Problem 11.5_2 Verify the following equation. e-H(t) * e¹H(-t) = ½e-²1.
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![**Problem 11.5_2**
Verify the following equation.
\[ e^{-t}H(t) * e^{t}H(-t) = \frac{1}{2}e^{-|t|}. \]
**Explanation:**
This problem involves verifying an equation related to convolution in signal processing or systems analysis.
- **\( e^{-t}H(t) \)**: Represents an exponentially decaying function multiplied by the Heaviside step function, \( H(t) \), which ensures that the function is zero for \( t < 0 \).
- **\( e^{t}H(-t) \)**: Represents an exponentially growing function, which is multiplied by \( H(-t) \) to ensure the function is zero for \( t > 0 \).
- **Convolution symbol \(*\)**: Indicates the convolution operation between two functions, which is a key concept in systems analysis, often used to analyze the output of linear time-invariant systems.
- **Result \( \frac{1}{2}e^{-|t|} \)**: This expression represents a potential output of the convolution operation, a function that describes exponentially decaying behavior but symmetric with respect to the vertical axis (due to the absolute value in the exponent).
The task is to demonstrate that the left-hand side convolution indeed simplifies to the right-hand side exponential function, using properties of the functions involved and convolution techniques.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F780f9839-f737-4aba-91a2-6210989911b1%2F97d8d163-b0d2-4b6b-a720-35a0bbb0d72f%2F5uhrji_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 11.5_2**
Verify the following equation.
\[ e^{-t}H(t) * e^{t}H(-t) = \frac{1}{2}e^{-|t|}. \]
**Explanation:**
This problem involves verifying an equation related to convolution in signal processing or systems analysis.
- **\( e^{-t}H(t) \)**: Represents an exponentially decaying function multiplied by the Heaviside step function, \( H(t) \), which ensures that the function is zero for \( t < 0 \).
- **\( e^{t}H(-t) \)**: Represents an exponentially growing function, which is multiplied by \( H(-t) \) to ensure the function is zero for \( t > 0 \).
- **Convolution symbol \(*\)**: Indicates the convolution operation between two functions, which is a key concept in systems analysis, often used to analyze the output of linear time-invariant systems.
- **Result \( \frac{1}{2}e^{-|t|} \)**: This expression represents a potential output of the convolution operation, a function that describes exponentially decaying behavior but symmetric with respect to the vertical axis (due to the absolute value in the exponent).
The task is to demonstrate that the left-hand side convolution indeed simplifies to the right-hand side exponential function, using properties of the functions involved and convolution techniques.
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