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 cosa cosb.

Answer 1

Answer:

Given in explanation.

Explanation:

First seperate the signal into components,

$(1 + 0.1 \cos5t)\cos100t=\cos100t+0.1 \cos5t\cos100t$

Now, we will use the identity given in the hint,

$\cos A \cos B = \frac{1}{2} (\cos(A+B) + \cos(A-B))\\\\\cos100t+0.1 \cos5t\cos100t=\cos100t + 0.05\cos105t + 0.05\cos95t$

Now the signal is decomposed into its different components.

Accordingly,

$at\:\:100/(2*\pi) = 50/\pi \:Hz \:(16\:Hz)\:\:The \:\:amplitude\:\: is\:\: 1 \\at\:\:105/(2*\pi) = 52.5/\pi \:Hz \:(17\:Hz)\:\:The \:\:amplitude\:\: is\:\: 0.05 \\at\:\:95/(2*\pi) = 47.5/\pi \:Hz \:(15\:Hz)\:\:The \:\:amplitude\:\: is\:\: 0.05$

Since there  is no other term within the cos function, the phase differences in all of the above cases are zero.