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Polyphase Resampling with Arbitrary (Non-Rational or Time-Varying) Rate

Module by: Phil Schniter. E-mail the author

Summary: Introduction to the concept of arbitrary rate resampling.

Though we have derived a computationally efficient polyphase resampler for rational factors Q=LM Q LM, the structure will not be practical to implement for large LL, such as might occur when the desired resampling factor QQ is not well approximated by a ratio of two small integers. Furthermore, we may encounter applications in which QQ is chosen on-the-fly, so that the number LL 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 QT Q T -rate samples x c mQT x c m Q T given the 1T 1 T -rate samples x c nT x c n T , where we assume that x c t x c t is bandlimited to 12T 1 2 T and QQ can be any positive real number. Consider, for a moment, the outputs of polyphase filters in an ideal zero-delay LL-branch polyphase interpolation bank (as in Figure 1).

Figure 1
Figure 1 (m10456fig1.png)

We know that, at time index nn, the pth pth and (p +1)th (p +1)th filter outputs equal x c (n+pL)T x c (n+p+1L)T x c n p L T x c n p 1 L T respectively. Because the highest frequency in x c t x c t is limited to 12T 1 2 T , the waveform cannot not change abruptly, and therefore cannot change significantly over a very small time interval. In fact, when LL is large, the waveform is nearly linear in the time interval between t=(n+pL)T t n p L T and t=(n+p+1L)T t n p 1 L T , so that, for any α 0 1 α 0 1 , x c (n+p+αL)T= x c 1(n+pL)T+α(n+p+1L)T 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+αL)T1 x c (n+pL)T+α x c (n+p+1L)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 x c t at any tR t by linearly interpolating adjacent-branch outputs of a polyphase filterbank with a large enough LL. The details are worked out below.

Assume an ideal LL-branch polyphase filterbank with dd-delay master filter and TT-sampled input, giving access to x c (n+pdL)T x c n p d L T for nZ n and p0L1 p 0 L 1 . By linearly interpolating branch outputs p p and p+1 p 1 at time nn, we are able to closely approximate x c (n+pd+αL)T x c n p d α L T for any α 0 1 α 0 1 . We would like to approximate ym= x c mTQdTL 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 mm, QQ, dd, and LL, we would like to find nZ n , p0L1 p 0 L 1 , and α 0 1 α 0 1 such that

(n+pd+αL)T=mTQdTL n p d α L T m T Q d T L
(1)
nL+p+α=mLQ=mQL=(mQ+mQmod1)L=mQL+mQmod1L+mQmod1Lmod1=mQL+mQmod1L+mLQmod1 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
(2)
where mQLZ m Q L , mQmod1L0L1 m Q 1 L 0 L 1 , mLQmod1 0 1 m L Q 1 0 1 .

Thus, we have found suitable nn, pp, and αα. Making clear the dependence on output time index mm, we write n m =mQ n m m Q p m =mQmod1L p m m Q 1 L α m =mLQmod1 α m m L Q 1 and generate output ym x c mTQdTL y m x c m T Q d T L via ym=1kk h p m kx n m k+αmkk h p m + 1 kx n m k 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 Figure 2.

Figure 2
Figure 2 (m10456fig2.png)

Note that our structure refers to polyphase filters H p m z H p m z and H p m + 1 z H p m + 1 z for p m 0L1 p m 0 L 1 . This specifies the standard polyphase bank H 0 z H L - 1 z H 0 z H L - 1 z plus the additional filter H L z H L z . Ideally the pth pth filter has group delay dpL d p L , so that H L z H L z should advance the input one full sample relative to H 0 z H 0 z , i.e., H L z=z H 0 z H L z z H 0 z . There are a number of ways to design/implement the additional filter.

  1. Design a master filter of length L N p +1 L N p 1 (where N p N p is the polyphase filter length), and then construct p,p0L: p0L p h p k h k L p p 0 L Note that h L k= h 0 k+1 h L k h 0 k 1 for 0k N p 2 0 k N p 2 .
  2. Set H L z= H 0 z 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 QQ with QmQm in the definitions for nmnm, pmpm, and αmαm.

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.

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