This appendix shows how the (1)k factors arise in Eq. (713) for an evenN multisection linearphase complex FSF. Substituting the positivefrequency, 0 k (N/2)–1, H(k)ejf(k) gain factors, with f(k) phase values from Eq. (711), into Eq. (710) gives
where the subscript "pf" means positive frequency. Focusing only on the numerator inside the summation in Eq. (G16), it is
Equation G17
showing how the (–1)k factors occur within the first summation of Eq. (713). Next we substitute the negativefrequency H(k)ejf(k) gain factors, (N/2)+1 k N–1, with f(k) phase values from Eq. (711''), into Eq. (710) giving
Equation G18
where the subscript "nf" means negative frequency. Again, looking only at the numerator inside the summation in Eq. (G18), it is
Equation G19
That ejpN factor in Eq. (G19) is equal to 1 when N is even, so we write
Equation G20
establishing both the negative sign before, and the (–1)k factor within, the second summation of Eq. (713). To account for the singlesection for the k = N/2 term (this is the Nyquist, or s/2, frequency, where w = p) we plug the H(N/2)ej0 gain factor, and k = N/2, into Eq. (78) giving
Equation G21
URL http://proquest.safaribooksonline.com/0131089897/app07lev1sec3
Amazon  


