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The auto-correlation function of a rectangular pulse of duration T is
- a)A rectangular pulse of duration T
- b)A rectangular pulse of duration 2T
- c)A triangular pulse of duration T
- d)A triangular pulse of duration 2T
Correct answer is option 'D'. Can you explain this answer?
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The auto-correlation function of a rectangular pulse of duration T isa...
Explanation: The auto-correlation function is the method of correlating the various instants of the signal with itself and that of a rectangular pulse of duration T is a triangular pulse of duration 2T.
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The auto-correlation function of a rectangular pulse of duration T isa...
Auto-Correlation Function of a Rectangular Pulse
Auto-correlation function is a statistical measure that describes the similarity between a signal and a delayed version of itself. In signal processing, it is used to analyze the periodicity and cyclic nature of a signal. When a rectangular pulse of duration T is considered, the auto-correlation function is given by:
ACF(τ) = (1/T) ∫_0^T-x ∫_0^T-x f(t) f(t-τ) dt dx
where f(t) is the rectangular pulse of duration T.
Interpretation of Auto-Correlation Function
The auto-correlation function gives information about the periodicity and cyclic nature of a signal. It is a measure of the similarity between the signal and a delayed version of itself. The following interpretations can be made from the auto-correlation function:
- When the auto-correlation function is maximum at τ = 0, it indicates that the signal is highly periodic and has a strong correlation with itself.
- When the auto-correlation function is maximum at τ = T, it indicates that the signal has a period of T and is periodic with a shift of T.
- When the auto-correlation function is minimum at τ = T/2, it indicates that the signal is anti-symmetric and has no periodicity.
Auto-Correlation Function of a Rectangular Pulse of Duration T
In the case of a rectangular pulse of duration T, the auto-correlation function is given by:
ACF(τ) = (1/T) ∫_0^T-x ∫_0^T-x f(t) f(t-τ) dt dx
where f(t) = 1 for 0 ≤ t ≤ T and f(t) = 0 for t < 0="" or="" t="" /> T.
Solving the integral, we get:
ACF(τ) = (1/T) ∫_0^(T-τ) x(T-x) dx
ACF(τ) = (T^2/3) - (τ^2/2) + (τ^3/3T)
The auto-correlation function is a triangular pulse of duration 2T, which is maximum at τ = 0 and minimum at τ = T.
Hence, the correct answer is option D, i.e., a triangular pulse of duration 2T.
Auto-correlation function is a statistical measure that describes the similarity between a signal and a delayed version of itself. In signal processing, it is used to analyze the periodicity and cyclic nature of a signal. When a rectangular pulse of duration T is considered, the auto-correlation function is given by:
ACF(τ) = (1/T) ∫_0^T-x ∫_0^T-x f(t) f(t-τ) dt dx
where f(t) is the rectangular pulse of duration T.
Interpretation of Auto-Correlation Function
The auto-correlation function gives information about the periodicity and cyclic nature of a signal. It is a measure of the similarity between the signal and a delayed version of itself. The following interpretations can be made from the auto-correlation function:
- When the auto-correlation function is maximum at τ = 0, it indicates that the signal is highly periodic and has a strong correlation with itself.
- When the auto-correlation function is maximum at τ = T, it indicates that the signal has a period of T and is periodic with a shift of T.
- When the auto-correlation function is minimum at τ = T/2, it indicates that the signal is anti-symmetric and has no periodicity.
Auto-Correlation Function of a Rectangular Pulse of Duration T
In the case of a rectangular pulse of duration T, the auto-correlation function is given by:
ACF(τ) = (1/T) ∫_0^T-x ∫_0^T-x f(t) f(t-τ) dt dx
where f(t) = 1 for 0 ≤ t ≤ T and f(t) = 0 for t < 0="" or="" t="" /> T.
Solving the integral, we get:
ACF(τ) = (1/T) ∫_0^(T-τ) x(T-x) dx
ACF(τ) = (T^2/3) - (τ^2/2) + (τ^3/3T)
The auto-correlation function is a triangular pulse of duration 2T, which is maximum at τ = 0 and minimum at τ = T.
Hence, the correct answer is option D, i.e., a triangular pulse of duration 2T.
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The auto-correlation function of a rectangular pulse of duration T isa)A rectangular pulse of duration Tb)A rectangular pulse of duration 2Tc)A triangular pulse of duration Td)A triangular pulse of duration 2TCorrect answer is option 'D'. Can you explain this answer?
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