GATE EC 2023 Set 1 — Question 45
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Communications → Analog Communications → Angle Modulation (FM & PM)
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Question
Let a frequency modulated (FM) signal
, where is a message signal of bandwidth . It is passed through a non-linear system with output . Let denote the FM bandwidth. The minimum value of required to recover from is
, where is a message signal of bandwidth . It is passed through a non-linear system with output . Let denote the FM bandwidth. The minimum value of required to recover from is
Correct answer
(B) (3)/(2) B_T
Solution
The FM signal is given by , where . The bandwidth of is . Its spectrum is centered at and occupies the frequency range .The output of the non-linear system is . Expanding the term:
The spectrum of consists of:
To recover without interference using a bandpass filter, the upper frequency limit of the component must be less than the lower frequency limit of the component:
Substituting :
For wideband FM, where , and . In this limit, the condition simplifies to:
.Thus, the minimum value of required is .
The spectrum of consists of:
1.A DC component at Hz.
2.The desired signal centered at with bandwidth .
3.A second-order FM component centered at with bandwidth .
According to Carson's rule:To recover without interference using a bandpass filter, the upper frequency limit of the component must be less than the lower frequency limit of the component:
Substituting :
For wideband FM, where , and . In this limit, the condition simplifies to:
.Thus, the minimum value of required is .
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