Answer to Question #90178 in Electrical Engineering for Shivam Nishad

Question

Derive mathematical expression of an A.M. wave and indicate the sideband components. Why is the S.S.B. transmission beneficial ?

Answer 1

Modulation is a technique which modifies waves of high frequency by the wave of low frequency.

Consider the carrier signal:

$$V_c(t)=V_c\text{sin}(\omega_ct),$$

the modulating signal:

$$V_m(t)=V_m\text{sin}(\omega_mt),$$

and the signal after modulation look like

$$V(t)=[V_c+V_m(t)]\text{sin}(\omega_ct).$$

Putting this all together gives

$$V(t)=V_c\text{sin}(\omega_ct)+\frac{V_m}{2}\text{cos}(\omega_ct-\omega_mt)-\frac{V_m}{2}\text{cos}(\omega_ct+\omega_mt).$$

Now multiply the terms with cosines by $V_c/V_c$:

$$V(t)=V_c\text{sin}(\omega_ct)+$$

$$+\frac{V_m}{2}\frac{V_c}{V_c}\text{cos}(\omega_ct-\omega_mt)-\frac{V_m}{2}\frac{V_c}{V_c}\text{cos}(\omega_ct+\omega_mt),$$

and substitute $V_m/V_c$ with $m_a$ - the amplitude modulation index.

$$V(t)=V_c\text{sin}(\omega_ct)+\frac{V_cm_a}{2}\text{cos}(\omega_ct-\omega_mt)-\frac{V_cm_a}{2}\text{cos}(\omega_ct+\omega_mt).$$

The sideband components are

$$\frac{\omega_c-\omega_m}{2\pi}=f_c-f_m$$

and

$$\frac{\omega_c+\omega_m}{2\pi}=f_c+f_m.$$

The single-sideband transmission reduces the bandwidth transmitted and it allows to double the number of channels within the same frequency band.