Hi Jason,
I shouldn't have hit the send button yesterday and went to bed right after.
Had a night of bad sleep; I realized my formulas only applied to the
case of real valued signals.
Your dongles won't be giving you cosines in complex baseband for a
signal they see at $f_\text{offset}$;
they will give you complex sinusoids
$e^{jf_{\text{offset},1}t}$ and $e^{jf_{\text{offset},2}t}$, respectively.
To be complete, there'd also be a phase offset, so it'd be
$e^{j(f_{\text{offset},1}t+\varphi_1)}$, and
$e^{j(f_{\text{offset},2}t+\varphi_2)}$.
Now, multiplication of these really just is addition of the exponents, so
$e^{j(f_{\text{offset},1}t+\varphi_1)}
e^{j(f_{\text{offset},2}t+\varphi_2)} =
e^{j((f_{\text{offset},1}+f_{\text{offset},1})t+\varphi_1+\varphi_2)}$
which means you'll only see the "sum frequency".
That's why you'd use the "multiply conjugate" block instead:
\documentclass{article} \usepackage[utf8x]{inputenc}
\usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb}
\usepackage{trfsigns} \DeclareMathOperator*{\argmin}{arg\,min}
\usepackage{tikz} \usepackage{circuitikz}
\usepackage[binary-units=true]{siunitx} \sisetup{exponent-product =
\cdot} \DeclareSIUnit{\dBm}{dBm} \newcommand{\imp}{\SI{50}{\ohm}}
\newcommand{\wrongimp}{\SI{75}{\ohm}} \pagestyle{empty} \begin{document}
\begin{align*} e^{j(f_{\text{offset},1}t+\varphi_1)}
\overline{e^{j(f_{\text{offset},2}t+\varphi_2)}} &=
e^{j(f_{\text{offset},1}t+\varphi_1)}
e^{-j(f_{\text{offset},2}t+\varphi_2)}\\ &=
e^{j(f_{\text{offset},1}t+\varphi_1)}
e^{j((-f_{\text{offset},2})t-\varphi_2)}\\
&=e^{j((f_{\text{offset},1}-f_{\text{offset},2})t+\varphi_1-\varphi_2)}
\end{align*} \end{document}
Regarding your splitter:
Usually, splitters don't introduce nonlinearities, so you should be fine.
Best regards,
Marcus
On 12.01.2016 19:47, Jason Matusiak wrote:
> Thanks Marcus, that helps a lot.
>
> Since I have to multiply the resulting offsets against each other, that
> means I will need to run a splitter from my sig-gen to the two dongles.
> Is there any concern that non-linearities in the two legs of the
> splitter would effect the results?
>
> Also, what should I do about the transition width on the LPF?
>
> Thanks for the thorough math explanation, that was a good lesson in what
> is going on.
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