Due to the minimum-phase spectral factorization, orthogonal PR-FIR filterbanks will not have linear-phase analysis and synthesis filters. Non-linear phase may be undesirable for certain applications. "Bi-orthogonal" designs are closely related to orthogonal designs, yet give linear-phase filters. The analysis-filter design rules for the bi-orthogonal case are

• Fz F z : zero-phase real-coefficient halfband such that Fz=∑n=-N-1N-1fnz-n F z n N 1 N 1 f n z n , where NN is even.
• z-N-1Fz= H 0 z H 1 -z z N 1 F z H 0 z H 1 z

It is straightforward to verify that these design choices satisfy the FIR perfect reconstruction condition detHz=cz-l H z c z l with c=1 c 1 and l=N-1 l N 1 : Furthermore, note that z-N-1Fz z N 1 F z is causal with real coefficients, so that both H 0 z H 0 z and H 1 z H 1 z can be made causal with real coefficients. (This was another PR-FIR requirement.) The choice c=1 c 1 implies that the synthesis filters should obey G 0 z=2 H 1 -z G 0 z 2 H 1 z G 1 z=-2 H 0 -z G 1 z -2 H 0 z From the design choices above, we can see that bi-orthogonal analysis filter design reduces to the factorization of a causal halfband filter z-N-1Fz z N 1 F z into H 0 z H 0 z and H 1 z H 1 z that have both real coefficients and linear-phase. Earlier we saw that linear-phase corresponds to root symmetry across the unit circle in the complex plane, and that real-coefficients correspond to complex-conjugate root symmetry. Simultaneous satisfaction of these two properties can be accomplished by quadruples of roots. However, there are special cases in which a root pair, or even a single root, can simultaneously satisfy these properties. Examples are illustrated in Figure 1:

The design procedure for the analysis filters of a bi-orthogonal perfect-reconstruction FIR filterbank is summarized below:

1. Design a zero-phase real-coefficient filter Fz=∑n=-N-1N-1fnz-n 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).
2. Compute the roots of Fz F z and partition into a set of root groups G 0 G 1 G 2 … G 0 G 1 G 2 … that have both complex-conjugate and unit-circle symmetries. Thus a root group may have one of the following forms: G i = a i a i ¯1 a i 1 a i ¯ G i a i a i 1 a i 1 a i ∀ a i ,| a i |=1: G i = a i a i ¯ a i a i 1 G i a i a i ∀ a i , a i ∈ℝ: G i = a i 1 a i a i a i G i a i 1 a i ∀ a i , a i =±1: G i = a i a i a i ± 1 G i a i Choose 1 a subset of root groups and construct H ^ 0 z H ^ 0 z from those roots. Then construct H ^ 1 -z H ^ 1 z from the roots in the remaining root groups. Finally, construct H ^ 1 z H ^ 1 z from H ^ 1 -z H ^ 1 z by reversing the signs of odd-indexed coefficients.
3. H ^ 0 z H ^ 0 z and H ^ 1 z H ^ 1 z are the desired analysis filters up to a scaling. To take care of the scaling, first create H ~ 0 z=a H ^ 0 z H ~ 0 z a H ^ 0 z and H ~ 1 z=b H ^ 1 z H ~ 1 z b H ^ 1 z where aa and bb are selected so that ∑n h ~ 0 n=1=∑n h ~ 1 n n h ~ 0 n 1 n h ~ 1 n . Then create H 0 z=c H ~ 0 z H 0 z c H ~ 0 z and H 1 z=c H ~ 1 z H 1 z c H ~ 1 z where cc is selected so that the property z-N-1Fz= H 0 z H 1 -z z N 1 F z H 0 z H 1 z is satisfied at DC (i.e., z=ⅇⅈ0=1 z 0 1 ). In other words, find cc so that ∑n h 0 n∑m h 1 n-1m=1 n h 0 n m h 1 n 1 m 1 .