The FIR perfect-reconstruction (PR) conditions leave some freedom in the choice of H 0 z H 0 z and H 1 z H 1 z . Orthogonal PR filterbanks are defined by causal real-coefficient even-length-NN analysis filters that satisfy the following two equations: To verify that these design choices satisfy the FIR-PR requirements for H 0 z H 0 z and H 1 z H 1 z , we evaluate detHz H z under the second condition above. This yields which corresponds to c=-1 c -1 and l=N-1 l N 1 in the FIR-PR determinant condition detHz=cz-l H z c z l . The remaining FIR-PR conditions then imply that the synthesis filters are given by The orthogonal PR design rules imply that H 0 ⅇⅈω H 0 ω is "power symmetric" and that H 0 ⅇⅈω H 1 ⅇⅈω H 0 ω H 1 ω form a "power complementary" pair. To see the power symmetry, we rewrite the first design rule using z=ⅇⅈω z ω and -1=ⅇ±ⅈπ -1 ± π , which gives The last two steps leveraged the fact that the DTFT of a real-coefficient filter is conjugate-symmetric. The power-symmetry property is illustrated in Figure 1:

Power complementarity follows from the second orthogonal PR design rule, which implies | H 1 ⅇⅈω|=| H 0 ⅇⅈπ-ω| H 1 ω H 0 ω . Plugging this into the previous equation, we find The power-complimentary property is illustrated in Figure 2: