GATE EC 2025 Set 1 — Question 14
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Communications → Analog Communications → Angle Modulation (FM & PM)
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Question
Consider a frequency-modulated (FM) signal , where and are, respectively, the amplitude and frequency (in Hz) of the carrier waveform. The frequency is in Hz, and assume that .
The peak frequency deviation of the FM signal in Hz is ________
The peak frequency deviation of the FM signal in Hz is ________
Correct answer
(A) 15f₁
Solution
The general form of an FM signal is , where is the instantaneous phase deviation.
From the given signal , we can identify the instantaneous phase deviation as:
The instantaneous frequency deviation is given by:
First, let's find the derivative of with respect to :
Now, substitute this into the formula for :
The peak frequency deviation, , is the maximum absolute value of .
The maximum value of a sum of cosine functions occurs when all cosine terms are at their maximum value (i.e., 1) simultaneously. This is possible here because , so if , then for some integer , which implies , and .So, the peak frequency deviation is:
The final answer is .
From the given signal , we can identify the instantaneous phase deviation as:
The instantaneous frequency deviation is given by:
First, let's find the derivative of with respect to :
Now, substitute this into the formula for :
The peak frequency deviation, , is the maximum absolute value of .
The maximum value of a sum of cosine functions occurs when all cosine terms are at their maximum value (i.e., 1) simultaneously. This is possible here because , so if , then for some integer , which implies , and .So, the peak frequency deviation is:
The final answer is .
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