The frequency response of a singlesection complex FSF is Hss(z) evaluated on the unit circle. We start by substituting ejw for z in Hss(z), because z = ejw defines the unit circle. Given an Hss(z) of
we replace the z terms with ejw, giving
Equation G7
Factoring out the halfangled exponentials e–jwN/2 and e–j(w/2 – pk/N), we have
Equation G8
Using Euler's identity, 2jsin(a) = eja – e–ja, we arrive at
Equation G9
Canceling common factors and rearranging terms in preparation for our final form, we have the desired frequency response of a singlesection complex FSF:
Equation G10
Next we derive the maximum amplitude response of a singlesection FSF when its pole is on the unit circle and H(k) = 1. Ignoring those phase shift factors (complex exponentials) in Eq. (G10), the amplitude response of a singlesection FSF is
Equation G11
We want to know the value of Eq. (G11) when w = 2pk/N, because that's the value of w at the pole locations, but Hss(e jw)w=2pk/N is indeterminate as
Equation G12
Applying the Marquis de L'Hopital's Rule to Eq. (G11) yields
Equation G13
The phase factors in Eq. (G10), when w = 2pk/N, are
Equation G14
Combining the result of Eqs. (G13) and (G14) with Eq. (G10), we have
Equation G15
So the maximum magnitude response of a singlesection complex FSF at resonance is H(k)N, independent of k.
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