Question

4 We previously computed the spectrum of the FM signal defined

by x_c1 (t)=A_c cos⁡〖(2πf_c t+β sin(2πf_m t).〗
Now assume that the modulated signal is given by
x₂ (t)=A cos⁡〖(2πf t+

Answer 1

Frequency modulation of a carrier wave in FM radio signals and the superposition of waves leading to concepts like beat frequency and average frequency.

Step-by-step explanation:

Frequency modulation (FM) signal analysis, a concept in Physics or Electrical Engineering within the realm of wave theory. An FM signal is one where the frequency of a carrier wave is varied in accordance with an audio signal, which is the input message signal. This variation happens without altering the amplitude of the carrier wave. This is in contrast to amplitude modulation (AM) where the amplitude of the carrier wave is modified while maintaining a constant frequency.

The equations provided represent waveforms with particular phase shifts and amplitudes, along with their respective cosine and sine components. The Fourier theorem is mentioned, which is essential in understanding how complex waveforms can be broken down into simpler sinusoidal forms. The concept of superposition is also introduced, where the resultant waveform from two waves interacting with each other can be determined. Factors such as beat frequency (fb) and average frequency (fave) emerge in the case of two waves with close frequencies that are superimposed.

Answer 2

The amplitude spectra of xc₁ (t) and xc₂ (t) are identical because the amplitude spectrum of an FM signal only depends on frequency deviation and the modulating frequency, not the modulating signal's shape. The phase spectrum can be computed using a Fourier transform.

Step-by-step explanation:

We previously computed the spectrum of the FM signal defined by xc₁ (t)= Ac cos(2πfct + βsin(2πfmt)). Now, let us assume the modulated signal is given by xc₂ (t) = Ac cos(2πfct + βcos(2πfmt)). The question is to show that the amplitude spectra of xc₁ (t) and xc₂ (t) are identical.

The amplitude spectrum of an FM signal is independent of the modulating signal's shape; it only depends on the frequency deviation and the modulating frequency. We know from Fourier analysis that shifting a signal in time, which is analogous to changing the phase of the modulating signal from sine to cosine, or vice versa, does not affect the amplitude spectrum of the modulated signal. As a result, the amplitude spectra of xc₁ (t) and xc₂ (t) are identical because both signals have the same frequency deviation and modulating frequency.

To compute the phase spectrum of xc₂ (t), one would need to perform a Fourier transform which would reveal the phase information as a function of frequency. Comparing this to the phase spectrum of xc₁ (t), we would observe that the two spectra would differ due to the different modulating functions (sine versus cosine), which will introduce different phase shifts for each of the frequency components of the modulating signal.

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