Transmission From Tower Of Height $h$:

The distance to the horizon $$d_T=\sqrt{2Rh_T}$$ $$d_M=\sqrt{2Rh_T} + \sqrt{2Rh_R}$$

Amplitude Modulation

The modulated signal $c_{m}(t)$ can be written as:

$$c_{m}(t)=A_{c} \sin \omega_{c} t+\frac{\mu A_{c}}{2} \cos \left(\omega_{c}-\omega_{m}\right) t-\frac{\mu A_{c}}{2} \cos \left(\omega_{c}+\omega_{m}\right)$$

Modulation Index:

$$m_{a}=\frac{\text { Change in amplitude of carrier wave }}{\text { Amplitude of original carrier wave }}=\frac{k A_{m}}{A_{c}}$$

Where: $\mathrm{k}=\mathrm{A}$ factor which determines the maximum change in the amplitude for a given amplitude $E_{m}$ of the modulating.

If $k=1$ then, $$m_{a}=\frac{A_{m}}{A_{c}}=\frac{A_{\text {max }}-A_{\text {min }}}{A_{\text {max }}-A_{\text {min }}}$$

If a carrier wave is modulated by several sine waves the total modulated index $m_{t}$ is given by

$$m_{t}=\sqrt{m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+\ldots}$$

Side Band Frequencies:

• $\left(f_{c}+f_{m}\right)=$ Upper side band (USB) frequency

• $\left(f_{c}-f_{m}\right)=$ Lower side band (LBS) frequency

Band width:

$$\text{Band width} : (f_c+f_m)-(f_c-f_m)=2f_m$$

Power in AM waves :

$$P=\frac{V_{\text {rms }}^{2}}{R}$$

(i) carrier power: $$P_{c}=\frac{\left(\frac{A_{c}}{\sqrt{2}}\right)^{2}}{R}=\frac{A_{c}^{2}}{2 R}$$

(ii) Total power of side bands: $$P_{s b}=\frac{\left(\frac{m_{a} A_{c}}{2 \sqrt{2}}\right)^{2}}{R}=\frac{\left(\frac{m_{a} A_{c}}{2 \sqrt{2}}\right)}{2 R}=\frac{m_{a}^{2} A_{c}^{2}}{4 R}$$

(iii) Total power of AM wave: $$P_{\text {Total }}=P_{c}+P_{a b}=\frac{A_{c}^{2}}{2 R}\left(1+\frac{m_{a}^{2}}{2}\right)$$

(iv) $$\frac{\mathrm{P} _{\mathrm{t}}}{\mathrm{P} _{\mathrm{c}}}=\left(1+\frac{\mathrm{m} _{\mathrm{a}}^2}{2}\right) \quad \text{and} \quad \frac{\mathrm{P} _{\mathrm{sb}}}{\mathrm{P} _{\mathrm{t}}}=\frac{\mathrm{m} _{\mathrm{a}}^2 / 2}{\left(1+\frac{\mathrm{m} _{\mathrm{a}}^2}{2}\right)}$$

(v) Maximum power in the AM (without distortion) will occur when $m_{a}=1$ i.e., $$P_{t}=1.5 P=3 P_{a b}$$

(vi) If $I_c=$ unmodulated current and $I_t=$ total or modulated current.

$$\Rightarrow \frac{P_{t}}{P_{c}}=\frac{I_{t}^{2}}{I_{c}^{2}} \Rightarrow \frac{I_{t}}{I_{c}}=\sqrt{\left(1+\frac{m_{a}^{2}}{2}\right)}$$

Frequency Modulation

• Frequency deviation $$\delta=\left(f_{\text {max }}-f_{c}\right)=f_{c}-f_{\text {min }}=k_{f} \cdot \frac{E_{m}}{2 \pi}$$
• Carrier swing (CS) $$=C S=2 \times \Delta f$$
• Frequency modulation index $\left(m_{f}\right)$

$$m_{f}=\frac{\delta}{f_{m}}=\frac{f_{\max }-f_{c}}{f_{m}}=\frac{f_{c}-f_{\text {min }}}{f_{m}}=\frac{k_{f} E_{m}}{f_{m}}$$

• Frequency spectrum $=\mathrm{FM}$ side band modulated signal consist of infinite number of side bands whose frequencies are

$$\left(f_{c} \pm f_{m}\right),\left(f_{c} \pm 2 f_{m}\right), \left(f_{c} \pm 3 f_{m}\right) \ldots$$

• Deviation ratio: $$ \text{Deviation ratio}=\frac{(\Delta f_{\max })}{\left(f_m \right)_{\max }}$$

• Percent modulation:

$$ m = \frac{ \Delta f_{actual} }{ \Delta f_{max }}$$