Question

Decompose the signal (1+0.1 cos5t) cos100t into a linear combination of sinusoidal functions, and find the amplitude, frequency, and phase of each component. Hint: use the identity for cosacosb.

Answer 1

Answer:

amplitudes : 1 , 0.05, 0.05

frequencies : 50/$\pi$,   105/$2\pi$,  95/2$\pi$

phases : $\pi /2 , \pi /2 , \pi /2$

Explanation:

signal  s(t) = ( 1 + 0.1 cos 5t )cos 100t

signal s(t) = cos100t + 0.1cos100tcos5t . using the identity for cosacosb

         s(t) = cos100t + $\frac{0.1}{2}$ [cos(100+5)t + cos (100-5)t]

          s(t) = cos 100t + 0.05cos ( 100+5)t + 0.05cos (100-5)t

               =  cos100t + 0.05cos(105)t + 0.05cos 95t

             = cos 2 $(\frac{50}{\pi } )t + 0.05cos2 (\frac{105}{2\pi } )t + 0.05cos2 (\frac{95}{2\pi } )t$ [ ∵cos (∅) = sin(/2 +∅ ]

= sin ( 2 $(\frac{50}{\pi } ) t$  + /2 ) + 0.05sin ( 2 $(\frac{105}{2\pi } ) t + /2 )$ + 0.05sin ( 2 $(\frac{95}{2\pi } )t + /2$ )

attached is the remaining part of the solution