Here we look at a low-pass normalized transfer function whose squared amplitude is shown in equation (4.3). We try to find a filter with the flattest possible frequency response in the passband when *x* is close to 0. To come as close as possible to the specification, the synthesized filter must have an amplitude diagram as flat as possible when *x* = 0. For that, we find the conditions that allow us to cancel the successive derivations of the function |*H*(*jx*)|^{2}.

We know that the squared amplitude function |*H*(*jx*)|^{2}, being analytic when *x*=0, can be developed by using the McLaurin series as follows:

This development introduces the successive derivatives of |*H*(*jx*)|^{2} used for *x*=0, or:

Moreover, |*H*(*jx*)|^{2} being a rational fraction (see equation (4.3)), we can approach it by a polynomial by proceeding to the division following increasing powers of polynomials that constitute the numerator and the denominator of |*H*(*jx*)|^{2}. In this way we obtain the following development:

Since the McLaurin development is unique in the convergence region, we can then identify the coefficients of equations (4.7) and ...

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