Question

The signal m(t) in the DSB-SC signal v(t)=m(t)cos(ωc​t+θ) is to be reconstructed by multiplying v(t) by a signal derived from v²(t).

(a) Show that v2(t) has a component at the frequency 2fc​. Find its amplitude.

Answer 1

To find the component of v²(t) at the frequency 2fc, square v(t) and extract the frequency component at 2fc. The resulting component is 0.5m²(t).

Step-by-step explanation:

To find the component of v²(t) at the frequency 2fc​, we need to square v(t) and take the frequency component at 2fc​. Let's start by squaring v(t):

v²(t) = (m(t)cos(ωc​t+θ))² = m²(t)cos²(ωc​t+θ)

Now, let's examine the cosine term. Using the trigonometric identity cos²(a) = 0.5(1 + cos(2a)), we can rewrite the cosine term as:

cos²(ωc​t+θ) = 0.5(1 + cos(2(ωc​t+θ))) = 0.5(1 + cos(2ωc​t+2θ))

Since the cosine function has a periodicity of 2π, the cosine term cos(2ωc​t+2θ) has a frequency component of 2fc​. Therefore, the component of v²(t) at the frequency 2fc​ is 0.5m²(t).