question_answer

A modulating signal is a square wave as shown in figure. The carrier wave is given by \[c(t)=2\sin (8\pi t)\] volt.

(i) Sketch the amplitude modulated waveform.

(ii) What is the modulation index?

Answer:

(ii) 0.5 Given, the equation of carrier wave, \[c\,\,(t)=2sin(8\pi t)\]             (i) According to the figure, Amplitude of modulating signal, \[{{A}_{m}}=1V\] Amplitude of carrier wave, \[{{A}_{c}}=2\]                                   [from Eq. (i)] \[{{T}_{m}}=1s\]                                                [from figure] From Eq. (i), we get                                 \[{{\omega }_{m}}=\frac{2\pi }{{{T}_{m}}}=\frac{2\pi }{1}=2\pi \,\,rad/s\]     ? (ii) \[c\,\,(t)=2sin\,\,(8\pi t)={{A}_{c}}\sin {{\omega }_{c}}t\]                 So,          \[{{\omega }_{c}}=8\pi \]                 From Eq. (ii), we get So,          \[{{\omega }_{c}}=4{{\omega }_{m}}\] Amplitude of modulated wave, \[A={{A}_{m}}+{{A}_{c}}=1+2=3\,V\] The sketch of the amplitude modulated waveform is shown below: For carrier signal, \[\omega =8\pi \]                         \[T=\frac{2\pi }{\omega }\]                         \[=\frac{2\pi }{8\pi }=\frac{1}{4}=0.25\,s\] (ii) According to the figure, Amplitude of modulating signal, \[{{A}_{m}}=1V\] Amplitude of carrier wave, \[{{A}_{c}}=2V\] Modulation index, \[\mu ={{\Alpha }_{m}}/{{A}_{C}}=1/2=0.5\]