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1

Note that the frequency response $$H(\omega)=\sum_{k=0}^na_k[\cos(\omega)]^k\tag{1}$$ is real-valued and even, and, consequently, the corresponding impulse response is also real-valued and even, i.e., it's a zero-phase type I FIR filter. The frequency response $(1)$ can also be written in the form $$H(\omega)=\sum_{k=0}^nb_k\cos(k\omega)\tag{2}$$ The ...

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So who is right? Both, I think. On first looks version  and version  use different definitions of $A(z)$.  conjugates the zeros and  conjugates the poles. Either one will probably work but the definition of the coefficients is different. Specifically the $a$ coefficients of version  or conjugates of those of version . So you have $$a_{n,i,... 1 Typically a transfer function is described as a function of the complex variable s as given by the Laplace Transform of the systems impulse response:$$H(s)= \frac{N(s)}{Q(s)} By expressing the transfer function of a linear system as a ratio of two such polynomials, we are able to describe the linear system uniquely in terms of the roots of those ...

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I don't think they get a special name -just the numerator and denominator function. Notice that the frequency response is defined as the ratio of output spectrum to input spectrum, but since the input signal is called X and the output Y, I don't think N and Q have a direct relationship with them (lest you're omitting some context from the material where you ...

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