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

amplitudes : 1 , 0.05, 0.05

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

phases :
`\pi /2 , \pi /2 , \pi /2`

Step-by-step 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 +
`(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
`((50)/(\pi ) )t + 0.05cos2 ((105)/(2\pi ) )t + 0.05cos2 ((95)/(2\pi ) )t` [ ∵cos (∅) = sin(/2 +∅ ]

= sin ( 2
`((50)/(\pi ) ) t` + /2 ) + 0.05sin ( 2
`((105)/(2\pi ) ) t + /2 )` + 0.05sin ( 2
`((95)/(2\pi ) )t + /2` )

attached is the remaining part of the solution