1.15 Polyphase resampling with arbitrary (non-rational or time

Though we have derived a computationally efficient polyphase resampler for rational factors $Q=\frac{L}{M}$ , 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 smallintegers. 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 tohandle both of these cases.

Say that our goal is to produce the $\frac{Q}{T}$ -rate samples $x_c(m\frac{Q}{T})$ given the $\frac{1}{T}$ -rate samples $x_c(nT)$ , where we assume that $x_c(t)$ is bandlimited to $\frac{1}{2T}$ and $Q$ can be any positive real number. Consider, for a moment, the outputs ofpolyphase filters in an ideal zero-delay $L$ -branch polyphase interpolation bank (as in ).

We know that, at time index $n$ , the $p^{th}$ and ${(p+1)}^{th}$ filter outputs equal $$x_c((n+\frac{p}{L})T)\land x_c((n+\frac{p+1}{L})T)$$ respectively. Because the highest frequency in $x_c(t)$ is limited to $\frac{1}{2T}$ , the waveform cannot not change abruptly, and therefore cannot change significantly over a very small timeinterval. In fact, when $L$ is large, the waveform is nearly linear in the time intervalbetween $t=(n+\frac{p}{L})T$ and $t=(n+\frac{p+1}{L})T$ , so that, for any $\alpha \in \left[0 , 1\right)$ , $$x_c((n+\frac{p+\alpha }{L})T)=x_c(1(n+\frac{p}{L})T+\alpha (n+\frac{p+1}{L})T)$$ $$x_c((n+\frac{p+\alpha }{L})T)\approx 1x_c((n+\frac{p}{L})T)+\alpha x_c((n+\frac{p+1}{L})T)$$ This suggests that we can closely approximate $x_c(t)$ at any $t\in \mathbb{R}$ by linearly interpolating adjacent-branch outputs of a polyphasefilterbank 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 $x_c((n+\frac{p-d}{L})T)$ for $n\in \mathbb{Z}$ and $p\in \{0, \dots , L-1\}$ . By linearly interpolating branch outputs $p$ and $p+1$ at time $n$ , we are able to closely approximate $x_c((n+\frac{p-d+\alpha }{L})T)$ for any $\alpha \in \left[0 , 1\right)$ . We would like to approximate $y(m)=x_c(m\frac{T}{Q}-d\frac{T}{L})$ 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 $n\in \mathbb{Z}$ , $p\in \{0, \dots , L-1\}$ , and $\alpha \in \left[0 , 1\right)$ such that

Thus, we have found suitable $n$ , $p$ , and $\alpha$ . Making clear the dependence on output time index $m$ , we write $$n_m=\lfloor \frac{m}{Q}\rfloor$$ $$p_m=\frac{m}{Q}\mod 1L$$ $${\alpha }_m=\frac{mL}{Q}\mod 1$$ and generate output $y(m)\approx x_c(m\frac{T}{Q}-d\frac{T}{L})$ via $$y(m)=1\sum h_{p_m}(k)x(n_m-k)+{\alpha }_m\sum h_{p_m+1}(k)x(n_m-k)$$ 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 $p_m\in \{0, \dots , L-1\}$ . This specifies the standard polyphase bank $\{H_0(z), \dots , H_{L-1}(z)\}$ plus the additional filter $H_L(z)$ .Ideally the $p^{th}$ filter has group delay $\frac{d-p}{L}$ , 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 $$\forall p, p\in \{0, \dots , L\}$$ Note that $h_L(k)=h_0(k+1)$ for $0\le k\le N_p-2$ .
• Set $H_L(z)=H_0(z)$ and advance the input stream to the last filter by one sample (relative to the other filters).

Finally, it should be mentioned that a more sophisticated interpolation could be used, e.g. , Lagrange interpolation involving more than two branch outputs. By makingthe interpolation more accurate, fewer polyphase filters would be necessary for the same overall performance, reducing thestorage requirements for polyphase filter taps. On the other hand, combining the outputs of more branches requires morecomputations per output point. Essentially, the different schemes tradeoff between storage and computation.