Question

This problem involves an investigation of the effects of roundoff on the location of the zeros and poles of the system transfer function and on the frequency response when the system is implemented as a cascade of second-order sections. You will have 4 second-order sections which we will call $H_1(z), H_2(z), H_3(z)$, and $H_4(z)$. Compute the equivalent overall system transfer function $$ H_2(z)=H_1(z) H_2(z) H_3(z) H_4(z)=\frac{\sum_{k=0}^8 b 2(k) z^{-k}}{1.0+\sum_{k=1}^8 a 2(k) z^{-k}} . $$

   This problem involves an investigation of the effects of roundoff on the location of
the zeros and poles of the system transfer function and on the frequency response
when the system is implemented as a cascade of second-order sections.
You will have 4 second-order sections which we will call $H_1(z), H_2(z), H_3(z)$, and $H_4(z)$. Compute the equivalent overall system transfer function
$$
H_2(z)=H_1(z) H_2(z) H_3(z) H_4(z)=\frac{\sum_{k=0}^8 b 2(k) z^{-k}}{1.0+\sum_{k=1}^8 a 2(k) z^{-k}} .
$$

Digital Signal Processing. Principles, Algorithms and System Design

Winser Alexander and… 1st Edition

Chapter 6, Problem 4

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Step 1

Let's assume that the transfer function for $H_1(z)$ is given by: $$ H_1(z) = \frac{b_1z^2 + b_2z + b_3}{a_1z^2 + a_2z + a_3} $$ Similarly, we can define the transfer functions for $H_2(z)$, $H_3(z)$, and $H_4(z)$ as: $$ H_2(z) = \frac{b_4z^2 + b_5z + Show more…

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This problem involves an investigation of the effects of roundoff on the location of the zeros and poles of the system transfer function and on the frequency response when the system is implemented as a cascade of second-order sections. You will have 4 second-order sections which we will call $H_1(z), H_2(z), H_3(z)$, and $H_4(z)$. Compute the equivalent overall system transfer function $$ H_2(z)=H_1(z) H_2(z) H_3(z) H_4(z)=\frac{\sum_{k=0}^8 b 2(k) z^{-k}}{1.0+\sum_{k=1}^8 a 2(k) z^{-k}} . $$

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