Question

Draw the double-sided amplitude spectrum of the following signals

x₁(t)=2cos(2πfₘ​t)×cos(2πf​t) where fₘ = 1kHz and f​=100kHz.

Answer 1

The double-sided amplitude spectrum of the given signal x₁(t) consists of two peaks at frequencies 99 kHz and 101 kHz, each with an amplitude of 1. This spectrum is the result of using a trigonometric identity to express the product of cosines as a sum of two cosine functions.

Step-by-step explanation:

Given the signal x₁(t)=2cos(2πfₘ​t)×cos(2πf​t) with fₘ = 1kHz and f = 100kHz, we can apply the trigonometric identity for the product of cosines to write this as a sum of two cosine functions:

x₁(t) = cos(2π(f - fₘ)t) + cos(2π(f + fₘ)t)

This represents two sinusoidal components, one at a frequency of (f - fₘ) and the other at (f + fₘ). Therefore, the double-sided amplitude spectrum will have two peaks: one at 99 kHz and another at 101 kHz, each with an amplitude of 1, as the original amplitude of 2 is equally divided between the two spectral lines.

The horizontal axis of the amplitude spectrum represents frequency with positive and negative parts indicating the sinusoids rotating in opposite directions, and the vertical axis represents the amplitude. The spectrum will be symmetric about the y-axis, showing lines of amplitude 1 at both +99 kHz and -99 kHz, as well as +101 kHz and -101 kHz.