prob_passband.tex

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\begin{document}

\title{Problems:  Passband Modulation}
\author{Prof.\ Sundeep Rangan}
\date{}

\maketitle

\begin{enumerate}


\item \emph{Passband conversion in time-domain.}  Suppose that the
complex baseband signal in time-domain is
\[
    u(t) = (a+bj)e^{-\alpha t}, \quad t \geq 0,
\]
where $a$, $b$, $\alpha$ are real.
\begin{enumerate}[(a)]
  \item What at the I and Q components?
  \item What is the real passband signal after upconversion with a carrier frequency $f_c$?
\end{enumerate}

\item \emph{Passband conversion in frequency domain.}
Let $u(t)$ be a complex baseband signal with the real and imaginary parts of
 the spectrum (Fourier Transform) shown in
Fig.~\ref{fig:Ubb}.  The constants are $f_0 = $ 5 MHz, $f_1 = $ 10 MHz, $A=8$ and $B=10$.

\begin{figure}[h]
\center
\begin{tikzpicture}[xscale=2,yscale=1]
    \pgfmathsetmacro{\fa}{-0.5}
    \pgfmathsetmacro{\fb}{1}
    \pgfmathsetmacro{\fm}{2}
    \pgfmathsetmacro{\A}{1}
    \pgfmathsetmacro{\tic}{0.2}

    % Draw the axes
    \draw [->] (-\fm,0) -- (\fm,0) node [right] {$f$ (MHz)};
    \draw [->] (0,-\tic) -- (0,1.3) node [right] {$\Real(U(f))$};
    \draw [-] (\fa,\tic) -- (\fa,-\tic) node [below] {$-f_0$};
    \draw [-] (\fb,\tic) -- (\fb,-\tic) node [below right] {$f_1$};
    \draw [-,dashed] (\fb-\tic,\A) -- (\fb+\tic,\A) node [right] {$A$};

    % Draw Real U(f)
    \draw [ultra thick,blue,-] (-\fm+0.2,0) -- (\fa,0) -- (\fa,\A) --
         (\fb,\A) -- (\fb,0) -- (\fm-0.2,0);

\end{tikzpicture}

\begin{tikzpicture}[xscale=2,yscale=1]
    \pgfmathsetmacro{\fa}{-0.5}
    \pgfmathsetmacro{\fb}{1}
    \pgfmathsetmacro{\fm}{2}
    \pgfmathsetmacro{\A}{1.3}
    \pgfmathsetmacro{\tic}{0.2}

    % Draw the axes
    \draw [->] (-\fm,0) -- (\fm,0) node [right] {$f$ (MHz)};
    \draw [->] (0,-\tic) -- (0,1.7) node [right] {$\Imag(U(f))$};
    \draw [-] (-\fb,\tic) -- (-\fb,-\tic) node [below] {$-f_1$};
    \draw [-] (\fb,\tic) -- (\fb,-\tic) node [below right] {$f_1$};
    \draw [-,dashed] (-\tic,\A) -- (\tic,\A) node [right] {$B$};

    % Draw Imag U(f)
    \draw [ultra thick,blue,-] (-\fm+0.2,0) -- (-\fb,0) -- (0,\A)
          -- (\fb,0) -- (\fm-0.2,0);

\end{tikzpicture}

\caption{Real and imaginary parts of complex baseband signal $U(f)$} \label{fig:Ubb}
\end{figure}

\begin{enumerate}[(a)]


\item Suppose that we create a real passband signal $u_p(t) = \Real(u(t)e^{2\pi if_c t})$
for a carrier frequency $f_c = 800$ MHz.  Draw the spectrum of $U_p(f)$.  Show
 both the real and imaginary parts and show both the positive and negative frequencies.

\item Is $u(t)$ an energy signal or power signal?  What is its energy or power (in linear scale)?
Leave your answer in terms of $A$, $B$, $f_0$ and $f_1$.  You do not need to
convert to dB scale.

\item A receiver attempts to downcovert the signal with a two step process:
\[
    v(t)= 2u(t)e^{-2\pi i f_c t}, \quad \hat{u}(t) = h_{LPF}(t) * v(t),
\]
where $h_{LPF}(t)$ has a frequency response,
\[
    H_{LPF}(f) = \begin{cases}
        C & \mbox{if } |f| < f_{LPF} \\
        0 & \mbox{if } |f| \geq f_{LPF}.
        \end{cases}
\]
For what values of $C$ and $f_{LPF}$ is $\hat{u}=u(t)$?
\end{enumerate}

\item \emph{Baseband equivalent filter.}
Consider a communication system with three steps:
\begin{itemize}
\item A complex baseband signal $u(t)$ is upconverted $u_p(t)=\Real(u(t)e^{2\pi if_ct})$
for some $f_c$.
\item The real passband channel is passed through a linear filter,
\[
    \frac{dy_p(t)}{dt} = b u_p(t)- ay_p(t),
\]
with constants $a$ and $b>0$.
\item The received signal is downconverted, $v(t)=2y_p(t)e^{-2\pi i f_ct}$ and
$y(t)=h_{\rm LPF}(t)*v(t)$ where $h_{\rm LPF}(t)$ is an ideal low-pass filter.
\end{itemize}
\begin{enumerate}[(a)]
\item What is the real passband frequency response, $H_p(f) = \frac{Y_p(f)}{U_p(f)}$?

\item What is the effective baseband frequency response $H(f) = \frac{Y(f)}{U(f)}$?

\item Find $a_1$ and $b_1$ such that
\[
    \frac{dy(t)}{dt} = b_1 x(t)- a_1y(t).
\]

\item Suppose that $2\pi f_c \gg a$, what is the power gain of $H(0)$ in dB?
\end{enumerate}

\item \emph{PSD and RX filtering.}
Suppose that a real passband signal has two components:
\[
    x(t)=x_0(t)+x_1(t),
\]
where $x_0(t)$ is a desired signal, and $x_1(t)$ is an interfering signal.  They have PSD
$S_i(f)=A_i\Rect((f-f_i)/W_i)$, $i=0,1$ with parameters:
\begin{itemize}
\item Desired signal: $f_0 = 2.50$ GHz, $W_0 = $ 20~MHz, total receive power $P_0$ = -100~dBm.
\item Interfering signal: $f_1 = 2.53$ GHz, $W_1 = $ 10~MHz, total receive power $P_1$ = -80~dBm.
\end{itemize}
\begin{enumerate}[(a)]
\item Find $A_i$ from $P_i$ using reasonable approximations.  State the units of $A_i$.
\item Draw $S_0(f)$ and $S_1(f)$.
\item A signal is downconverted with mixing $v(t)=2x(t)e^{2\pi i f_ct}$ and $u(t)=h(t)*v(t)$.
Find $f_c$ and a filter magnitude response $|H(f)|^2$ such that:
\begin{itemize}
\item The component from desired signal is centered at 0 and amplified to -60 dBm.
\item The component from interfering signal attenuated to below -110 dBm.
\end{itemize}
There is no single correct answer.  Draw $|H(f)|^2$ and the PSD of $u(t)$.
\end{enumerate}


\begin{figure}
  \centering
  \includegraphics[width=10cm]{transceiver_60GHz}
  \caption{Schematic diagram of a SiGe RF front-end from Floyd, \emph{et.~al}, 2005. }\label{fig:transceiver60}
\end{figure}


\item \emph{FFT bins.}  A RX receives a complex baseband signal $x(t)$ and samples
$x(t)$ at a sampling rate $f_s=$\,\SI{20}{MHz}.  It then takes an $N$-point FFT
$X[k]$, $k=0,1,\ldots,N-1$ with $N=1024$.  For each of the following signals $x(t)$, indicate
which bin $k$ will   $|X[k]|^2$ have the maximum value?
\begin{enumerate}[(a)]
  \item $x(t)$ is a complex exponential with frequency $f_0=$\,\SI{4}{MHz}.
  \item $x(t)$ is a complex exponential with frequency $f_0=$\,\SI{-4}{MHz}.
  \item A real passband signal is $x_p(t) = A\cos(2\pi ft)$ with $f=$\,\SI{2.002}{GHz}.
  Then $x(t)$ comes from downconverting $x_p(t)$ with carrier $f_c=$\,\SI{2}{GHz}.
  \item $x(t)$ comes from downconverting $x_p(t)$ as in part (c).  But, the
  carrier frequency $f_c$ is nominally \SI{2}{GHz}, but with an error of \SI{+10}{ppm}
  too high.
\end{enumerate}


\item \emph{Circuit implementations.}  Fig.~\ref{fig:transceiver60} shows the circuit schematic
of the RF front-end a SiGe (silicon Germanium) bipolar transceiver presented in:
\begin{quote}
Floyd, Brian A., et al. "SiGe bipolar transceiver circuits operating at 60 GHz." \emph{IEEE journal of solid-state circuits} 40.1 (2005): 156-167.
\end{quote}
Feel free to look up terms in the paper or any other source to answer the following questions:
\begin{enumerate}
\item Which block is the local oscillator?
\item What is the carrier frequency?  Why?
\item What is the role of the "Differential branch line directional coupler"?
\item How is the frequency tuned?
\end{enumerate}

\end{enumerate}

\end{document}