Priliminaries

Before we introduce the LS framework, we need to show the key notations and basic knowledge that will be used in following sections.

Key Notations

../_images/NotationTable.png

SSVEP reference and template signals

The well-known SSVEP reference signal is constructed as a set of sine-cosine signals, which can be presented as

\[\begin{split}\mathbf{Y}_m=\left[ \begin{array}{c} \sin\left(2\pi f_m t+\theta_m\right)\\ \cos\left(2\pi f_m t+\theta_m\right)\\ \vdots\\ \sin\left(2\pi N_h f_m t+N_h\theta_m\right)\\ \cos\left(2\pi N_h f_m t+N_h\theta_m\right) \end{array} \right]^T,\end{split}\]

where \(f_m\) and \(\theta_m\) denote the frequency and phase of the \(m\)-th stimulus.

The SSVEP template signals are normally computed as the average of signals over all calibration trials, i.e.,

\[\overline{\mathbf{X}}_m=\frac{1}{N_t}\sum_{n=1}^{N_t}\mathbf{X}_{n,m}.\]

QR factorization

The QR factorization of a square matrix \(\mathbf{A}\) decomposes \(\mathbf{A}\) into the production of an orthonormal matrix \(\mathbf{Q}^{(\mathbf{A})}\), and an upper triangular matrix \(\mathbf{R}^{(\mathbf{A})}\), which can be expressed as

\[\mathbf{A}=\mathbf{Q}^{(\mathbf{A})}\mathbf{R}^{(\mathbf{A})}.\]

To simplify expressions in following sections, we define a new operation of \(\mathbf{A}\):

\[\,\mathcal{P}\!\left(\mathbf{A}\right) = \mathbf{A}\left(\mathbf{A}^T\mathbf{A}\right)^{-1}\mathbf{A}^T=\mathbf{Q}^{(\mathbf{A})}\left(\mathbf{Q}^{(\mathbf{A})}\right)^T.\]

Singular value decomposition (SVD)

The SVD of a matrix \(\mathbf{A}\) decomposes \(\mathbf{A}\) into the production of an orthonormal matrix \(\mathbf{U}^{(\mathbf{A})}\), a rectangular diagonal matrix \(\mathbf{D}^{(\mathbf{A})}\), and another orthonormal matrix \(\mathbf{V}^{(\mathbf{A})}\), which can be

\[\mathbf{A} = \mathbf{U}^{(\mathbf{A})}\mathbf{D}^{(\mathbf{A})}\left(\mathbf{V}^{(\mathbf{A})}\right)^T.\]

Let \(\mathbf{\Sigma}^{(\mathbf{A})}=\mathbf{A}^T\mathbf{A}=\mathbf{V}^{(\mathbf{A})}\left(\mathbf{D}^{(\mathbf{A})}\right)^2\left(\mathbf{V}^{(\mathbf{A})}\right)^T\), we define a new operation to simplify expressions in following section:

\[\,\mathcal{S}\!\left(\mathbf{A}\right) = \left(\mathbf{\Sigma}^{(\mathbf{A})}\right)^{-\frac{1}{2}}=\mathbf{V}^{(\mathbf{A})}\left(\mathbf{D}^{(\mathbf{A})}\right)^{-1}\left(\mathbf{V}^{(\mathbf{A})}\right)^T.\]