c.) d.) g[n] = 3x[n] + 10[n − 2] + 8x[n − 4] h[n] = (−0.4)^u[n] + 2( −0.75)^='u[n = 1] =
c.) d.) g[n] = 3x[n] + 10[n − 2] + 8x[n − 4] h[n] = (−0.4)^u[n] + 2( −0.75)^='u[n = 1] =
Introductory Circuit Analysis (13th Edition)
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Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
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![### BC:7.1 Problem Statement
For each of the following Linear Time-Invariant (LTI) systems, determine the **z-domain transfer function**, identifying its **zeros and poles**. The following parameters are provided for these expressions:
- **\( h[n] \)** is the impulse response.
- **\( x[n] \)** is the input signal.
- **\( y[n] \)** is the output signal.
Your task is to express your answer in one of the following forms:
- As a **z polynomial of finite order**.
- As the **ratio of two z polynomials of finite order**.
### Assumptions:
1. Assume the input signal commences from **\( n = 0 \)** or later.
2. Assume that **\( y[n] = 0 \)** for **\( n < 0 \)**.
### Additional Task:
- **State whether the system is FIR or IIR and briefly justify your answer.**
This problem involves understanding the fundamental concepts of LTI systems, including transfer functions, zeros, and poles, and determining whether a given system is Finite Impulse Response (FIR) or Infinite Impulse Response (IIR).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F028a22e8-f094-478d-95cf-e0334af4f4a7%2Fa0f76c32-044c-4145-bbce-2806fd361e72%2Fw44u5ed_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### BC:7.1 Problem Statement
For each of the following Linear Time-Invariant (LTI) systems, determine the **z-domain transfer function**, identifying its **zeros and poles**. The following parameters are provided for these expressions:
- **\( h[n] \)** is the impulse response.
- **\( x[n] \)** is the input signal.
- **\( y[n] \)** is the output signal.
Your task is to express your answer in one of the following forms:
- As a **z polynomial of finite order**.
- As the **ratio of two z polynomials of finite order**.
### Assumptions:
1. Assume the input signal commences from **\( n = 0 \)** or later.
2. Assume that **\( y[n] = 0 \)** for **\( n < 0 \)**.
### Additional Task:
- **State whether the system is FIR or IIR and briefly justify your answer.**
This problem involves understanding the fundamental concepts of LTI systems, including transfer functions, zeros, and poles, and determining whether a given system is Finite Impulse Response (FIR) or Infinite Impulse Response (IIR).
![### Transcription and Analysis
#### Part c
The equation given is:
\[ y[n] = 3x[n] + 10x[n-2] + 8x[n-4] \]
**Explanation:**
- This equation represents a discrete-time linear system.
- \( y[n] \) is the output signal.
- \( x[n] \) is the input signal at time \( n \).
- The equation includes terms with time delays, indicating a relation between the current output and past inputs:
- \( 3x[n] \) refers to the input at the current time.
- \( 10x[n-2] \) refers to the input delayed by 2 time units.
- \( 8x[n-4] \) refers to the input delayed by 4 time units.
#### Part d
The equation provided is:
\[ h[n] = (-0.4)^n u[n] + 2(-0.75)^{n-1} u[n-1] \]
**Explanation:**
- This equation describes the impulse response of a discrete-time system.
- \( h[n] \) is the impulse response at time \( n \).
- \( u[n] \) and \( u[n-1] \) are unit step functions, indicating causality.
- The expressions \( (-0.4)^n \) and \( 2(-0.75)^{n-1} \) represent coefficients that may attenuate or amplify the response at different time steps:
- \( (-0.4)^n u[n] \) denotes the contribution of the current and future steps to the response based on the exponent \( n \).
- \( 2(-0.75)^{n-1} u[n-1] \) indicates the contribution from a previous step (shifted by one unit into the future in this equation context).
These equations are fundamental in understanding signal processing and systems analysis within the context of electrical engineering and computer science education.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F028a22e8-f094-478d-95cf-e0334af4f4a7%2Fa0f76c32-044c-4145-bbce-2806fd361e72%2Ftbrq0gr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Transcription and Analysis
#### Part c
The equation given is:
\[ y[n] = 3x[n] + 10x[n-2] + 8x[n-4] \]
**Explanation:**
- This equation represents a discrete-time linear system.
- \( y[n] \) is the output signal.
- \( x[n] \) is the input signal at time \( n \).
- The equation includes terms with time delays, indicating a relation between the current output and past inputs:
- \( 3x[n] \) refers to the input at the current time.
- \( 10x[n-2] \) refers to the input delayed by 2 time units.
- \( 8x[n-4] \) refers to the input delayed by 4 time units.
#### Part d
The equation provided is:
\[ h[n] = (-0.4)^n u[n] + 2(-0.75)^{n-1} u[n-1] \]
**Explanation:**
- This equation describes the impulse response of a discrete-time system.
- \( h[n] \) is the impulse response at time \( n \).
- \( u[n] \) and \( u[n-1] \) are unit step functions, indicating causality.
- The expressions \( (-0.4)^n \) and \( 2(-0.75)^{n-1} \) represent coefficients that may attenuate or amplify the response at different time steps:
- \( (-0.4)^n u[n] \) denotes the contribution of the current and future steps to the response based on the exponent \( n \).
- \( 2(-0.75)^{n-1} u[n-1] \) indicates the contribution from a previous step (shifted by one unit into the future in this equation context).
These equations are fundamental in understanding signal processing and systems analysis within the context of electrical engineering and computer science education.
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Step 1: We need to determine transfer function, poles and zeros and FIR or IIR system.
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