Suppose we wish to transmit the signal x(t) = sin 200 πt

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Suppose we wish to transmit the signal x(t) = sin 200 πt + 2 sin 400πt using a modulation that create the signal g(t) = x (t) sin 400 πt. If the product g(t) sin 400 πt is passed through an ideal LPF with cut-off frequency 400π and pass band gain of 2, the signal obtained at the output of the LPF is—

1. sin 200 πt
2. 1 2 sin 200 πt
3. 2 sin 200 πt
4. 0

Correct Option: A

Given that,
x(t) = sin 200πt + 2 sin 400πt
g(t) = x(t)·sin 400πt
or
g(t) = (sin 200πt + 2 sin 400πt) sin 400πt
If the product say
y(t) = g(t) sin 400 nt
y(t) = g(t)·sin 400πt
= (sin 200πt + 2 sin 400πt) sin² 400πt

= (sin 200πt + 2 sin 400πt)	(1 − cos 800πt)
	2

=	1	(sin 200πt) – sin (200πt) (cos 800πt) + sin (200πt) – sin (400πt) (cos 800πt)
	2

=	1	sin (200πt) −	1	[sin (100πt) −sin (900πt)] + sin (400πt) −	1	[sin (1200πt) −sin (400πt)]
	2		4		4

If this signal is passed through LPF with frequency 400π and gain 2, the output will be sin (200πt)
Hence alternative (A) is the correct choice.