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1. Design a digital Butterworth lowpass filter with the following specifications: Maximum attenuation in the passband = Ap = 0.6 dB Minimum attenuation in the stopband = A₁ = 15 dB Passband cutoff frequency = fp = 1830 Hz Stopband cutoff frequency = fs = 3020 Hz Sampling rate = f sampling = 10000 Hz excess tolerance = stopband (a) Find the digital cutoff frequencies 0p, 05. (b) Find the prewarped cutoff frequencies @p, 00s. (c) Find the cutoff frequencies 2p and 2, of the normalized lowpass filter.

1. Design a digital Butterworth lowpass filter with the following specifications: Maximum attenuation in the passband = Ap = 0.6 dB Minimum attenuation in the stopband = A₁ = 15 dB Passband cutoff frequency = fp = 1830 Hz Stopband cutoff frequency = fs = 3020 Hz Sampling rate = f sampling = 10000 Hz excess tolerance = stopband (a) Find the digital cutoff frequencies 0p, 05. (b) Find the prewarped cutoff frequencies @p, 00s. (c) Find the cutoff frequencies 2p and 2, of the normalized lowpass filter.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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### Designing a Digital Butterworth Lowpass Filter #### Filter Specifications: To design a digital Butterworth lowpass filter with the following specifications: - **Maximum attenuation in the passband** \(A_p\) = 0.6 dB - **Minimum attenuation in the stopband** \(A_s\) = 15 dB - **Passband cutoff frequency** \(f_p\) = 1830 Hz - **Stopband cutoff frequency** \(f_s\) = 3020 Hz - **Sampling rate** \(f_{\text{sampling}}\) = 10,000 Hz - **Excess tolerance** = Stopband #### Steps for Design Calculation: (a) **Find the digital cutoff frequencies** \(\theta_p\), \(\theta_s\). (b) **Find the prewarped cutoff frequencies** \(\omega_p\), \(\omega_s\). (c) **Find the cutoff frequencies** \(\Omega_p\) and \(\Omega_s\) **of the normalized lowpass filter**. (d) **Determine the order \(n\) of the normalized lowpass filter**. (e) **Find the recalculated values** \(A_{s1}\) and \(\Omega_c\). (f) **Locate the poles and zeros for the normalized lowpass filter**. (g) **Locate the frequency transformed lowpass poles and zeros**. (h) **Identify the digital lowpass poles and zeros**. Each of these steps involves specific formulae and calculations fundamental in signal processing and digital filter design. This step-by-step approach ensures designing an efficient filter that meets the required specifications. Understanding Butterworth filters and their characteristic of maximizing flatness in the passband, making them a popular choice in various applications such as audio processing, communication systems, and more, is crucial for practical implementation.
Transcribed Image Text:### Designing a Digital Butterworth Lowpass Filter #### Filter Specifications: To design a digital Butterworth lowpass filter with the following specifications: - **Maximum attenuation in the passband** \(A_p\) = 0.6 dB - **Minimum attenuation in the stopband** \(A_s\) = 15 dB - **Passband cutoff frequency** \(f_p\) = 1830 Hz - **Stopband cutoff frequency** \(f_s\) = 3020 Hz - **Sampling rate** \(f_{\text{sampling}}\) = 10,000 Hz - **Excess tolerance** = Stopband #### Steps for Design Calculation: (a) **Find the digital cutoff frequencies** \(\theta_p\), \(\theta_s\). (b) **Find the prewarped cutoff frequencies** \(\omega_p\), \(\omega_s\). (c) **Find the cutoff frequencies** \(\Omega_p\) and \(\Omega_s\) **of the normalized lowpass filter**. (d) **Determine the order \(n\) of the normalized lowpass filter**. (e) **Find the recalculated values** \(A_{s1}\) and \(\Omega_c\). (f) **Locate the poles and zeros for the normalized lowpass filter**. (g) **Locate the frequency transformed lowpass poles and zeros**. (h) **Identify the digital lowpass poles and zeros**. Each of these steps involves specific formulae and calculations fundamental in signal processing and digital filter design. This step-by-step approach ensures designing an efficient filter that meets the required specifications. Understanding Butterworth filters and their characteristic of maximizing flatness in the passband, making them a popular choice in various applications such as audio processing, communication systems, and more, is crucial for practical implementation.
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