Combination of discrete/continuous signals

[MathematicalPhysicist]

I am looking for literature in theoretical engineering that covers a topic of a signal which is both discrete and continuous.

For example $x[n,t) = t/n$ where $t$ ranges over $[0,\infty)\cap \mathbb{R}$ and $n$ is discrete, i.e takes values in $\mathbb{Z}$.

I believe that this isn't covered in the usual books of Oppenhiemer, but I may be wrong.
Thanks in advance!

 

[jasonRF]

It isn't covered in the typical signals and systems books I am familiar with, but most things you would want to do with such a signal should be pretty straightforward to write down. For example, if $x$ is an input to a linear time-invariant system with impulse response $h[n,t)$, then the output would be
$$
y[n,t) = \int_{-\infty}^\infty \sum_{m=-\infty}^{\infty} h[m,\tau) x[n-m,t-\tau) \, d\tau
$$
Likewise, to get into the frequency domain you can do a combination of a 1D continuous Fourier transform and a 1D discrete time Fourier transform. etc.

If you are not comfortable with that, one direct approach is to treat it like a 2D continuous signal defined by
$$
x_c(v,t) = \sum_{n=-\infty}^\infty x[n,t) \delta(v-n).
$$
Then if you have a continuous linear time-invariant system with impulse response $h_c(v,t)$, the output is
$$
\begin{eqnarray*}
y(v,t) & = & \int_{-\infty}^\infty \int_{-\infty}^\infty h_c(\nu,\tau) x_c(v-\nu,t-\tau) \, d\tau \, d\nu \\
& = & \int_{-\infty}^\infty \int_{-\infty}^\infty h_c(\nu,\tau) \sum_{m=-\infty}^\infty x[m,t-\tau) \delta(v-\nu-m) \, d\tau \, d\nu \\
& = & \int_{-\infty}^\infty h_c(v-m,\tau) \sum_{m=-\infty}^\infty x[m,t-\tau) \, d\tau \\
& = & \int_{-\infty}^\infty \sum_{m=-\infty}^{\infty} h_c(m,\tau) x[v-m,t-\tau) \, d\tau ,
\end{eqnarray*}
$$
which is of course only defined when $v$ is an integer. You can of course take 2D continuous Fourier transforms of $x_c$ as well. etc.

Is there some particular application or messy situation you are looking at?

jason

 

[MathematicalPhysicist]

@jasonRF not anything in particular.
I just wonder what has already been done in this topic of combined signals.