# Let x(t) be a signal with Nyquist rate ω0. Determine the Nyquist rate for y(t) = x(t)cos(ω0t)

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1. ω0
2. 0
3. 0
4. 2π

Option 3 : 3ω0
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## Detailed Solution

CONCEPT:

When we multiply any signal with cos (ω0t) then that signal shifts by ω0

x (t) → X (ω)

$$x\left( t \right)\cos \left( {{\omega _0}t} \right) \to \frac{1}{2}\left\{ {X\left( {\omega - {\omega _0}} \right) + X\left( {\omega + {\omega _0}} \right)} \right\}$$

If X(ω)

Then x(t) cos ω0t = y(t)

Given:

Nyquist Rate of x(t) is ω0

Let ωm is maximum frequency of x(t)

m = ω0

$${\omega _m} = \frac{{{\omega _0}}}{2}$$

Now y (t) = x (t) cos ω0 t

So maximum frequency of y (t) = ω0 + ωm

$${\omega _0} + \frac{{{\omega _0}}}{2}$$

$$\frac{{3{\omega _0}}}{2}$$

Nyquist Rate of y (t) = $$2 \times \frac{{3{\omega _0}}}{2} = 3{\omega _0}$$
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