A signal processing engineer is designing a digital filter. The filter’s transfer function, $H(z)$, is a polynomial in the complex variable $z$, given by $H(z) = z^4 - 2.5z^3 + az^2 - 2.5z + 1$, where $a$ is a real number. It is known that the filter has reciprocal zeros, meaning if $z_0$ is a zero, then $\frac{1}{z_0}$ is also a zero.
Determine the value of $a$, and hence find all zeros of the polynomial $H(z)$. Justify your reasoning.
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Create Free Account Log inThis is a free VCE Units 3 & 4 Mathematical Methods practice question worth 7 marks, testing your understanding of Polynomial Equation Solutions. It falls under Algebra, number and structure in Unit 4: Mathematical Methods Unit 4. Submit your answer above to receive instant AI-powered marking and personalised feedback.
Continues the study of functions, algebra, calculus, and introduces probability and statistics.
Covers algebra of functions, inverse functions, and solutions of equations and systems of equations.
solution of polynomial equations with real coefficients of degree $n$ having up to $n$ real solutions, including numerical solutions
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