# Timing syncronisation using polyphase decimation filter

This question is regarding application of polyphase decimation filter in symbol time correction. How does choosing a perticular "polyphase branch" in a polyphase decimation filter result in timing adjustment ?

Suppose I have a 256:1 decimation filter with a sampling frequency of $256\textrm{ kHz}$ with total of $256\cdot 8$ taps (or $256$ filter banks with $8$ tap each). These are linear phase FIR filters, each filter bank having a data rate of $1\textrm{ kHz}$. Consider following two scenario:

1. I start with polyphase filter bank index $1$ and carry on convolving input data and roll over polyphase bank index at $256$.
2. I start with polyphase filter bank index $2$ and roll over polyphase bank index at $256$.

How does this result in timing adjustment ?

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The phases of a polyphase filter can be interpreted as a fractional delays. Consider your filter with $8\cdot 256$ taps in the up-sampled domain. In order to avoid aliasing the filter needs to be a low pass that doesn't allow any content above $1/256$ of the sample rate.
Since there is no energy above $1/256\cdot f_s$ in the transfer function you can actually down-sample the impulse response by a factor of $256$ without aliasing. That means that all phases have the same magnitude response in the down-sampled domain. The only difference between the phases is a linear phase that corresponds to a fractional delay. If phase $0$ has a delay of "$0$" samples, then phase $1$ will have a delay of $1/256$ samples etc. in the down sampled domain.