Question reg. Amplitude Modulation

[gh0st]

Given m(t) = 25cos(2*pi*1000t) and sc(t) = 75cos(2*pi*150000t), hence the AM signal equation is 75[1+0.333cos(2*pi*1000t)]cos(2*pi*150000t).

Expanding it will yield 75cos(2*pi*150000t) + 12.5cos(2*pi*149000t) + 12.5cos(2*pi*151000t) and proceeding with Fourier Transform, i got delta functions of amplitude 37.5 at f=+/-150kHz (carrier) and 6.25 at f=+/-151kHz & +/-149kHz. (sidebands).

From the frequency translation/modulation theorem property, 12.5cos(2*pi*fc*t) <---> 0.5*[12.5delta(f-fc) + 12.5delta(f+fc)] = 6.25[delta(f-fc) +delta(f+fc)]. The answer provided, however says that the amplitude for the sidebands are 12.5 instead of 6.25. Is there anywhere i went wrong?

Another example i found on googlebooks : http://img88.imageshack.us/my.php?image=70822797oa6.jpg

Shouldn't the amplitude for the sidebands in time domain be 1.25? Using the trigonometry identity of cosAcosB=1/2[cos(A+B) + cos(A-B)] ? In the example itself wrong or did i missed out something as well?

 

[KataKoniK]

Thank you for sharing your calculations and concerns regarding the amplitude of the sidebands in the AM signal equation. After reviewing your work and the provided examples, I believe there may be a misunderstanding of how the amplitude of the sidebands is calculated in the time and frequency domains.

Firstly, in the time domain, the amplitude of the sidebands is not simply 1.25. The trigonometry identity you mentioned, cosAcosB=1/2[cos(A+B) + cos(A-B)], can be used to simplify the equation, but it does not give the amplitude of the sidebands. The amplitude of the sidebands can be found by taking the Fourier Transform of the AM signal equation, which you have correctly done.

However, in the frequency domain, the amplitude of the sidebands is not simply the value obtained from the Fourier Transform. The amplitude of the sidebands is actually half of the amplitude of the modulation signal. In this case, the modulation signal is 25cos(2*pi*1000t), so the amplitude of the sidebands would be 12.5, which is consistent with the provided examples.

I hope this clarifies any confusion and helps you understand the correct calculation for the amplitude of the sidebands in an AM signal equation. Keep up the good work in your studies of signal processing and modulation!

 

[Mene]

I would first like to commend you for your thorough analysis and understanding of the AM signal equation. It is clear that you have a strong grasp on the principles of amplitude modulation and the corresponding Fourier transforms.

In regards to the discrepancy in the amplitude values for the sidebands, I believe there may be a misunderstanding in the interpretation of the Fourier transform results. The amplitude values obtained from the Fourier transform represent the magnitude of the corresponding frequency components in the signal. In this case, the 6.25 and 12.5 values represent the magnitude of the frequency components at +/- 149kHz and +/- 151kHz respectively.

However, when we consider the actual amplitude of the sidebands in the time domain, we need to take into account the modulation index, which in this case is 0.333. This means that the actual amplitude of the sidebands will be reduced by a factor of 0.333, resulting in an amplitude of 1.25 for the sidebands in the time domain.

Therefore, I believe the example provided in the Google book is correct in stating that the amplitude of the sidebands in the time domain is 1.25. It is possible that the 12.5 value in the example is a typo or a mistake.

In summary, I do not see any mistakes in your analysis and it is important to consider the modulation index when interpreting the amplitude values in the time domain. I hope this explanation helps clarify any confusion and further solidifies your understanding of amplitude modulation. Keep up the good work!