Introduction It is possible to design combinations of analysis and synthesis filters such that the aliasing from downsampling/upsampling is completely cancelled. Below we derive aliasing-cancellation conditions for two-channel filterbanks. Though the results can be extended to M-channel filterbanks in a rather straightforward manner, the two-channel case offers a more lucid explanation of the principle ideas (see ).

Aliasing Cancellation Conditions The aliasing cancellation conditions follow directly from the input/output equations derived below. Let i 0 1 denote the filterbank branch index. Then U i z 1 2 p 0 1 H i z 1 2 p X z 1 2 p Y z i 0 1 G i z U i z 2 i 0 1 G i z 1 2 p 0 1 H i z p X z p 1 2 i 0 1 G i z H i z X z H i z X z 1 2 X z X z H 0 z H 1 z H 0 z H 1 z G 0 z G 1 z where H z H 0 z H 1 z H 0 z H 1 z . H z is often called the aliasing component matrix. For aliasing cancellation, we need to ensure that X z does not contribute to the output Y z . This requires that H 0 z H 1 z G 0 z G 1 z H 0 z G 0 z H 1 z G 1 z 0 which is guaranteed by G 0 z G 1 z H 1 z H 0 z or by the following pair of conditions for any rational Cz G 0 z C z H 1 z G 1 z C z H 0 z Under these aliasing-cancellation conditions, we get the input/output relation Y z 1 2 H 0 z H 1 z H 1 z H 0 z C z X z where T z 1 2 H 0 z H 1 z H 1 z H 0 z C z represents the system transfer function. We say that "perfect reconstruction" results when y n x n l for some l , or equivalently when T z z l . The aliasing-cancellation conditions remove one degree of freedom from our filterbank design; originally, we had the choice of four transfer functions H 0 z H 1 z G 0 z G 1 z , whereas now we can choose three: H 0 z H 1 z C z .