The frequency response of a guaranteedstable complex Nsection FSF, when r < 1, is Hgs,cplx(z) with the z variable in Eq. (718) replaced by ejw, giving
To temporarily simplify our expressions, we let q = w – 2pk/N, giving
Equation G23
Factoring out the halfangled exponentials, and accounting for the r factors, we have
Equation G24
Converting all the terms inside parentheses to exponentials (we'll see why in a moment), we have
Equation G25
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 G26
Replacing angle q with w – 2pk/N, canceling the –2 factors, we have
Equation G27
Rearranging and combining terms we conclude with
Equation G28
(Whew! Now we see why this frequency response expression is not usually found in the literature.)
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