Question

(i)Draw the plot of amplitude versus '$$\omega$$' for an amplitude modulated wave whose carrier wave $$(\omega_c)$$ is carrying two modulating signals, $$\omega_1$$ and $$\omega_1 (\omega_2 > \omega_1)$$
(ii) Is the plot symmetrical about $$\omega_c$$? Comment especially about plot in region $$\omega < \omega_c$$
(iii) Extrapolate and predict the problem one can expect if more waves are to be modulated.
(iv) Suggest solutions to the above problem. In the process can one understand another advantage of modulation in terms of bandwidth?

Answer 1

(i) Let the two modulating signals Am1sinωm1t and Am2sinωm2t be superimposed on carrier signal

Acsimωct. The signal produced is x(t)=Am1sinωm1t+Am2sinωm2t+Acsinωct

To produce amplitude modulated wave the signal x(t) is passed through a square law device which produces

an output given by

$$y(t)=B[A_{m_1}sinω_{m_1}+A_{m_2}sinω_{m_2}t+A_csinω_ct]+C[A_{m_1}sinω_{m_1}t+A_{m_2}sinω_{m_2}t+A_csimω_ct]^2$$

$$B=[A_{m_1}sinω_{m_1}t+A_{m_2}sinω_{m_2}t+A_csinω_ct]+C[(A_{m_1}sinω_{m_1}t+A_{m_2}sinω_{m_2}t)2+A^2_csin^2ω_ct$$

$$+2A_csinω_ct(A_{m_1}sinω_{m_1}t+A_{m_2}sinω_{m_2}t)]$$

$$=B[A_{m_1}sinω_{m_1}t+A_{m_2}sinω_{m_2}t+A_csinω_ct]$$

$$+C[A^2_{m_1}sin^2ω_{m_1}t+A^2_{m_2}sin^2ω_{m_2}t+2A_{m_1}A_{m_2}sinω_{m_1}tsinω_{m_2}t$$

$$+A^2_csin^2ω_ct+2A_c(A_{m_1}sinω_{m_1}tsinω_ct+Am^2sinω_{m_2}tsinω_ct)]$$

$$=B[A{m_1}sinω_{m_1}t+A_{m_2}sinω_{m_2}t+A_csinω_ct]$$

$$+C[A^2_{m_1}sin^2ω_{m_1}t+A^2_{m_2}sin^2ω_{m_2}t+A_{m_1}A_{m_2}{cos(ω_{m_2}−ω_{m_1})t−cos(ω_{m_2}+ω_{m_1})}+A^2_csin^2ω_ct$$

$$+A_cA_{m_1}{cos(ω_c−ω_{m_1})t−cos(ω_c+ω_{m_1})t}+A_cA_{m_2}{cos(ω_c−ω_{m_2})t−cos(ω_c+ω_{m_2})t}]$$

In the above amplitude modulated waves, the frequencies present are

$$ω_{m_1},ω_{m_2},ω_c,(ω_{m_2}−ω_{m_1}),(ω_{m_2}+ω_{m_1}),(ω_c−ω_{m_1}),(ω_c+ω_{m_1}),(ω_c−ω_{m_2}) and (ω_c+ω_{m_2})$$

The plot of amplitude versus ω is shown in

figure

(ii) From figure , we note that frequency

spectrum is not symmetrical about $$ω_c$$.

Crowding of spectrum is present for $$ω<ω_c.$$

(iii) if more waves are to be modulated then

there will be more crowding in the modulating

signal in the region $$ω<ω_c.$$ That will result

more chances of mixing of signals.

(iv) To accommodate more signals, we should increase band width and frequency of carrier waves $$ω_c$$. This

shows that large carrier frequency enables to carry more information (i.e., more $$ω_m$$) and the same will

inturn increase band width.