**Consider a Causal IIR Filter**A causal Infinite Impulse Response (IIR) filter is characterized by the following difference equation:\[ y[n] - 0.25y[n-1] = x[n] + 0.5x[n-1] \]To analyze this filter, we will address the following points:### (i) System Function and Stability**Determine the system function (including its region of convergence) and assess the filter's stability.****Solution:**1. **System Function (H(z))**: Using the z-transform of the difference equation, we can express the system function \( H(z) \) as: \[ Y(z) - 0.25Y(z)z^{-1} = X(z) + 0.5X(z)z^{-1} \] Rearrange this to solve for \( H(z) \): \[ Y(z) (1 - 0.25z^{-1}) = X(z) (1 + 0.5z^{-1}) \] \[ H(z) = \frac{Y(z)}{X(z)} = \frac{1 + 0.5z^{-1}}{1 - 0.25z^{-1}} \] Writing in standard form: \[ H(z) = \frac{1 + 0.5z^{-1}}{1 - 0.25z^{-1}} \] 2. **Region of Convergence (ROC)**: For causal systems, the ROC is \( |z| > 0.25 \).3. **Stability**: A system is stable if the ROC includes the unit circle \( |z| = 1 \). Here, the ROC \( |z| > 0.25 \) does indeed include the unit circle, hence the filter is stable.### (ii) Filter Response to a Given Input Sequence**Using z-transforms, determine the response \( y[n] \) of this filter to the two-sided input sequence:**\[ x[n] = 3.0u[-n - 1] + 2.0(0.75)^n u[n] \]**Solution:**1. The given input sequence consists of two terms: - \( 3.0u[-n - 1] \): A step function starting at \( n = -1

Causal IIR filters are digital systems which depends on present input, past inputs and past outputs.…

Consider the (causal) IIR filter described by the difference equation y[n] – 0.25y[n – 1] = x[n] + 0.5x[n – 1] (i) Determine the system function of this filter (including its region of convergence). Is the filter stable? (ii) Using z-transforms, determine the response y[n] of this filter to the two-sided input sequence x[n] = 3.0u[–n - 1] + 2.0(0.75)"u[n] (iii) Determine the impulse response of a noncausal LTI system that satisfies the same difference equation.

Consider the (causal) IIR filter described by the difference equation y[n] – 0.25y[n – 1] = x[n] + 0.5x[n – 1] (i) Determine the system function of this filter (including its region of convergence). Is the filter stable? (ii) Using z-transforms, determine the response y[n] of this filter to the two-sided input sequence x[n] = 3.0u[–n - 1] + 2.0(0.75)"u[n] (iii) Determine the impulse response of a noncausal LTI system that satisfies the same difference equation.

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...

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