# Radiometric Identification of Signals by Matched Whitening Transform
## 2 Framework for Radiometric Identification
### Development of a Whitening Measure  
There are several issues with tying the unknown data to its own whitening matrix. First, the IQ components of the real data are already quite decorrelated so whitening may not bring significant additional decorrelation. Second, the subspace defied in (1) is created offline from the training data. However, the test data are different even if coming from the same population as the training data. If the data different than the training set are used, the whitening of the data will be approximate. The core property is that the covariance matrix of the unknown data will resemble the identity matrix if projected on its own subspace more than on any other. Third, how to measure \"whiteness\". This is a problem in covariance matrix matching [49]. There are any number of metrics to measure the distances between two symmetric, positive definite covariance matrices. They include KL divergence, Euclidean distance, squared Frobenius norm, Bhattacharyya distance, Bregman matrix divergence and LogDet [50], among others. In this work we use the Forstner-Moonen metric [49] as a similarity measure of two covariance matrices. As a point of reference, the well-cited Correlation Matrix  
Figure 1: The “unknown” data, shown in black, are repeatedly projected on the spaces spanned by the eigenvectors of other groups. When the principal axis of the projected data aligns with the reference axis, a match is declared. In this example, the source of the unknown data is the group shown in black. (**a**) Three multivariate normal data sets simulating different signal sources, (**b**) Black data is whitened by the red whitening matrix, (**c**) Black data is whitened by the green whitening matrix, (**d**) Black data is whitened by its own whitening matrix. Figure 2: Radiometric identification by whitening transformations. The branch with the most whitened data reveals the source of the unknown signal. Distance(CMD) metric [51] and the Kullback-Leibler measures are studied. There is no one definition for similarity but three are monotonic with correlation and hence are valid measures. We have superimposed CMD, KL and Forstner-Moonen plots for comparison. The graphs appear later on in Figure 3a. As expected, the pairwise distance increases with increasing correlation, meaning that the covariance matrix of correlated variables is at farther distances from a diagonal covariance matrix. It is noteworthy that the KL measure is virtually coincident with the Forstner-Moonen metric hence justifying its use as a similarity index. Let \(\mathbf{A}\) and \(\mathbf{B}\) be the reference and measured covariance matrices. The proposed distance measure is defined by,  
\[d(\mathbf{A},\mathbf{B})=\sqrt{\sum_{i=1}^{n}ln^{2}\lambda_{i}(\mathbf{A}, \mathbf{B})} \tag{3}\]  
where \(\lambda_{i}(\mathbf{A},\mathbf{B})\), the joint eigenvalues of \(\mathbf{A}\) and \(\mathbf{B}\), are the roots of \(|\lambda\mathbf{A}-\mathbf{B}|=0\). In the context of the whitening transform, the reference covariance matrix is the identify matrix \(\mathbf{A}=\mathbf{I}\) and \(\mathbf{B}=cov(\mathbf{Y}_{i})\) is the covariance matrix of the unknown data whitened by \(\mathbf{W}_{i}\). Therefore, the joint eigenvalues reduce to simply the eigenvalues of the measured covariance matrix \(\mathbf{B}\) of the unknown data. The classifier built on (3) is a Majority, or Plurality, Vote Classifier [52] governed by rules \(h_{1},h_{2},\ldots,h_{m}\). The rules are membership functions. Given the measurements \(\mathbf{X}_{i}\) from an unknown source,  
\[h_{j}(\mathbf{X}_{i}) =1 \mathbf{X}_{i}\in c_{j} \tag{4}\] \[=0 \mathbf{X}_{i}\
ot\in c_{j}\]  
The membership functions work off the Forstner-Moonen distance as follows,  
\[h_{j}(\mathbf{X}_{i})= \begin{cases}1&d(\Sigma_{j},I)<d(\Sigma_{k},I)\quad \forall k\
eq j\\ 0&\text{otherwise}\end{cases}  \tag{5}\]  
Figure 3: Two illustrations of the effectiveness of the Forstner-Moonen distance.