Section G.4. MULTISECTION COMPLEX FSF FREQUENCY RESPONSE
G 4 MULTISECTION COMPLEX FSF FREQUENCY RESPONSE
The frequency response of a guaranteed-stable complex N-section FSF, when r < 1, is Hgs,cplx(z) with the z variable in Eq. (7-18) replaced by ejw, giving
To temporarily simplify our expressions, we let q = w – 2pk/N, giving
Equation G-23
Factoring out the half-angled exponentials, and accounting for the r factors, we have
Equation G-24
Converting all the terms inside parentheses to exponentials (we'll see why in a moment), we have
Equation G-25
The algebra gets a little messy here because our exponents have both real and imaginary parts. However, hyperbolic functions to the rescue. Recalling when a is a complex number, sinh(a) = (ea – e–a)/2, we have
Equation G-26
Replacing angle q with w – 2pk/N, canceling the –2 factors, we have
Equation G-27
Rearranging and combining terms we conclude with
Equation G-28
(Whew! Now we see why this frequency response expression is not usually found in the literature.)
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