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Mismatched Filters

The following signal set is employed to transmit equally likely signals over an additive white Gausssian noise channel with spectral height $\frac{N_0}{2}$ .

$\begin{picture} (490,200)(55,580) \setlength {\unitlength}{0.01in} % \thick... ... \put(385,600){\line( 1, 2){ 80}} \put(465,760){\line( 1,-2){ 40}}\end{picture}$

1.: Sketch and accurately label the block diagram of the receiver which minimizes the probability of error.
2.: Compute the probability of error of your receiver from part (a).
3.: Consider now the following receiver:

$\begin{picture} (100,30) \setlength {\unitlength}{1mm} \put(0,10){\vector(1... ...ebox(30,7){if $R \gt 0$, say $s_0$}} \put(90,10){\vector(1,0){10}}\end{picture}$
Compute the probability of error of this receiver if g(t) is given by
$\begin{displaymath} g(t) = \left\{ \begin{array} {cl} 1 & \mbox{for $0 \leq t < 1$}\\ -1 & \mbox{for $1 \leq t < 2$} \end{array} \right. \end{displaymath}$
4.: Repeat part (c) for $g(t) = \sin(\pi t)$ .
5.: Compare the probabilities of error in parts (b)-(d).

Hint: You may need

$\begin{displaymath} \int x \sin(ax)\;dx = \frac{\sin(ax)}{a^2} - \frac{x \cos(ax)}{a}.\end{displaymath}$

Prof. Bernd-Peter Paris
3/3/1998