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
; they will give you complex sinusoids
and 
, respectively.To be complete, there'd also be a phase offset, so it'd be
, and 
.Now, multiplication of these really just is addition of the exponents, so
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}](pngXVnMVrwIbK.png "\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.