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Convolution of x(t + 5) with impulse function δ(t - 7) is equal to
- a)x(t - 2)
- b)x(t + 12)
- c)x(t - 12)
- d)(t + 2)
Correct answer is option 'A'. Can you explain this answer?
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Convolution of x(t + 5) with impulse function δ(t - 7) is equal toa)x(...
Convolution of | x(t + s) with d(t - 7)
x(t - 7 + 5) 
x(t - 2).Most Upvoted Answer
Convolution of x(t + 5) with impulse function δ(t - 7) is equal toa)x(...
Convolution of x(t - 5) with impulse function δ(t - 7) is equal to x(t - 2).
To understand why the correct answer is option 'A', let's break down the convolution process and apply it to the given functions step by step:
1. Understanding the impulse function δ(t - 7):
The impulse function δ(t - 7) represents an infinitesimally narrow pulse located at t = 7. It has an area of 1 and behaves as a unit impulse at t = 7.
2. Shifting x(t - 5) to the right by 5 units:
The function x(t - 5) represents a signal that is shifted to the right by 5 units. This means that all the values of x(t - 5) are shifted 5 units to the right compared to the original signal x(t).
3. Convolution integral:
The convolution integral between x(t - 5) and δ(t - 7) is given by:
y(t) = ∫[x(τ - 5) * δ(t - τ)] dτ
Where y(t) represents the convolution output.
4. Evaluating the convolution integral:
To find the convolution y(t), we need to evaluate the integral by substituting the given functions and solving it:
y(t) = ∫[x(τ - 5) * δ(t - τ)] dτ
= x(t - 5) * δ(t - 7) (since δ(t - τ) = 0 for all τ ≠ 7)
= x(t - 5) * δ(t - 7)
= x(t - 5) (since δ(t - 7) behaves as a unit impulse)
Hence, the convolution of x(t - 5) with δ(t - 7) is equal to x(t - 5).
5. Shifting x(t - 5) to the left by 3 units:
Since the convolution is equal to x(t - 5), we can further simplify it by shifting x(t - 5) to the left by 3 units:
y(t) = x(t - 5) = x(t - 5 - (-3)) = x(t - 2)
Therefore, the convolution of x(t - 5) with δ(t - 7) is equal to x(t - 2).
Hence, the correct answer is option 'A': x(t - 2).
To understand why the correct answer is option 'A', let's break down the convolution process and apply it to the given functions step by step:
1. Understanding the impulse function δ(t - 7):
The impulse function δ(t - 7) represents an infinitesimally narrow pulse located at t = 7. It has an area of 1 and behaves as a unit impulse at t = 7.
2. Shifting x(t - 5) to the right by 5 units:
The function x(t - 5) represents a signal that is shifted to the right by 5 units. This means that all the values of x(t - 5) are shifted 5 units to the right compared to the original signal x(t).
3. Convolution integral:
The convolution integral between x(t - 5) and δ(t - 7) is given by:
y(t) = ∫[x(τ - 5) * δ(t - τ)] dτ
Where y(t) represents the convolution output.
4. Evaluating the convolution integral:
To find the convolution y(t), we need to evaluate the integral by substituting the given functions and solving it:
y(t) = ∫[x(τ - 5) * δ(t - τ)] dτ
= x(t - 5) * δ(t - 7) (since δ(t - τ) = 0 for all τ ≠ 7)
= x(t - 5) * δ(t - 7)
= x(t - 5) (since δ(t - 7) behaves as a unit impulse)
Hence, the convolution of x(t - 5) with δ(t - 7) is equal to x(t - 5).
5. Shifting x(t - 5) to the left by 3 units:
Since the convolution is equal to x(t - 5), we can further simplify it by shifting x(t - 5) to the left by 3 units:
y(t) = x(t - 5) = x(t - 5 - (-3)) = x(t - 2)
Therefore, the convolution of x(t - 5) with δ(t - 7) is equal to x(t - 2).
Hence, the correct answer is option 'A': x(t - 2).
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Convolution of x(t + 5) with impulse function δ(t - 7) is equal toa)x(t - 2)b)x(t + 12)c)x(t - 12)d)(t + 2)Correct answer is option 'A'. Can you explain this answer?
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