Design of Orthogonal PR-FIR Filterbanks via Halfband Spectral Factorization Recall that analysis-filter design for orthogonal PR-FIR filterbanks reduces to the design of a real-coefficient causal FIR prototype filter H 0 z that satisfies the power-symmetry condition H 0 ω 2 H 0 ω 2 1 Power-symmetric filters are closely related to "halfband" filters. A zero-phase halfband filter is a zero-phase filter F z with the property F z F -z 1 When, in addition, F z has real-valued coefficients, its DTFT is "amplitude-symmetric": F ω F ω 1 The amplitude-symmetry property is illustrated in :

If, in addition to being real-valued, Recall that zero-phase filters have real-valued DTFTs. F ω is non-negative, then F ω constitutes a valid power response. If we can find H 0 z such that H 0 ω 2 F ω , then this H 0 z will satisfy the desired power-symmetry property H 0 ω 2 H 0 ω 2 1 . First, realize F ω is easily modified to ensure non-negativity: construct q n f n ε δ n for sufficiently large ɛ, which will raise F ω by ε uniformly over ω (see ).

The resulting Q z is non-negative and satisfies the amplitude-symmetry condition Q ω Q ω 1 2 ε . We will make up for the additional gain later. The procedure by which H 0 z can be calculated from the raised halfband Q z , known as spectral factorization, is described next. Since q n is conjugate-symmetric around the origin, the roots of Q z come in pairs a i 1 a i . This can be seen by writing Q z in the factored form below, which clearly corresponds to a polynomial with coefficients conjugate-symmetric around the 0 th -order coefficient. Q z n N 1 N 1 q n z n A i 1 N 1 1 a i z 1 a i z where A ℝ + . Note that the complex numbers a i 1 a i are symmetric across the unit circle in the z-plane. Thus, for ever root of Q z inside the unit-circle, there exists a root outside of the unit circle (see ).

Let us assume, without loss of generality, that a i 1 . If we form H 0 z from the roots of Q z with magnitude less than one: H 0 z A i 1 N 1 1 a i z then it is apparent that H 0 ω 2 Q ω . This H 0 z is the so-called minimum-phase spectral factor of Q z . Actually, in order to make H 0 ω 2 Q ω , we are not required to choose all roots inside the unit circle; it is enough to choose one root from every unit-circle-symmetric pair. However, we do want to ensure that H 0 z has real-valued coefficients. For this, we must ensure that roots come in conjugate-symmetric pairs, i.e., pairs having symmetry with respect to the real axis in the complex plane ().

Because Q z has real-valued coefficients, we know that its roots satisfy this conjugate-symmetry property. Then forming H 0 z from the roots of Q z that are strictly inside (or strictly outside) the unit circle, we ensure that H 0 z has real-valued coefficients. Finally, we say a few words about the design of the halfband filter F z . The window design method is one technique that could be used in this application. The window design method starts with an ideal lowpass filter, and windows its doubly-infinite impulse response using a window function with finite time-support. The ideal real-valued zero-phase halfband filter has impulse response (where n ): f ¯ n 2 n n which has the important property that all even-indexed coefficients except f ¯ 0 equal zero. It can be seen that this latter property is implied by the halfband definition F ¯ z F ¯ z 1 since, due to odd-coefficient cancellation, we find 1 F ¯ z F ¯ z ⇔ 2 m f ¯ 2 m z 2 m f ¯ 2 m 1 2 δ m Note that windowing the ideal halfband does not alter the property f ¯ 2 m 1 2 δ m , thus the window design F z is guaranteed to be halfband feature. Furthermore, a real-valued window with origin-symmetry preserves the real-valued zero-phase property of f ¯ n above. It turns out that many of the other popular design methods (e.g., LS and equiripple) also produce halfband filters when the cutoff is specified at 2 radians and all passband/stopband specifications are symmetric with respect to ω 2 .

Design Procedure Summary We now summarize the design procedure for a length-N analysis lowpass filter for an orthogonal perfect-reconstruction FIR filterbank: Design a zero-phase real-coefficient halfband lowpass filter F z n N 1 N 1 f n z n where N is a positive even integer (via, e.g., window designs, LS, or equiripple). Calculate ε , the maximum negative value of F ω . (Recall that F ω is real-valued for all ω because it has a zero-phase response.) Then create "raised halfband" Q z via q n f n ε δ n , ensuring that Q ω 0 , forall ω. Compute the roots of Q z , which should come in unit-circle-symmetric pairs a i 1 a i . Then collect the roots with magnitude less than one into filter H ^ 0 z . H ^ 0 z is the desired prototype filter except for a scale factor. Recall that we desire H 0 ω 2 H 0 ω 2 1 Using Parseval's Theorem, we see that h ^ 0 n should be scaled to give h 0 n for which n 0 N 1 h 0 n 2 1 2 .