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 H 0 z that satisfies the power-symmetry condition Power-symmetric filters are closely related to "halfband" filters. A zero-phase halfband filter is a zero-phase filter Fz F z with the property When, in addition, Fz F z has real-valued coefficients, its DTFT is "amplitude-symmetric": The amplitude-symmetry property is illustrated in Figure 1:

If, in addition to being real-valued, 1 Fⅇⅈω F ω is non-negative, then Fⅇⅈω F ω constitutes a valid power response. If we can find H 0 z H 0 z such that | H 0 ⅇⅈω|2=Fⅇⅈω H 0 ω 2 F ω , then this H 0 z H 0 z will satisfy the desired power-symmetry property | H 0 ⅇⅈω|2+| H 0 ⅇⅈπ-ω|2=1 H 0 ω 2 H 0 ω 2 1 .

First, realize Fⅇⅈω F ω is easily modified to ensure non-negativity: construct qn=fn+εδn q n f n ε δ n for sufficiently large ɛɛ, which will raise Fⅇⅈω F ω by εε uniformly over ωω (see Figure 2).

The resulting Qz Q z is non-negative and satisfies the amplitude-symmetry condition Qⅇⅈω+Qⅇⅈπ-ω=1+2ε Q ω Q ω 1 2 ε . We will make up for the additional gain later. The procedure by which H 0 z H 0 z can be calculated from the raised halfband Qz Q z , known as spectral factorization, is described next.

Since qn q n is conjugate-symmetric around the origin, the roots of Qz Q z come in pairs a i 1 a i ¯ a i 1 a i . This can be seen by writing Qz Q z in the factored form below, which clearly corresponds to a polynomial with coefficients conjugate-symmetric around the 0 th 0 th -order coefficient. where A∈ ℝ + A ℝ + . Note that the complex numbers a i 1 a i ¯ a i 1 a i are symmetric across the unit circle in the z-plane. Thus, for ever root of Qz Q z inside the unit-circle, there exists a root outside of the unit circle (see Figure 3).

Let us assume, without loss of generality, that | a i |<1 a i 1 . If we form H 0 z H 0 z from the roots of Qz Q z with magnitude less than one: then it is apparent that | H 0 ⅇⅈω|2=Qⅇⅈω H 0 ω 2 Q ω . This H 0 z H 0 z is the so-called minimum-phase spectral factor of Qz Q z .

Actually, in order to make | H 0 ⅇⅈω|2=Qⅇⅈω 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 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 (Figure 4).

Because Qz Q z has real-valued coefficients, we know that its roots satisfy this conjugate-symmetry property. Then forming H 0 z H 0 z from the roots of Qz Q z that are strictly inside (or strictly outside) the unit circle, we ensure that H 0 z H 0 z has real-valued coefficients.

Finally, we say a few words about the design of the halfband filter Fz 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∈ℤ n ): which has the important property that all even-indexed coefficients except f ¯ 0 f ¯ 0 equal zero. It can be seen that this latter property is implied by the halfband definition F ¯ z+ F ¯ z-1=1 F ¯ z F ¯ z 1 since, due to odd-coefficient cancellation, we find Note that windowing the ideal halfband does not alter the property f ¯ 2m=12δm f ¯ 2 m 1 2 δ m , thus the window design Fz 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 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 π2 2 radians and all passband/stopband specifications are symmetric with respect to ω=π2 ω 2 .