Q$1)[60$p$]$ Given the following message and carrier signal

$m(t)=cos2000t+2cos6000t+cos10000t$

$c(t)=20cos320000t$

The signal is passed through following nonlinear device and a subsequent filter to generate DSB$-$AM signal. The nonlinear device has the input$-$output relation $y(t)=x^2(t)+10x(t).$

a$)$ Determine $y(t).$

b$)$ Determine type of filter $($BPF$,$ HPF$,$ LPF$)$ and the bandwidth to obtain a DSB$-$AM signal $($Conventional AM$)$

c$)$ What is the modulation index?

d$)$ Determine power of $m(t),u(t),$ and the ratio of power in side bands to carrier.

e$)$ Assume that modulation signal $u(t)$ is going to be demodulated by an envelope detector. The resistor is selected as $R=1K.$ What is the minimum and maximum value of capacitor for a good detector.