$5\mathrm{.}28$ Civen $m(n)=e^{-\frac{t^2}{100}},f_c={10}^{4}Hz,k=6000,$ and $k_p=8000.$

$($a$)$ Find $y,$ the frequency deviation for FM and PM$.$

$($b$)$ Eetimate the bendaithe of the H and PM waves. Hint: Find $M(n)$ end fad its $3$ dB bandwidth.

$5\mathrm{.}29($o$)$ Applying Fernevals theorem, ahow that

${\displaystyle \underset{n=}{\overset{}{}}}{J}_i^2()=1$

$($b$)$ Powe dasL

HW #$3$

$5\mathrm{.}4\mathrm{.}3$ Let $s(t)$ be an angle$-$modulated signal that a receiver obtains

Problems

$301$

$s(t)=2cos[{10}^{7}t+2sin(2000t+0\mathrm{.}3)-3cos(100t)]$

$($a$)$ Find the bandwidth of this FM signal.

$($b$)$ If $s(t)$ is sent to an $($ideal$)$ envelope detector, find the detector output signal.

$($c$)$ If $s(t)$ is first differentiated before the envelope detector, find the detector output signal.

$($d$)$ Explain which detector output can be processed to yield the message signal $m(t)$ and find the message signal $m(t)$ if $k_y=200.$

$5\mathrm{.}4-4$ Use small$-$error analysis of PLL to show that a first$-$order loop $H(s)=1$ cannot track an incoming signal whose instantaneous frequency varies linearly with time ${}_i(t)=k{d}^2.$ This signal can be tracked within a constant phase if $H(s)=\frac{s+a}{s}.$ It can be tracked with zero phase error if $H(s)=\frac{s^2+as+b}{s^2}.$