Polyphase Resampling with Arbitrary (Non-Rational or Time-Varying) Rate 2.9 2002/01/14 2005/10/18 17:23:16.288 GMT-5 Phil Schniter schniter@ee.eng.ohio-state.edu Adan Galvan jago@rice.edu Erin Maloney emaloney@rice.edu Phil Schniter schniter@ee.eng.ohio-state.edu non-rational polyphase resampling Introduction to the concept of arbitrary rate resampling. Though we have derived a computationally efficient polyphase resampler for rational factors Q LM , the structure will not be practical to implement for large L, such as might occur when the desired resampling factor Q is not well approximated by a ratio of two small integers. Furthermore, we may encounter applications in which Q is chosen on-the-fly, so that the number L of polyphase branches cannot be chosen a priori. Fortunately, a slight modification of our exisiting structure will allow us to handle both of these cases. Say that our goal is to produce the Q T -rate samples x c m Q T given the 1 T -rate samples x c n T , where we assume that x c t is bandlimited to 1 2 T and Q can be any positive real number. Consider, for a moment, the outputs of polyphase filters in an ideal zero-delay L-branch polyphase interpolation bank (as in ).

We know that, at time index n, the pth and (p +1)th filter outputs equal x c n p L T x c n p 1 L T respectively. Because the highest frequency in x c t is limited to 1 2 T , the waveform cannot not change abruptly, and therefore cannot change significantly over a very small time interval. In fact, when L is large, the waveform is nearly linear in the time interval between t n p L T and t n p 1 L T , so that, for any α 0 1 , x c n p α L T x c 1 α n p L T α n p 1 L T x c n p α L T 1 α x c n p L T α x c n p 1 L T This suggests that we can closely approximate x c t at any t by linearly interpolating adjacent-branch outputs of a polyphase filterbank with a large enough L. The details are worked out below. Assume an ideal L-branch polyphase filterbank with d-delay master filter and T-sampled input, giving access to x c n p d L T for n and p 0 … L 1 . By linearly interpolating branch outputs p and p 1 at time n, we are able to closely approximate x c n p d α L T for any α 0 1 . We would like to approximate y m x c m T Q d T L in this manner - note the inclusion of the master filter delay. So, for a particular m, Q, d, and L, we would like to find n , p 0 … L 1 , and α 0 1 such that n p d α L T m T Q d T L n L p α m L Q m Q L m Q m Q 1 L m Q L m Q 1 L m Q 1 L 1 m Q L m Q 1 L m L Q 1 where m Q L , m Q 1 L 0 … L 1 , m L Q 1 0 1 . Thus, we have found suitable n, p, and α. Making clear the dependence on output time index m, we write n m m Q p m m Q 1 L α m m L Q 1 and generate output y m x c m T Q d T L via y m 1 αm k k h p m k x n m k αm k k h p m + 1 k x n m k The arbitrary rate polyphase resampling structure is summarized in .

Note that our structure refers to polyphase filters H p m z and H p m + 1 z for p m 0 … L 1 . This specifies the standard polyphase bank H 0 z … H L - 1 z plus the additional filter H L z . Ideally the pth filter has group delay d p L , so that H L z should advance the input one full sample relative to H 0 z , i.e., H L z z H 0 z . There are a number of ways to design/implement the additional filter. Design a master filter of length L N p 1 (where N p is the polyphase filter length), and then construct p h p k h k L p p 0 … L Note that h L k h 0 k 1 for 0 k N p 2 . Set H L z H 0 z and advance the input stream to the last filter by one sample (relative to the other filters). In certain applications the rate of resampling needs to be adjusted on-the-fly. The arbitrary rate resampler easily facilitates this requirement by replacing Q with Qm in the definitions for nm, pm, and αm. Finally, it should be mentioned that a more sophisticated interpolation could be used, e.g., Lagrange interpolation involving more than two branch outputs. By making the interpolation more accurate, fewer polyphase filters would be necessary for the same overall performance, reducing the storage requirements for polyphase filter taps. On the other hand, combining the outputs of more branches requires more computations per output point. Essentially, the different schemes tradeoff between storage and computation.