Question #90178
Derive mathematical expression of an A.M. wave and indicate the sideband components. Why is the S.S.B. transmission beneficial ?
1
2019-06-10T04:56:39-0400

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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