text: # 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
  Free
  CT 1: Current Affairs (Government Policies and Schemes)
  51592
  10 Questions 10 Marks 10 Mins
  
  ## 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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