Question

Make an analysis of amplitude modulated wave. Plot the frequency spectrum.

Answer 1

A carrier wave may be represented as,

$e_c=E_{c}cos{\omega }_{c}t$ ...(1)

where $e_c,E_c$ and ${\omega }_c$ represent the instantaneous voltage, amplitude and angular frequency of the carrier wave respectively.
In amplitude modulation, the amplitude EcEc of the carrier wave is varied in accordance with the intensity of the audio signal as shown in figure. The modulating signal may be represented as,

$e_c=E_{s}cos{\omega }_{s}t$ ...(2)
where $e_s,E_s$ and ${\omega }_s$ represent instantaneous voltage, amplitude and angular frequency of the signal respectively.
Amplitude modulated wave is obtained by varying $E_c$ of equation (1) in accordance with $E_s$. Thus, amplitude modulated wave is,

$e=(E_c+E_{s}cos{\omega }_{s}t)cos{\omega }_{c}t$

$e=E_c(1+\frac{E_s}{E_c}cos{\omega }_{c}t)cos{\omega }_{c}t$

$=E_c[1+mcos{\omega }_{s}t]cos{\omega }_{c}t$

where mm is the modulation factor which is equal to $\frac{E_s}{E_c}$.

$\therefore e=E_{c}cos{\omega }_{c}t+m{E}_{c}cos{\omega }_{c}t.cos{\omega }_{s}t$ ...(3)

$=E_{c}cos{\omega }_{c}t+\frac{m{E}_c}{2}[2cos{\omega }_{c}t.cos{\omega }_{s}t]$

$=E_{c}cos{\omega }_{c}t+\frac{m{E}_c}{2}\left[cos\right({\omega }_c+{\omega }_s)t+cos({\omega }_c-{\omega }_s\left)t\right]$

$=E_{c}cos{\omega }_{c}t+\frac{m{E}_c}{2}cos({\omega }_c+{\omega }_s)t+\frac{m{E}_c}{2}cos({\omega }_c-{\omega }_s)t$ ...(4)

This expression shows that the modulated wave contains three components :
(i) $E_{c}cos{\omega }_{c}t$: This component is same as the carrier wave.
(ii) $\frac{m{E}_c}{2}cos({\omega }_c+{\omega }_s)t$ : This component has a frequency greater than that of the carrier and is called as the Upper Side Band (USB).
(iii) $\frac{m{E}_c}{2}cos({\omega }_c-\omega )s)t$ : This component has a frequency lesser than that of the carrier and is called as the Lower Side Band (LSB).

Frequency spectrum : The lower side band term and upper side band term are located in the frequency spectrum on either side of the carrier at a frequency interval of ${\omega }_s$ as shown in figure. The magnitude of both the upper and lower side bands is $\frac{m}{2}$ times the carrier amplitude $E_c$. If the modulation factor mm is equal to unity, then each side band has amplitude equal to half of the carrier amplitude.