Consider the following signal s(t) = (1 + 0.1 cos(5t))cos (100t). a) Decompose s
ID: 3641378 • Letter: C
Question
Consider the following signal s(t) = (1 + 0.1 cos(5t))cos (100t).a) Decompose s(t) into a linear combination of sinusoidal functions and find the
amplitude, frequency, and phase of each component.
b) Find the period (or frequency) of s(t)
c) Find the total power for s(t).
Explanation / Answer
The give signal can be decomposed into the following components: => (1 + 0.1cos5t)cos100t => cos100t + 0.1cos5tcos100t Using the formula: cosAcosB = 1/2{cos(A+B) + cos(A-B)} we get, => cos100t + (0.1/2)cos105t + (0.1/2)cos95t => cos100t + (0.05)cos105t + 0.05cos95t Now the above is the decomposed signal into its different components. Accordingly, => at 100/(2*pi) = 50/pi Hz (~16Hz) The amplitude is 1(one). => at 105/(2*pi) = 52.5/pi Hz (~17Hz) The amplitude is 0.05. => at 95/(2*pi) = 47.5/pi Hz (~15 Hz) The amplitude is 0.05. The phase difference in all of the above cases remains zero as there is no other term within the cos function. Notes: 1.The decomposition of the given signal does not contain any sine function terms which anyway can be readily obtained by appropriate conversions. 2.The 'j' term (imaginary part) does not exist in this case. 3. The amplitude at all other frequencies except for the ones listed above are zero.