\documentstyle[11pt]{article}
% frob with the margins
\addtolength{\topmargin}{-1in}
\addtolength{\footskip}{-1in}
\addtolength{\textheight}{2in}
\addtolength{\oddsidemargin}{-.5in}
\addtolength{\evensidemargin}{-.5in}
\addtolength{\textwidth}{1in}
% no page numbers
\pagestyle{empty}
% counter for page numbers
\newcounter{pg}
\setcounter{pg}{2}

% DEFINE THE PROBLEM SET NUMBER
\def\psnum{7}

% title command
\newcommand{\mytitle}[1]{\begin{center} {\Large 6.003 Problem Set \psnum --
Juliet Bourne -- #1 -- TA R. Kim} \\ \rule{6in}{.5mm} \end{center}}
% problem command
\newcommand{\prob}[1]{\vspace{.15in}{\noindent\large\bf Problem #1} \par
\vspace{-.1in} \noindent\rule{2in}{.1mm}}
% subproblem command
\newcommand{\subprob}[1]{\vspace{.15in} \noindent{\large\bf #1.}}
% equations commands
\newcommand{\beq}{\vspace*{-.3in}\begin{eqnarray*}}
\newcommand{\eeq}{\end{eqnarray*}}
% newpage setup command 
\newcommand{\mynewpage}{\newpage\noindent
\begin{center}
{\Large 6.003 Problem Set \psnum -- Juliet Bourne -- {\bf Page \thepg}}
\rule{6in}{.5mm}
\end{center}
\addtocounter{pg}{1}}
% see-page command
\newcommand{\seepage}{\noindent See page \rule{.2in}{.1mm} for the
computer output.}

\begin{document}

\mytitle{4/25/90}

\prob{13.1 CSAS}

\subprob{a}

\beq
{\cal X}(f) & = & \int^\infty_{- \infty} x(t) cas (2\pi ft) dt \\
\mbox{} & = & \int^\infty_{- \infty} x(t) \cos (2\pi ft) dt + 
\int^\infty_{- \infty} x(t) \sin (2\pi ft) dt \\
\mbox{} & = & \frac{1}{2} \int^\infty_{- \infty} x(t) (e^{j2\pi ft} + 
e^{-j2\pi ft}) dt + \frac{1}{2j} \int^\infty_{- \infty} x(t) (e^{j2\pi ft} - 
e^{-j2\pi ft}) dt \\
\mbox{} & = & \frac{1}{2} (1 - j) \int^\infty_{- \infty} x(t) e^{j2\pi ft} dt 
+ \frac{1}{2} (1 + j) \int^\infty_{- \infty} x(t) e^{-j2\pi ft} dt 
\eeq

\subprob{b} i)

\beq 
x(t) & = & \delta (t - \tau) \\
{\cal X}(f) & = & \int^\infty_{- \infty} \delta (t - \tau) cas (2\pi ft) dt \\
\mbox{} & = & cas (2\pi f\tau)
\eeq

\vspace{1in}

\noindent ii)

\beq
x(t) & = & e^{-\alpha t} u(t) \\
{\cal X}(f) & = & \int^\infty_{- \infty} e^{-\alpha t} u(t) cas (2\pi ft) dt \\
\mbox{} & = & \int^\infty_0 e^{-\alpha t} cas (2\pi ft) dt \\
\mbox{} & = & \int^\infty_0 e^{-\alpha t} \cos (2\pi ft) dt + 
\int^\infty_0 e^{-\alpha t} \sin (2\pi ft) dt \\
\mbox{} & = & \frac{1}{2} \int^\infty_0 e^{-\alpha t} (e^{j2\pi ft} + 
e^{-j2\pi ft}) dt + \int^\infty_0 e^{-\alpha t} (e^{-j2\pi ft} - e^{j2\pi
ft}) dt \\
\mbox{} & = & \frac{1}{2} \int^\infty_0 e^{(-\alpha + j2\pi f)t} +
\frac{1}{2} \int^\infty_0 e^{(-\alpha - j2\pi f)t} + \frac{1}{2}
\int^\infty_0 e^{(-\alpha - j2\pi f)t} - \frac{1}{2} \int^\infty_0
e^{(-\alpha + j2\pi f)t} \\
{\cal X}(f) & = & \int^\infty_0 e^{(-\alpha - j2\pi f)t} dt \\
\mbox{} & = & \frac{1}{-\alpha - j2\pi f} \left[ e^{(-\alpha - j2\pi
f)t} \right]^\infty_0 \\
\mbox{} & = & \frac{1}{-\alpha -j2\pi f}
\eeq

\mynewpage

\subprob{c}

\beq
y(t) & = & x(t) * h(t) \\
\mbox{} & = & \int^\infty_{- \infty} x(\tau) h(t - \tau) d\tau \\
{\cal Y}(f) & = & \int^\infty_{- \infty} \left( \int^\infty_{- \infty}
x(\tau) h(t - \tau) d\tau \right) cas (2\pi ft) dt \\
\eeq

\prob{13.7 CSAS}

\subprob{a}

\beq
X(f) & = & \int^\infty_{- \infty} e^{j2\pi f_o t x(t)} e^{-j2\pi
ft} dt \\
\mbox{} & = & x(t) e^{- j2\pi (f - f_o)t} dt \\
\mbox{} & = & X(f - f_o)
\eeq

\subprob{b}

\beq
x(t) = 1 & \rightarrow & X(f) = \delta (f) \\
e^{j2\pi f_o t} x(t) = e^{j2\pi f_o t} & \rightarrow & X(f - f_o) =
\delta (f - f_o) \\
x(t) & = & \int^\infty_{- \infty} X(f) e^{-j2\pi ft} dt \\
\mbox{} & = & \int^\infty_{- \infty} \delta (f - f_o) e^{-j2\pi ft} dt \\
\mbox{} & = & e^{-j2\pi (- f_o) t} \\
x(t) & = & e^{j2\pi f_o t}
\eeq

\subprob{c}

\beq
F\{\sin 2\pi f_ot\} & = & \frac{1}{2j} F\{e^{j2\pi f_ot}\} - e^{j2\pi
f_ot}\} \\ 
\mbox{} & = & \frac{1}{2j} F\{e^{j2\pi f_ot}\} - \frac{1}{2j}
F\{e^{-j2\pi f_ot}\} \\
\mbox{} & = & \frac{1}{2j} \delta (f - f_o) - \frac{1}{2j} \delta (f +
f_o) 
\eeq

\beq
F\{\cos 2\pi f_ot\} & = & \frac{1}{2} F\{e^{j2\pi f_ot} + e^{-j2\pi
f_ot}\} \\
\mbox{} & = & \frac{1}{2} ( \delta (f - f_o) + \delta (f + f_o) )
\eeq

\subprob{d} Using the times-$e^{j2\pi f_ot}$:

\beq 
F\{e^{j2\pi f_ot}u(t)\} & = & F\{u(t)\}(f - f_o) \\
\mbox{} & = & \frac{\delta (f - f_o)}{2} + \frac{1}{j2\pi (f - f_o)}
\\
\eeq

\mynewpage

\beq
F\{\cos (2\pi f_ot) u(t)\} & = & \frac{1}{2} F\{e^{j2\pi f_ot} u(t) +
e^{-j2\pi f_o t} u(t)\} \\
\mbox{} & = & \frac{1}{2} \left[ \frac{\delta (f - f_o)}{2} +
\frac{1}{j2\pi (f - f_o)} + \frac{\delta (f + f_o)}{2} +
\frac{1}{j2\pi (f + f_o)} \right] \\
\mbox{} & = & \frac{1}{2} \left[ \frac{\delta (f - f_o) + \delta (f +
f_o)}{2} + \frac{1}{j2\pi} \left( \frac{1}{f - f_o} + \frac{1}{f +
f_o} \right) \right] \\
\mbox{} & = & \frac{\delta (f - f_o) + \delta (f - f_o)}{4} +
\frac{1}{j4\pi} \left( \frac{f + f_o + f - f_o}{f^2 - f_o^2} \right)
\\
\mbox{} & = & \frac{\delta (f - f_o) + \delta (f + f_o)}{4} +
\frac{1}{j2\pi} \left( \frac{f}{f^2 - f_o^2} \right)
\eeq

\beq
F\{\sin (2\pi f_o t) u(t)\} & = & \frac{1}{2j} F\{e^{j2\pi f_ot} u(t) -
e^{-j2\pi f_ot} u(t) \} \\
\mbox{} & = & \frac{1}{2j} \left[ \frac{\delta (f - f_o)}{2} +
\frac{1}{j2\pi (f - f_o)} - \frac{\delta (f + f_o)}{2} -
\frac{1}{j2\pi (f + f_o)} \right] \\
\mbox{} & = & \frac{\delta (f - f_o) - \delta (f + f_o)}{4j} -
\frac{1}{4\pi} \left( \frac{f + f_o - f + f_o}{f^2 - f_o^2} \right) \\
\mbox{} & = & \frac{\delta (f - f_o) - \delta (f + f_o)}{4j} -
\frac{1}{2\pi} \left( \frac{f_o}{f^2 - f_o^2} \right)
\eeq

\subprob{e}

\beq
x(t) = e^{j2\pi f_ot} & \Rightarrow & X(f) = \delta (f - f_o) \\
Y(f) & = & H(f) X(f) \\
\mbox{} & = & H(f) \delta (f - f_o) \\
Y(f) & = & H(f_o)
\eeq

\prob{13.8 CSAS}

\subprob{a}

\beq
\frac{dx}{dt} & = & \frac{d}{dt} \left[ \int^\infty_{- \infty} X(f)
e^{j2\pi ft} df \right] \\
\mbox{} & = & \int^\infty_{- \infty} \left[ \frac{\delta}{\delta t}
X(f) e^{j2\pi ft} \right] df \\
\mbox{} & = & \int^\infty_{- \infty} X(f) j2\pi f e^{j2\pi ft} df \\
\mbox{} & = & j2\pi f X(f)
\eeq

\subprob{b}

\beq
X(f) & = & \int^\infty_{- \infty} x(t) e^{-j2\pi ft} dt \\
\frac{dX(f)}{df} & = & \int^\infty_{- \infty} \frac{\delta}{\delta f}
\left[ x(t) (-j2\pi) e^{-j2\pi ft} \right] dt \\
- \frac{1}{j2\pi} \frac{dX(f)}{df} & = & \int^\infty_{- \infty} t x(t)
e^{j2\pi ft} dt \\
F\{t x(t)\} & = & - \frac{1}{j2\pi} \frac{dX(f)}{df}
\eeq

\mynewpage

\subprob{c}

\noindent i)

\beq
F\{\cos 2\pi f_ot\} & = & \frac{1}{2\pi f_o} F\left\{ \frac{d}{dt}
\sin 2\pi f_o t\right\} \\
\mbox{} & = & \frac{1}{2\pi f_o} j2\pi f_o \left( \frac{\delta (f - f_o) -
\delta (f + f_o)}{2j} \right) \\
\mbox{} & = & \frac{1}{2} ( \delta (f - f_o) - \delta (f + f_o) )
\eeq

\noindent ii)

\beq
F\{\dot{\delta}(t)\} & = & F\left\{ \frac{d}{dt} \delta (t) \right\} \\
\mbox{} & = & j2\pi f (1) \\
\mbox{} & = & j2\pi f
\eeq

\noindent iii)

\beq
F\{te^{-\alpha t}u(t)\} & = & - \frac{1}{j2\pi} \left[ \frac{d}{df}
F\{ e^{-\alpha t} u(t)\} \right] \\
\mbox{} & = & - \frac{1}{j2\pi} \left[ \frac{d}{df} \frac{1}{\alpha +
j2\pi f} \right] \\
\mbox{} & = & - \frac{1}{j2\pi} \left[ \frac{d}{df} ( \alpha + j2\pi
f)^{-1} \right] \\
\mbox{} & = & \frac{1}{j2\pi} \left[ (-1) (\alpha + j2\pi f)^{-2}
(j2\pi) \right] \\
\mbox{} & = & \frac{1}{(\alpha + j2\pi f)^2}
\eeq

\noindent iv)

\beq
F\{t\} & = & - \frac{1}{j2\pi} \frac{d}{df} F\{1\} \\
\mbox{} & = & \frac{j}{2\pi} \left[ \frac{d}{df} \delta (t) \right] \\
\mbox{} & = & \frac{j\dot{\delta} (t)}{2\pi}
\eeq

\prob{13.13 CSAS}

\subprob{a}

\beq
x(t) & = & e^{-(1/\tau)t} u(t) + e^{-(1/\tau)(-t)} u(-t) \\
X(f) & = & \frac{1}{1/\tau + j2\pi f} + \frac{1}{\left|-1\right|}
F\{e^{-(1/\tau)(-t/-1)} u(-t/-1) \} \\
\mbox{} & = & \frac{1}{1/\tau + j2\pi f} + \frac{1}{1/\tau + j2\pi f}
\\
\mbox{} & = & \frac{2}{1/\tau + j2\pi f}
\eeq

\mynewpage

\subprob{b}

\beq
sgn t & = & \lim_{\tau \rightarrow \infty} x(t) \\
F\{sgn t\} & = & \lim_{\tau \rightarrow \infty} X(f) \\
\mbox{} & = & \lim_{\tau \rightarrow \infty} \frac{2}{1/\tau + j2\pi
f} \\
\mbox{} & = & \frac{2}{j2\pi f} \\
\mbox{} & = & \frac{1}{j\pi f}
\eeq

\subprob{c}

\vspace{1in}

\beq
u(t) & = & \frac{1}{2} (1 + sgn t) \\
F\{ u(t) \} & = & \frac{1} (\delta (t) + \frac{1}{j\pi f}) \\
\mbox{} & = & \frac{\delta (t)}{2} + \frac{1}{j2\pi f}
\eeq

\mynewpage

\prob{8.1 PE}

\subprob{a}

\vspace{2in}

\subprob{b} \seepage

\subprob{d}

\mynewpage

\prob{8.2 PE}

\subprob{a}

\vspace{1in}

\subprob{b}

\vspace{1in}

\subprob{c}

\vspace{1in}

\subprob{d}

\vspace{1in}

\subprob{e}

\vspace{1in}

\subprob{f}

\vspace{1in}

\subprob{g} 

\vspace{1in}

\mynewpage

\subprob{h}

\vspace{1in}

\subprob{i}

\vspace{1in}

\subprob{j}

\vspace{1in}

\subprob{k}

\end{document}


