A causal LTI system has the transfer function H(s) given below: 28 H(s) = (s+1-j100) (s+1+j100)* 1) Determine a differential equation for this system. 2) Sketch the pole-zero plot for this system. 3) Sketch the frequency response magnitude of this system. Do not use a log scale, i.e., do not sketch the Bode plot.
A causal LTI system has the transfer function H(s) given below: 28 H(s) = (s+1-j100) (s+1+j100)* 1) Determine a differential equation for this system. 2) Sketch the pole-zero plot for this system. 3) Sketch the frequency response magnitude of this system. Do not use a log scale, i.e., do not sketch the Bode plot.
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![**Title: Analysis of a Causal LTI System with Given Transfer Function**
**Transfer Function:**
A causal LTI (Linear Time-Invariant) system has the transfer function \( H(s) \) defined as:
\[
H(s) = \frac{2s}{(s + 1 - j100)(s + 1 + j100)}
\]
**Tasks:**
1. **Determine a Differential Equation for this System:**
To find the differential equation, express the transfer function in terms of time-domain variables by taking the inverse Laplace transform. This involves equating \( H(s) \) to the ratio of polynomials in \( s \), corresponding to derivatives in the time domain.
2. **Sketch the Pole-Zero Plot for this System:**
- **Poles**: The system's poles are located at \( s = -1 + j100 \) and \( s = -1 - j100 \).
- **Zero**: There is a zero at the origin, \( s = 0 \).
The pole-zero plot involves marking these poles and the zero on the complex plane.
3. **Sketch the Frequency Response Magnitude of this System:**
To sketch the frequency response magnitude, calculate the magnitude of \( H(j\omega) \) over a range of frequencies. This involves substituting \( s = j\omega \) and plotting the results on a linear scale.
4. **What Type of Filter is this System?**
Analyze the position and frequency response characteristics to determine if it behaves like a low-pass, high-pass, band-pass, or band-stop filter.
5. **Determine the Output \( y(t) \) of this System when the Input is \( x(t) = u(t) - e^{-3t}u(t) \):**
- \( u(t) \): Unit step function.
- \( e^{-3t}u(t) \): Exponential decay starting from \( t = 0 \).
Use convolution or Laplace transform properties to find the response \( y(t) \).
6. **Is this System Stable? Why or Why Not?**
Evaluate stability by analyzing the location of poles:
- A system is stable if all poles have negative real parts.
- With poles at \( s = -1 \pm j100 \), both poles have negative real parts.
Thus, the system is stable.
This analysis](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F39a3543a-bde3-4119-be75-304b57c879c6%2Fe177efd3-99d2-4a38-972f-0a3c79414a09%2Fdyj2uz_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Analysis of a Causal LTI System with Given Transfer Function**
**Transfer Function:**
A causal LTI (Linear Time-Invariant) system has the transfer function \( H(s) \) defined as:
\[
H(s) = \frac{2s}{(s + 1 - j100)(s + 1 + j100)}
\]
**Tasks:**
1. **Determine a Differential Equation for this System:**
To find the differential equation, express the transfer function in terms of time-domain variables by taking the inverse Laplace transform. This involves equating \( H(s) \) to the ratio of polynomials in \( s \), corresponding to derivatives in the time domain.
2. **Sketch the Pole-Zero Plot for this System:**
- **Poles**: The system's poles are located at \( s = -1 + j100 \) and \( s = -1 - j100 \).
- **Zero**: There is a zero at the origin, \( s = 0 \).
The pole-zero plot involves marking these poles and the zero on the complex plane.
3. **Sketch the Frequency Response Magnitude of this System:**
To sketch the frequency response magnitude, calculate the magnitude of \( H(j\omega) \) over a range of frequencies. This involves substituting \( s = j\omega \) and plotting the results on a linear scale.
4. **What Type of Filter is this System?**
Analyze the position and frequency response characteristics to determine if it behaves like a low-pass, high-pass, band-pass, or band-stop filter.
5. **Determine the Output \( y(t) \) of this System when the Input is \( x(t) = u(t) - e^{-3t}u(t) \):**
- \( u(t) \): Unit step function.
- \( e^{-3t}u(t) \): Exponential decay starting from \( t = 0 \).
Use convolution or Laplace transform properties to find the response \( y(t) \).
6. **Is this System Stable? Why or Why Not?**
Evaluate stability by analyzing the location of poles:
- A system is stable if all poles have negative real parts.
- With poles at \( s = -1 \pm j100 \), both poles have negative real parts.
Thus, the system is stable.
This analysis
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