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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!

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  • $\begingroup$ @MattL.: Thanks for pointing that out. That was a fault of citing block which is not doing great with equations... $\endgroup$
    – jojeck ♦
    Commented Oct 6, 2014 at 12:15
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    $\begingroup$ i wish we would bump the number of dimensions down one, so that we get something i can grok, and then maybe extend it back to the original number of dimensions. i might like to see it as $$ y(t) = \int\limits_{-\infty}^{\infty} h(t,u) x(t-u) \ du $$ $h(t,u)$ is the impulse response at time $u$ of an impulse applied at time $t$. then i gotta idea about what happens if if $h(t+T,u+T) = h(t,u)$ . you can break it up into a summation of inputs and a time-aliased impulse response $h(t)$. i think that's how it goes. $\endgroup$
    – robert bristow-johnson
    Commented Nov 7, 2014 at 2:17
  • $\begingroup$ probably it's more complicated than i had thought, there's the variation of $h(t,u)$ w.r.t. $t$ within a single period of length $T$. $\endgroup$
    – robert bristow-johnson
    Commented Nov 7, 2014 at 2:20

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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.

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