources. Following the above observations, the corresponding confusion matrices can now be computed and are shown in Table 2. The numbers indicate the percent of correct votes cast for each source over 72 frames of the test data. Note that the mode classifier in (6) looks for a plurality of the votes to pick a winner. It's a hard voting scheme. For example, Paradise has received only 77.1% of the votes but the unknown signal is still correctly classified to Paradise. Therefore, Table 2 indicates 100% correct classification. Confusion matrices can be used in a soft voting scheme as well by keeping the actual vote percentages. Next, we investigate the impact of smaller data sets and added noise above and beyond what is already in the data. The total sample size is now \(10^{7}\) which are broken into blocks of a quarter million samples each translating to less than 100 msec.