%
% IMS 2012 Conference example for Beamer
% Tibault Reveyrand - http://www.microwave.fr
%
% http://www.microwave.fr/LaTeX.html
% ---------------------------------------
%
% This example file comes from Polymtl template created by Manuel Vonthron
% Manuel Vonthron - <manuel DOT vonthron AT acadis DOT org>
%
%
\documentclass{beamer}
\hypersetup{pdfpagemode=FullScreen} % makes your presentation go automatically to full screen
\usepackage{textpos}
\usepackage{mathabx}
\usetheme[numbering={true}]{mtt}
\setbeamercovered{transparent=20}
\begin{document}
\title[TH2A-2]{A new method to measure pulsed RF time domain waveforms with a sub-sampling system}
%\subtitle{subtitle}
\author{T. Reveyrand$^1$ and Z. Popovi\'{c}$^2$}
\date{\today}
\institute{ \tiny{$^1$ XLIM - CNRS, UMR 6172, 123 Av. A. Thomas, 87060 Limoges Cedex, France} \\
\vspace{-0.2em}{\tiny tibo@xlim.fr} \\
$^2$ ECEE, University of Colorado at Boulder, 425 UCB, CO 80309, USA \\
\vspace{-0.2em}{\tiny zoya@colorado.edu} \\
}
\begin{frame}[plain]
\titlepage
\end{frame}
\begin{frame}{Table of contents}
\tableofcontents
\end{frame}
%\section*{Introduction}
\begin{frame}{Introduction}
\begin{columns}
\begin{column}{0.55\paperwidth}
Designing very high efficiency Power Amplifiers requires transistors level characterizations such as :\\
\begin{itemize}
\item Large-signal measurements ;
\item RF time-domain measurements ;
\item Pulsed mode for radar applications ;
\end{itemize}
\end{column}
\begin{column}{0.40\paperwidth}
\vspace{-15pt}
\begin{figure}
\includegraphics[height=6.8cm]{fig/intro.pdf}
\end{figure}
\end{column}
\end{columns}
\end{frame}
\section{Bench}
\subsection{Setup}
\begin{frame}{Large-Signal Measurement Setup}
\begin{figure}
\includegraphics[height=5.5cm]{fig/bench.pdf}
\end{figure}
\end{frame}
\subsection{Calibration}
\begin{frame}{Calibration Procedure (CW)}
\begin{columns}
\begin{column}{0.45\paperwidth}
\begin{itemize}
\item SOLT
\item Absolute Power
\item Absolute Phase
\end{itemize}
\vspace{0.5cm}
\scalebox{0.8}{
$\begin{pmatrix}a1\\b1\\a2\\b2\end{pmatrix}=\left\| K \right\|.e^{j.\phi}.\begin{bmatrix}1&\beta_1&0&0\\\gamma_1&\delta_1&0&0\\0&0&\alpha_2 & \beta_2\\0&0&\gamma_2&\delta_2\end{bmatrix}.\begin{pmatrix}R1\\R2\\R3\\R4\end{pmatrix}$
}
\end{column}
\begin{column}{0.50\paperwidth}
\begin{figure}
\includegraphics[height=5.5cm]{fig/calibration.pdf}
\end{figure}
\end{column}
\end{columns}
\end{frame}
\section{Receivers}
\begin{frame}{Receivers for CW measurements}
\begin{itemize}
\item NVNA approach : frequency domain
\begin{figure}
\includegraphics[height=2cm]{fig/receiver_NVNA.pdf}
\end{figure}
\item LSNA approach : subsampling
\begin{figure}
\includegraphics[height=3.2cm]{fig/receiver_LSNA.pdf}
\end{figure}
\end{itemize}
\end{frame}
\subsection{Mixer based (NVNA)}
\begin{frame}{Mixer based pulsed measurements (NVNA)}
\begin{figure}
\includegraphics[height=6.5cm]{fig/NVNA.pdf}
\end{figure}
\begin{textblock*}{4cm}[0,0](200pt,-200pt)
$P_{pulse}=P_{meas}.{\left(\frac{T}{\tau}\right)}^2$
\end{textblock*}
\end{frame}
\subsection{Sampler based (LSNA)}
\begin{frame}{Sampler based pulsed measurements (LSNA)}
$f_{RF}=1.5 GHz$ \hfill $\tau_{pulse}=10\mu s$\hfill $T_{IF}=8\mu s$
\begin{figure}
\includegraphics[height=2.5cm]{fig/fig1.pdf}
\end{figure}
%\pause
\begin{figure}
\includegraphics[height=3.5cm]{fig/fig2d.pdf}
\end{figure}
\end{frame}
\section{Algorithm}
\begin{frame}{About inner-products}
According to a dictionary
$$\mathcal{D} = {\left\{\psi_k\right\}}_{k\in\Gamma} $$
$x\left(t\right)$ can be represented by its inner-products coefficients \\
$$\left\langle x,\psi_k\right\rangle = \int_{-\infty}^{+\infty}x\left(t\right).\overline\psi_k\left(t\right).dt$$
If $x\left(t\right)$ is sparse in $\mathcal{D}$ then
$$x\left(t\right)\approx\sum_{k\in\Lambda\subset\Gamma}\left\langle x,\psi_k\right\rangle . \psi_k$$
\end{frame}
\subsection{Frequency Analysis}
\begin{frame}{What is a Fourier Transform ?}
\begin{columns}
\begin{column}{58mm}
\begin{itemize}
\item $ \mathcal{D}={\left\{\psi_f\left(t\right)=e^{j.2.\pi.f.t}\right\}}$
\item $X\left(f\right)=\left\langle x,\psi_f\right\rangle$
\end{itemize}
\begin{itemize}
\item $x\left(t\right)=\int_{-\infty}^{+\infty} X\left(f\right) e^{j.2.\pi.f.t} df$
\item $X\left(f\right)=\int_{-\infty}^{+\infty} x\left(t\right) e^{-j.2.\pi.f.t} dt$
\end{itemize}
\begin{itemize}
\item $x\left(t\right)\approx\sum_{k} X\left(k.f_0\right) e^{j.2.\pi.k.f_0.t} $
\end{itemize}
\vspace{10pt}
Standard LSNA uses boxcar window
\end{column}
\begin{column}{58mm}
Projection basis :
\begin{figure}
\includegraphics[height=4.5cm]{fig/fourier.pdf}
\end{figure}
\end{column}
\end{columns}
\end{frame}
\subsection{Time-Frequency Analysis}
\begin{frame}{The Short Time Fourier Transform }
Rectangular STFT is well suited for harmonic analysis
\vspace{10pt}
\begin{columns}
\begin{column}{58mm}
\begin{itemize}
\item $\mathcal{D}=\left\{\psi_{k,\tau}\left(t\right)\right\}$
\item $\psi_{k,\tau}\left(t\right) = P_k.\psi_k\left(t-\tau\right)$
\end{itemize}
\begin{itemize}
\item $\psi_k\left(t\right)~=~\Pi\left(f_0.t\right) e^{j.2.\pi.k.f_0.t}$
\item $ P_k=f_0.e^{j.2.\pi.k.f_0.\tau}$
\end{itemize}
\begin{itemize}
\item $X\left(k.f_0,\tau\right)=\overline P_k.x\left(t\right)\convolution\overline\psi_k\left(t\right) $
\end{itemize}
\end{column}
\begin{column}{58mm}
Projection basis :
\begin{figure}
\includegraphics[height=4.5cm]{fig/psi.pdf}
\end{figure}
\end{column}
\end{columns}
\begin{center}
\fbox{$ X\left(k.f_0,\tau\right)=\overline P_k.\mathcal{F}^{-1}\left\{ X\left(f\right).\overline\Psi_k\left(f\right) \right\}$}
\end{center}
\end{frame}
\begin{frame}{LSNA software modifications}
\begin{columns}
\begin{column}{58mm}
Standard procedure
\begin{figure}
\includegraphics[height=3cm]{fig/algo1.pdf}
\end{figure}
%\pause
\vspace{-10pt}
\begin{figure}
\includegraphics[height=2.5cm]{fig/fig2d.pdf}
\end{figure}
\end{column}
\begin{column}{58mm}
New procedure
\begin{figure}
\includegraphics[height=6cm]{fig/algo2.pdf}
\end{figure}
\end{column}
\end{columns}
\end{frame}
\section{Results}
\begin{frame}{Experimental view of the algorithm}
$f_{RF}=1.5 GHz$ \hfill $\tau_{pulse}=10\mu s$\hfill $T_{\psi}=8\mu s$\hfill $k\in\left\{1,2,3,4\right\}$
\begin{figure}
\includegraphics[height=6.5cm]{fig/fig3_c.pdf}
\end{figure}
\end{frame}
\begin{frame}{LSNA pulsed measurements on a PA ($T=100\mu s$)}
$f_{RF}=1.5 GHz$ \hfill $T_{\psi}=8\mu s$ \hfill $k\in\left\{1,2,3,4\right\}$
\begin{figure}
\includegraphics[height=6.5cm]{fig/example.pdf}
\end{figure}
\begin{textblock*}{20mm}[0,0](100pt,-185pt)$\tau=10 \mu s$\end{textblock*}
\begin{textblock*}{20mm}[0,0](250pt,-185pt)$\tau=50 \mu s$\end{textblock*}
\begin{textblock*}{20mm}[0,0](100pt,-95pt)$\tau=100 \mu s$\end{textblock*}
\begin{textblock*}{20mm}[0,0](250pt,-95pt)$CW$\end{textblock*}
\end{frame}
\appendix
%\section*{Conclusion}
\begin{frame}{Conclusion}
\begin{itemize}
\item Standard LSNA harware can measure pulsed RF
\item Minimal software modification (FFT procedure)
\item Compatible with CW and pulsed signals
\item Adaptive method
\begin{itemize}
\item No trigger
\item Pulse's width and period ($\tau$, $T$) not required
\end{itemize}
\item Both 'Average' and 'Envelope Transcient' modes availables
\end{itemize}
Future work :
\begin{itemize}
\item Narrow pulses (double aliasing)
\item Other types of modulation
\end{itemize}
\end{frame}
%\begin{frame}[allowframebreaks]{References}
% \tiny
% \usebibitemtemplate{\color{structure}\insertbiblabel}
% \usebibliographyblocktemplate{\color{structure}}{\color{black}}{\color{structure!75}} {\color{structure!75}}
% \bibliographystyle{unsrt}
% \bibliography{CU-Boulder.bib}
%\end{frame}
\end{document}
