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A sinusoidal message signal having root mean square value of 4 V and frequency of 1 kHz is fed to phase modulator with phase deviation constant 2 rad/volt. If the carrier signal is c(t ) = 2 cos(2π106 t ) , the maximum instantaneous frequency of the phase modulated signal (rounded off to one decimal place) is ________ Hz.
Correct answer is '1011313.71'. Can you explain this answer?
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Most Upvoted Answer
A sinusoidal message signal having root mean square value of 4 V and f...
The formula for phase modulation is given by:
s(t) = Acos(2πfct + β∫m(t)dt)
Where:
s(t) = modulated signal
A = amplitude of the carrier signal
fc = frequency of the carrier signal
β = phase deviation constant
m(t) = message signal
In this case, the carrier signal is c(t) = 2cos(2πfct), where fc is the carrier frequency. Let's assume fc = 1 kHz.
The message signal has a root mean square (RMS) value of 4 V. The RMS value is equal to the peak value divided by the square root of 2. Therefore, the peak value of the message signal is:
Vm = RMS * √2 = 4 * √2 = 5.657 V
The frequency of the message signal is 1 kHz.
Now, we can calculate the phase deviation constant using the formula:
β = 2π * Δφ / ΔV
Where Δφ is the phase deviation and ΔV is the corresponding voltage deviation.
In this case, Δφ = 2 rad and ΔV = 1 V (since the phase deviation constant is given as 2 rad/volt).
Therefore, β = 2π * 2 / 1 = 4π rad/V
Now, we can express the message signal as m(t) = Vm * sin(2πfmt), where fm is the frequency of the message signal.
Given that fm = 1 kHz, we can write m(t) = 5.657 * sin(2π * 1 kHz * t).
Finally, we can express the modulated signal as:
s(t) = Acos(2πfct + β∫m(t)dt) = 2cos(2π * 1 kHz * t + 4π ∫[0,t] 5.657 * sin(2π * 1 kHz * τ)dτ)
To simplify the expression, we can integrate the sine function:
∫[0,t] sin(2π * 1 kHz * τ)dτ = -cos(2π * 1 kHz * τ) / (2π * 1 kHz) evaluated from 0 to t
= (-cos(2π * 1 kHz * t) + cos(0)) / (2π * 1 kHz)
= (-cos(2π * 1 kHz * t) + 1) / (2π * 1 kHz)
Substituting this back into the modulated signal expression:
s(t) = 2cos(2π * 1 kHz * t + 4π [(-cos(2π * 1 kHz * t) + 1) / (2π * 1 kHz)])
= 2cos(2π * 1 kHz * t - 4cos(2π * 1 kHz * t) + 4)
= 2cos(2π * 1 kHz * t - 4cos(2π * 1 kHz * t) + 4)
This is the final expression for the modulated signal.
s(t) = Acos(2πfct + β∫m(t)dt)
Where:
s(t) = modulated signal
A = amplitude of the carrier signal
fc = frequency of the carrier signal
β = phase deviation constant
m(t) = message signal
In this case, the carrier signal is c(t) = 2cos(2πfct), where fc is the carrier frequency. Let's assume fc = 1 kHz.
The message signal has a root mean square (RMS) value of 4 V. The RMS value is equal to the peak value divided by the square root of 2. Therefore, the peak value of the message signal is:
Vm = RMS * √2 = 4 * √2 = 5.657 V
The frequency of the message signal is 1 kHz.
Now, we can calculate the phase deviation constant using the formula:
β = 2π * Δφ / ΔV
Where Δφ is the phase deviation and ΔV is the corresponding voltage deviation.
In this case, Δφ = 2 rad and ΔV = 1 V (since the phase deviation constant is given as 2 rad/volt).
Therefore, β = 2π * 2 / 1 = 4π rad/V
Now, we can express the message signal as m(t) = Vm * sin(2πfmt), where fm is the frequency of the message signal.
Given that fm = 1 kHz, we can write m(t) = 5.657 * sin(2π * 1 kHz * t).
Finally, we can express the modulated signal as:
s(t) = Acos(2πfct + β∫m(t)dt) = 2cos(2π * 1 kHz * t + 4π ∫[0,t] 5.657 * sin(2π * 1 kHz * τ)dτ)
To simplify the expression, we can integrate the sine function:
∫[0,t] sin(2π * 1 kHz * τ)dτ = -cos(2π * 1 kHz * τ) / (2π * 1 kHz) evaluated from 0 to t
= (-cos(2π * 1 kHz * t) + cos(0)) / (2π * 1 kHz)
= (-cos(2π * 1 kHz * t) + 1) / (2π * 1 kHz)
Substituting this back into the modulated signal expression:
s(t) = 2cos(2π * 1 kHz * t + 4π [(-cos(2π * 1 kHz * t) + 1) / (2π * 1 kHz)])
= 2cos(2π * 1 kHz * t - 4cos(2π * 1 kHz * t) + 4)
= 2cos(2π * 1 kHz * t - 4cos(2π * 1 kHz * t) + 4)
This is the final expression for the modulated signal.
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Community Answer
A sinusoidal message signal having root mean square value of 4 V and f...
Given
Message signal frequency, fm=1 kHz
Carrier signal, c(t ) = 2 cos(2π106 t )
Phase sensitivity factor, kp=2 rad/V
Assuming message signal, m(t) = Am sin(2πfmt )
RMS value of message signal m(t) is,
So, phase modulation equation is,
Instantaneous frequency is given as,
So, maximum value of instantaneous frequency is,
Hence, the maximum instantaneous frequency of the phase modulated signal is 1011313.7 Hz.
Question_Type: 4
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A sinusoidal message signal having root mean square value of 4 V and frequency of 1 kHz is fed to phase modulator with phase deviation constant 2 rad/volt. If the carrier signal is c(t ) = 2 cos(2π106 t ) , the maximum instantaneous frequency of the phase modulated signal (rounded off to one decimal place) is ________ Hz.Correct answer is '1011313.71'. Can you explain this answer?
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A sinusoidal message signal having root mean square value of 4 V and frequency of 1 kHz is fed to phase modulator with phase deviation constant 2 rad/volt. If the carrier signal is c(t ) = 2 cos(2π106 t ) , the maximum instantaneous frequency of the phase modulated signal (rounded off to one decimal place) is ________ Hz.Correct answer is '1011313.71'. Can you explain this answer? for Electronics and Communication Engineering (ECE) 2024 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about A sinusoidal message signal having root mean square value of 4 V and frequency of 1 kHz is fed to phase modulator with phase deviation constant 2 rad/volt. If the carrier signal is c(t ) = 2 cos(2π106 t ) , the maximum instantaneous frequency of the phase modulated signal (rounded off to one decimal place) is ________ Hz.Correct answer is '1011313.71'. Can you explain this answer? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A sinusoidal message signal having root mean square value of 4 V and frequency of 1 kHz is fed to phase modulator with phase deviation constant 2 rad/volt. If the carrier signal is c(t ) = 2 cos(2π106 t ) , the maximum instantaneous frequency of the phase modulated signal (rounded off to one decimal place) is ________ Hz.Correct answer is '1011313.71'. Can you explain this answer?.
A sinusoidal message signal having root mean square value of 4 V and frequency of 1 kHz is fed to phase modulator with phase deviation constant 2 rad/volt. If the carrier signal is c(t ) = 2 cos(2π106 t ) , the maximum instantaneous frequency of the phase modulated signal (rounded off to one decimal place) is ________ Hz.Correct answer is '1011313.71'. Can you explain this answer? for Electronics and Communication Engineering (ECE) 2024 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about A sinusoidal message signal having root mean square value of 4 V and frequency of 1 kHz is fed to phase modulator with phase deviation constant 2 rad/volt. If the carrier signal is c(t ) = 2 cos(2π106 t ) , the maximum instantaneous frequency of the phase modulated signal (rounded off to one decimal place) is ________ Hz.Correct answer is '1011313.71'. Can you explain this answer? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A sinusoidal message signal having root mean square value of 4 V and frequency of 1 kHz is fed to phase modulator with phase deviation constant 2 rad/volt. If the carrier signal is c(t ) = 2 cos(2π106 t ) , the maximum instantaneous frequency of the phase modulated signal (rounded off to one decimal place) is ________ Hz.Correct answer is '1011313.71'. Can you explain this answer?.
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