Question

An amplitude-modulated (AM) cosine wave is represented by the formula x(t)=[3+sin(πt)]cos(13πt+π/2). (a) Use phasors to show that x(t) can be expressed in the form x(t)=A₁cos(ω₁t+ϕ₁)+A₂cos(ω₂t+ϕ₂)+A₃cos(ω₃t+ϕ₃) where ω₁<ω₂<ω₃. In addition, find values for the parameters Aᵢ,ϕᵢ, and ωᵢ for i=1,2,3.

Answer 1

An AM wave x(t) is expanded using phasor notation into a sum of cosines. This results in identifying amplitudes, phase shifts, and angular frequencies for the carrier and sideband components of the AM signal.

Step-by-step explanation:

The given amplitude-modulated (AM) cosine wave x(t)=[3+sin(πt)]cos(13πt+π/2) can be expanded using trigonometric identities and phasor notation into a sum of three cosine waves representing its sidebands and carrier frequency. We know that the product of a sinusoidal factor and a cosine factor yields a sum of cosines, and applying this principle:

• First, we express sin(πt) as a phasor: eiπt - e-iπt / 2i
• Then multiply the phasor by the carrier, cos(13πt + π/2), which can be written as phasors ei(13πt + π/2) + e-i(13πt + π/2) / 2
• Next, distribute and apply Euler's formula to expand the modulation into the sum of cosines, resulting in three distinct cosine terms with their respective amplitude (Ai), phase shift (ϕi), and angular frequency (ωi)

Through this process, we find the sidebands and carrier frequency with distinct angular frequencies. For the given question, ω1 and ω3 represent the lower and upper sideband frequencies, which are the carrier frequency ± the modulation frequency, and ω2 is the carrier frequency.

This leads to the expression x(t) = A1cos(ω1t+ϕ1) + A2cos(ω2t+ϕ2) + A3cos(ω3t+ϕ3), with the specific values for Ai, ϕi, and ωi calculated based on the initial equation.