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A sinusoidal signal x(t) = 2sin(100t + π/3) is passed through a square law device defined by input-output relation y(t) = x2(t), then the DC component in the signal is:
- a)4
- b)2
- c)1
- d)2√3 / √2
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
A sinusoidal signalx(t) = 2sin(100t + π/3)is passed through a squar...
The square law device produces an output signal that is equal to the square of the input signal. When a sinusoidal signal is passed through a square law device, the output signal will contain both the fundamental frequency and harmonics at higher frequencies. The DC component of the output signal is the average value of the output signal over time, and it is equal to the sum of the DC component of the input signal and the DC component of the harmonics.
In this case, the input signal is x(t) = 2sin(100t + π/3). The DC component of this signal is 0, since the average value of a sinusoidal signal over one period is 0. The output signal is y(t) = x^2(t) = (2sin(100t + π/3))^2 = 4cos^2(100t + π/3) = 4(1 - sin^2(100t + π/3)) = 4(1 - (2sin^2(100t + π/3)/2)) = 4(1 - sin^2(100t + π/3)/2). The DC component of this signal is 4/2 = 2.
Therefore, the correct answer is 2
Most Upvoted Answer
A sinusoidal signalx(t) = 2sin(100t + π/3)is passed through a squar...
The square law device produces an output signal that is equal to the square of the input signal. When a sinusoidal signal is passed through a square law device, the output signal will contain both the fundamental frequency and harmonics at higher frequencies. The DC component of the output signal is the average value of the output signal over time, and it is equal to the sum of the DC component of the input signal and the DC component of the harmonics.
In this case, the input signal is x(t) = 2sin(100t + π/3). The DC component of this signal is 0, since the average value of a sinusoidal signal over one period is 0. The output signal is y(t) = x^2(t) = (2sin(100t + π/3))^2 = 4cos^2(100t + π/3) = 4(1 - sin^2(100t + π/3)) = 4(1 - (2sin^2(100t + π/3)/2)) = 4(1 - sin^2(100t + π/3)/2). The DC component of this signal is 4/2 = 2.
Therefore, the correct answer is 2
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A sinusoidal signalx(t) = 2sin(100t + π/3)is passed through a squar...
Understanding the Sinusoidal Signal
The given sinusoidal signal is:
- x(t) = 2sin(100t + π/3)
This signal oscillates between -2 and +2, with a frequency determined by the coefficient of t.
Passing through the Square Law Device
The square law device has the relationship:
- y(t) = x^2(t)
To find the output signal, we need to compute x^2(t):
- y(t) = (2sin(100t + π/3))^2
- y(t) = 4sin^2(100t + π/3)
Using Trigonometric Identity
We can simplify sin^2 using the identity:
- sin^2(θ) = (1 - cos(2θ))/2
Applying this to our equation:
- y(t) = 4 * (1 - cos(200t + 2π/3))/2
- y(t) = 2(1 - cos(200t + 2π/3))
- y(t) = 2 - 2cos(200t + 2π/3)
Identifying the DC Component
The DC component of a signal is defined as the constant term (the part that does not oscillate). From the expression of y(t):
- The term "2" is the DC component.
- The term "-2cos(200t + 2π/3)" oscillates and does not contribute to the DC component.
Conclusion
Thus, the DC component in the signal y(t) is:
- 2
The correct answer is option 'B'.
The given sinusoidal signal is:
- x(t) = 2sin(100t + π/3)
This signal oscillates between -2 and +2, with a frequency determined by the coefficient of t.
Passing through the Square Law Device
The square law device has the relationship:
- y(t) = x^2(t)
To find the output signal, we need to compute x^2(t):
- y(t) = (2sin(100t + π/3))^2
- y(t) = 4sin^2(100t + π/3)
Using Trigonometric Identity
We can simplify sin^2 using the identity:
- sin^2(θ) = (1 - cos(2θ))/2
Applying this to our equation:
- y(t) = 4 * (1 - cos(200t + 2π/3))/2
- y(t) = 2(1 - cos(200t + 2π/3))
- y(t) = 2 - 2cos(200t + 2π/3)
Identifying the DC Component
The DC component of a signal is defined as the constant term (the part that does not oscillate). From the expression of y(t):
- The term "2" is the DC component.
- The term "-2cos(200t + 2π/3)" oscillates and does not contribute to the DC component.
Conclusion
Thus, the DC component in the signal y(t) is:
- 2
The correct answer is option 'B'.
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