28-3 Find and sketch all possible inverse Laplace transforms x(t) for X(s) = (s-3)(s+2)* Sketch the ROC in each case.
Transfer function
A transfer function (also known as system function or network function) of a system, subsystem, or component is a mathematical function that modifies the output of a system in each possible input. They are widely used in electronics and control systems.
Convolution Integral
Among all the electrical engineering students, this topic of convolution integral is very confusing. It is a mathematical operation of two functions f and g that produce another third type of function (f * g) , and this expresses how the shape of one is modified with the help of the other one. The process of computing it and the result function is known as convolution. After one is reversed and shifted, it is defined as the integral of the product of two functions. After producing the convolution function, the integral is evaluated for all the values of shift. The convolution integral has some similar features with the cross-correlation. The continuous or discrete variables for real-valued functions differ from cross-correlation (f * g) only by either of the two f(x) or g(x) is reflected about the y-axis or not. Therefore, it is a cross-correlation of f(x) and g(-x) or f(-x) and g(x), the cross-correlation operator is the adjoint of the operator of the convolution for complex-valued piecewise functions.
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![**Inverse Laplace Transform and Region of Convergence**
In this exercise, we are tasked with finding and sketching all possible inverse Laplace transforms \( x(t) \) for the function:
\[ X(s) = \frac{2s - 3}{(s - 3)(s + 2)^2} \]
Additionally, for each inverse transform, we must sketch the Region of Convergence (ROC).
### Steps to Solve:
1. **Partial Fraction Decomposition:**
- Begin by expressing \( X(s) \) in partial fraction form to simplify the inverse Laplace transform process.
2. **Determine Inverse Transforms:**
- Use known Laplace transform pairs to find \( x(t) \).
- Consider different time-domain conditions (e.g., causal, anti-causal).
3. **Sketch the Transform:**
- Illustrate \( x(t) \) for each case obtained from the inverse transformation.
4. **Region of Convergence:**
- For each \( x(t) \), identify and sketch the ROC in the complex plane to illustrate the range of \( s \) for which the Laplace transform converges.
This exercise involves analytical skills to decompose the given function and knowledge of Laplace transform properties to correctly identify possible inverse functions and corresponding ROCs.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F584ffa08-94ad-4eb4-a87d-fd258db9f802%2Ff1c2873b-edf6-4ba8-95b9-8002b2fa714f%2Fp1zzudm_processed.png&w=3840&q=75)
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