Polynomial analysis poles and zeros

Consider a simple quadratic equation defined in terms of a real variable x  

We will now show how polynomial analysis can be applied to the transfer function. A polynomial defined in terms the complex variable z takes on just the same form as when defined 

This analysis can be extended to any digital LTI filter. As the general form transfer function, H z) is defined as the ratio of two polynomials 

We will now show how polynomial analysis can be applied to the transfer function. A polynomial defined in terms of the complex variable z takes on just the same form as when defined in terms ofx. The z form is actually less misleading, because in general the roots will be complex (e.g. /(z) = z + z - 0.5 has roots 0.5 + 0.5J and 0.5 - 0.5J.). The transfer function is defined in terms of negative powers of z - we can convert a normal polynomial into one in negative powers by multiplying ly z. So a second-order polynomial is 

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The exponential form of Eq. 6-28 is a nonrational transfer function that cannot be expressed as a rational function, a ratio of two polynominals in s. Consequently, (6-28) cannot be factored into poles and zeros, a convenient form for analysis, as discussed in Section 6.1. However, it is possible to approximate by polynomials using either a Taylor series expansion or a Fade approximation. 

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