## 3 Theory  
The application of SVD allows for the decomposition of the dimensionality of a matrix into fundamental components of the original signal, comparable to a form of compressed sensing. The assumptions made when performing SVD processing are that the signal of interest can be deconstructed into a linear combination of a few key components from a defined dictionary [37]. This methodology is applicable to micro-Doppler classification, as the spectrogram matrix is a suitable candidate for sparse representation due to only a few key features containing a significant proportion of the vital information. SVD signal processing enables a sparse representation of this data through subspace base analysis. The SVD as well as eigenvalue decomposition (EVD) processing techniques  
Figure 3: Image of Soprano radar deployed outside of building. Figure 2: Geometry of the experimental setup showing individual walking up and down centrally within living room space and radar deployed at 43 cm distance. have been previously used for subspace estimation, despite their computationally demanding algorithms. Modifications to these techniques include sliding window adaptive implementations [38] and a K-SVD representation [39] of the model. For the analysis within this paper, a simple representation of the SVD model is used to analyse the real data produced by the radar in order to extract features to input into a classifier. The advantage of the SVD processing technique is that physical empirical features from the micro-Doppler signature, such as period and bandwidth, do not have to be manually extracted prior to classification, making the processing automated and easily repeatable. This theorem is based on the concept that any matrix \(\mathbf{A}\) is defined as,  
\[\mathbf{A}\in\mathbf{R}^{m\times n} \tag{1}\]  
When matrix \(A\) has rank of one, the matrix can be represented by,  
\[\mathbf{A}=u\sigma v^{T} \tag{2}\]  
where \(u\in\mathbf{R}^{m}\), \(v\in\mathbf{R}^{n}\), and \(\sigma>0\). This can be expanded to when matrix \(\mathbf{A}\) has an arbitrary rank of \(r\), where the matrix can be represented by a sum or rank one matrices. \[\mathbf{A}=\sum_{i=1}^{r}\sigma_{i}u_{i}v_{i}^{T} \tag{3}\]  
where \(r\) is the rank of the matrix, \(u_{i}\) and \(v_{i}\) values are all mutually orthogonal and the \(\sigma_{i}\) values are all positive and labelled as singular values of the matrix \(\mathbf{A}\). It is the matrices \(v\) and \(u\) that have been evaluated for each of the micro-Doppler datasets that are available. The \(\sigma\) values are only scaling values for the unity matrices \(u\) and \(v\), and are, hence, not used as part of the classification process. Features from within these matrices are extracted in order to perform a classification process. The classifiers that have been applied to the data are Discriminant Analysis (DA), which is a form of a Bayesian classifier, and Random Forest (RF) classifier types. The DA classifier is based on the assumption that the features' vectors inputs are both Gaussian distributed and statistically independent. The two categories of DA classifiers that have been applied are linear and quadratic; for the linear case, the model has the same covariance matrix for each class, only the means vary, while for the quadratic case, both means and covariance of each class vary [40; 41]. During the training process the mean and variance of these feature vectors are estimated as part of the classification process. Then, the posterior probability of each sample under test belonging to each class can be evaluated in order to assign the sample to a given class which gives the highest probability. This is achieved by minimizing the expected classification cost, which is defined in [4  
With \(\mu_{k}\) representing the mean and \(\Sigma_{k}\) representing the covariance.