Information Signal (message) $\rightarrow E_m = E_{mo} \sin(\omega_m t)$

where:

$E_{mo}$ : Amplitude of message

$\omega_m$ : angular frequency of message

Carrier wave $\rightarrow E_c = E_{co} \sin ({\omega_c t})$

Amplitude modulated $\rightarrow E_{am} = (E_{co} + E_m ) \sin ({\omega_c t)}$

Amplitude modulated wave has 3 frequencies

$\omega_c$ - $\omega_m$ : Lower side band

$\omega_c$ : carrier frequency

$\omega_c$ + $\omega_m$ : upper side band

Amplitude modulation index ($\mu$) = $\frac {A_m}{A_c}$ = $\frac{E_{mo}}{E_{co}}$ = $\frac {\text {Amplitude of modulately wave (message)}}{\text{Amplitude of Carrier wave}}$

$\mu$ = $\frac{E_{mo}}{E_{co}}$ = $\frac{2 E_{mo}}{2 E_{co}}$ = $\frac {2 E_{mo} + E_{co} - E_{co}}{2 E_{co} + E_{mo} - E_{mo}}$

= $\frac {(E_{co} + E_{mo}) - (E_{co} - E_{mo})}{ (E_{co} + E_{mo}) + (E_{co} - E_{mo})}$

$\mu$ = $\frac{(E_{a m})^{max} - (E_{a m})^{min }} {(E_{a m})^{max } + (E_{a m})^{min }}$

$E_{a_m}=[E_{c_0}+E_{mo} \sin (\omega_{m} t)] \sin (\omega_c t)$

Maximum of amplitude modulated Electric field = $E_{c_0}+E_{m 0}$

Minimum of amplitude modulated field =$E_{c_0}-E_{mo}$.

The envelope of amplitude modulated wave represents the original modulating wave.

$\mu = 1$, efficient amplitude modulated transfer.

Generation of Amplitude modulated wave

After adder: x(t) = m(t) + $m_c (t)$

After square law device: y(t) = B x (t) + C $x^2 (t)$

$E_{mod}$ = m(t) = $E_{mo} \sin(\omega_m t)$

$E_c (t)$ = $m_c (t)$= $E_{co} \sin (\omega_c t)$

After adder:

$x(t)=E_{mo} \sin (\omega_m t)+ E_{co} \sin (\omega_c t)$

After square law device:

$y(t)= B x(t)+ C x^2 (t)$

= $B[E_{mo} \sin (\omega_m t)+E_{co} \sin (\omega_c t)] + c [E_{mo} \sin (\omega_m t)+E_{co} \sin (\omega_c t)]^2$

y(t)=$B E_{mo} \sin (\omega_m t)+B E_{co} \sin (\omega_c t)$ + $c[E_{mo}^2 \sin ^2(\omega_{m} t)+E_{co}^2 \sin ^2 (\omega_c t)+2 E_{mo} E_{co} \sin (\omega_m t) \times \sin \ (\omega_c t)]$

$\sin^2 (\omega t) = \frac {1 - \cos (2 \omega t)}{2}$ ;

$\sin (x) \sin(y) = \frac{1}{2} \times [ -\cos (x+y) + cos (x-y)]$

$y(t)=B E_{mo} \sin (\omega_{m} t)+B E_{co} \sin (\omega_c t)$ + $c [E_{mo}^2(\frac{1-\cos (2 \omega_m t)}{2})$ +$E_{co}^2(\frac{1-\cos (2 \omega_c t)}{2})$ + $E_{mo} E_{co} \cos {(\omega_c-\omega_m) t}$

$-E_{mo} E_{co} \cos {(\omega_c+\omega_m) t}]$

= $B E_{mo} \sin ({\omega_m t}) t)$+$B E_{co} \sin (\omega_c t)$ -$\frac{C E_{mo}^2}{2} \cos (2 \omega_{w} t)$ - $\frac{c E_{co}^2}{2} \cos (2 \omega_c t )$ + $c E_{mo} E_{c_0} \ cos {(\omega_c-\omega_{m}) t} - c$ $E_{mo} E_{co} \cos {(\omega_c+\omega_{m}) t}$ + $\frac{c (E_{mo}^2 + E_{co}^2)}{2}$

We saw that after passing through the square law device, the resultant wave has 6 frequencies.

1. $\omega_m$

2. $\omega_c$

3. 2$\omega_m$

4. 2$\omega_c$

5. $(\omega_m-\omega_c)$

6. $(\omega_c-\omega_m)$

$\omega_c, \omega_c + \omega_m$

$\omega_c$- $\omega_m$

At output of band-pass filter

y(t) = $B E_{co} \sin (\omega_c t)$ + $c E_{mo} E_{co} \cos [(\omega_c - \omega_m)t]$ - $c E_{mo} E_{co} \cos[(\omega_c + \omega_m)t]$

In frequency modulation the frequency of the carrier wave is charged in acceleration with the modulating(message) wave.

Carrier wave $E_c$ = $E_{co} \sin (\omega_c t)$

Message wave $E_{mod}$ = $E_{mo} \sin (\omega_m t)$

$E_{fm}$ = $E_{co} \sin ( \omega_c t + \theta)$ = $E{co} \sin (\omega_c t + \theta)$

$\theta \rightarrow$ in accordance with message wave.

$E_{fm} = E_{co} \sin (\omega_c t + \theta)$

If $\omega_i$ is the instantaneous frequency of the frequency modulated wave.

$\omega_i = \omega_c + \Delta \omega \sin (\omega_m t )$

Angular frequency $\omega = 2 \pi \nu = \frac {d \theta }{dt}$

d$\theta$ = $\omega$ dt $\Rightarrow \theta = \int{d\theta} = \int{\omega d t }$ =$\int {\omega_i dt} = \int{[\omega_c + \Delta \omega sin (\omega_m t) ]}dt$

Equation of frequency modulated wave

$E_{fm} = E_{co} \sin [\omega_c t - \frac {\Delta \omega}{\omega_m} \cos (\omega_m t)]$

Frequency modulation index = $\frac {\Delta \omega}{\omega_m}$ = $\frac{\Delta f }{f_m}$