Mumbai University > Computer Engineering > Sem 3 > Electronic Circuits and Communication Fundamentals

Marks: 10 Marks

Year: May 2015

For sinusoidal AM, the modulating waveform is of the form

$$e_m(t)=E_{m \max} \cos (2\pi f_mt+\phi_m)$$

In general the fixed phase angle 4,,, is unrelated to the fixed phase angle 4k for the carrier, showing that these two signals are independent of each other in time. However, the amplitude modulation results are independent of these phase angles, which may therefore be set equal to zero to simplify the algebra and trigonometry used in the analysis. The equation for the sinusoidal modulated wave is therefore

$$e(t)=(E_{c\max}+E_{m\max}\cos2\pi f_mt)\cos2\pi f_ct$$

Since in this particular case $E_{\max}=E_{c \max}+E_{m \max} and E_{\min}=E_{c \max}-E_{m \max})$ the modulation index is given by

$$m=\dfrac{E_{\max}-E_{min}}{E_{\max}+E_{min}} \\ =\dfrac{E_{m \max}}{E_{c \max}}$$

The equation for the sinusoidal amplitude modulated wave may therefore be written as

$$e(t)=E_{c\max}(1+m\cos2\pi f_mt)\cos2\pi f_ct$$

Figure shows the sinusoidal modulated waveforms for three different values of m

Although the modulated waveform contains two frequencies fc and fm the modulation process generates new frequencies that are the sum and difference of these. The spectrum is found by expanding the equation for the sinusoidal modulated AM as follows:

Figure: Sinusoidally amplitude odulated waveforms for (a) m=0.5 (undermodulated), (b) m=1 (fully modulated), and (c) m>1 (overmodulated)

$$e(t)=E_{c\max}(1+m\cos2\pi f_mt)\cos2\pi f_ct \\ =E_{c\max}\cos2\pi f_ct+mE_{c\max}\cos2\pi f_mt \times \cos 2\pi f_ct \\ =E_{c\max}\cos2\pi f_ct+\dfrac{m}{2} E_{c\max}\cos2\pi (f_c-f_m)t+\dfrac{m}{2}E_{c\max}\cos2\pi (f_c+f_m)t$$

It is left as an exercise for the student to derive this result making use of the trigonometric identity

$$\cos(A \pm B)=\cos A \cos B \mp \sin A \sin B$$

Equation above shows that the sinusoidal modulated wave consists of three components : a carrier wave of amplitude Ec max, and frequency fc, a lower side frequency of amplitude mEc max/2 and frequency fc — fm, and an upper side frequency of amplitude mEc max/2 and frequency fc. +fm. The amplitude spectrum is shown in Fig.