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Question #90178
Derive mathematical expression of an A.M. wave and indicate the sideband components. Why is the S.S.B. transmission beneficial ?

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.

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