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Design of Orthogonal PR-FIR Filterbanks via Halfband Spectral Factorization

Module by: Phil Schniter. E-mail the author

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 H 0 z that satisfies the power-symmetry condition

| H 0 ejω|2+| H 0 ej(πω)|2=1 H 0 ω 2 H 0 ω 2 1
(1)
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
Fz+F-z=1 F z F -z 1
(2)
When, in addition, Fz F z has real-valued coefficients, its DTFT is "amplitude-symmetric":
Fejω+Fej(πω)=1 F ω F ω 1
(3)
The amplitude-symmetry property is illustrated in Figure 1:

Figure 1
Figure 1 (ortho_f3.png)

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

First, realize Fejω 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 Fejω F ω by εε uniformly over ωω (see Figure 2).

Figure 2
Figure 2 (ortho_f4.png)

The resulting Qz Q z is non-negative and satisfies the amplitude-symmetry condition Qejω+Qej(πω)=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.

Qz= n =(N1)N1qnzn=A i =1N1(1 a i z-1)(1 a i *z) Q z n N 1 N 1 q n z n A i 1 N 1 1 a i z 1 a i z
(4)
where A + A + . 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).

Figure 3
Figure 3 (ortho_f5.png)

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:

H 0 z=A i =1N11 a i z-1 H 0 z A i 1 N 1 1 a i z
(5)
then it is apparent that | H 0 ejω|2=Qejω 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 ejω|2=Qejω 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).

Figure 4
Figure 4 (ortho_f6.png)

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 nZ n ):

f ¯ n=sinπ2nπn f ¯ n 2 n n
(6)
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
1= F ¯ z+ F ¯ z-1=2 m = f ¯ 2mz(2m) f ¯ 2m=12δm 1 F ¯ z F ¯ z 2 m f ¯ 2 m z 2 m f ¯ 2 m 1 2 δ m
(7)
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 .

Design Procedure Summary

We now summarize the design procedure for a length-NN analysis lowpass filter for an orthogonal perfect-reconstruction FIR filterbank:

  1. Design a zero-phase real-coefficient halfband lowpass filter Fz= n =(N1)N1fnzn F z n N 1 N 1 f n z n where NN is a positive even integer (via, e.g., window designs, LS, or equiripple).
  2. Calculate ε ε, the maximum negative value of Fejω F ω . (Recall that Fejω F ω is real-valued for all ωω because it has a zero-phase response.) Then create "raised halfband" Qz Q z via qn=fn+εδn q n f n ε δ n , ensuring that Qejω0 Q ω 0 , forall ωω.
  3. Compute the roots of Qz Q z , which should come in unit-circle-symmetric pairs a i 1 a i * a i 1 a i . Then collect the roots with magnitude less than one into filter H ^ 0 z H ^ 0 z .
  4. H ^ 0 z H ^ 0 z is the desired prototype filter except for a scale factor. Recall that we desire | H 0 ejω|2+| H 0 ej(πω)|2=1 H 0 ω 2 H 0 ω 2 1 Using Parseval's Theorem, we see that h ^ 0 n h ^ 0 n should be scaled to give h 0 n h 0 n for which n =0N1 h 0 2n=12 n 0 N 1 h 0 n 2 1 2 .

Footnotes

  1. Recall that zero-phase filters have real-valued DTFTs.

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