Usefulness of Matrix Notation for Linear Periodically Time Variant Transformations I am quit puzzled about a notation I found in Gardner's 1986 book titled Introduction to Random Processes. It is in the chapter on Cyclostationary Processes at section 12.4 which pertains to Linear Periodically Time-Variant Transformations:


  (...) many modulation systems can be modelled as the scalar response
  of a multi-input LPTV transformation with stationary excitation. This
  includes amplitude modulation, phase and frequency modulation,
  quadrature amplitude modulation, pulse-amplitude modulation, all
  synchronous digital modulations such as phase-shift keying,
  frequency-shift keying, and so on. Consequently, the study of
  cyclostationarity is facilitated by general formulas that describes
  cyclic spectra in terms of the parameters of LPTV transformations.
  This includes cyclic spectra that are generated by LPTV
  transformations of processes that exhibit no cyclostationarity, as
  well as cyclic spectra that are transformed by LPTV transformations of
  processes that do exhibit cyclostationarity. Let us consider the LPTV
  transformation $$ Y(t) = \int_{-\infty}^\infty
\textbf{h}(t,u)\textbf{X}(u)du $$ for which $\textbf{X}(t)$ is a
  (column) vector excitation, $Y(t)$ is a scalar response, and
  $\textbf{h}(t, u)=\textbf{h}(t+T, u+T)$ is the (row) vector of
  impulse-response functions that specify the transformation.


The subject of LPTV Transformations is new for me and I don’t understand the usefulness of using non-scalars for $\textbf{h}(t, u)$ and $\textbf{X}(t)$.

Is this a commonly used notation?

Why should a matrix product be used?

I have been trying to make sense of this for some days now and I have really made no progress whatsoever. I hope someone here has an idea...

With thanks!
I don't have the book, so my answer unfortunately cannot be very detailed. However, just based on the excerpt, this could be useful if you are modulating a number of different data streams $\mathbf{X}(u)$ onto a single transmit signal. For example, this would be the case if you have a number of frequency bins that you will use for different data, as would be done for OFDM (orthogonal frequency devision multiplexed) transmissions or DMT (discrete multitone) as is used in DSL modems. I'm personally curious why $\mathbf{h}(t,u)$ is not a matrix and $Y(t)$ a vector, since it is perfectly plausible to have a system with multiple transmitting antennas.