Signal Processing Magic (3) -- Farrow Structure

Farrow structure was invented by C. W Farrow. It minimizes the number of multiplication used in fractional interpolation. Farrow structure is often deployed when there is fractional sampling rate change in the signal path. For example, ADC rate is set as 120MHz but baseband signal processing only supports 100MHz. Therefore, a filter needs to be designed for supporting rate conversion from 120MHz to 100MHz. Farrow structure is one candidate for this kind of job.

The setting of Farrow depends on the polynomial it supports. Let us take an example of using cubic Lagrange for fractional interpolation. The output y[n] can be written as combination of input x[n] as:
\[y[n]=C_{-2}x[n-2]+C_{-1}x[n-1]+C_{0}x[n]+C_{1}x[n+1]\]

Assuming fractional delay is \(\tau\), using Lagrange formula, \(C_{-2}\) equals to the division of \((-\tau-(-1))(-\tau)(-\tau-1)\) by \((-2-(-1))(-2)(-2-1)\). Thus,
\[C_{-2}=\frac{\tau^{3}}{6}-\frac{\tau}{6}\]
Extend the same formula to other coefficients, we have
\[C_{-1}=-\frac{\tau^{3}}{2}+\frac{\tau^{2}}{2}+\tau\]
\[C_{0}=\frac{\tau^{3}}{2}-\tau^{2}-\frac{\tau}{2}+1\]
\[C_{1}=-\frac{\tau^{3}}{6}+\frac{\tau^{2}}{2}-\frac{\tau}{3}\]
Rewriting the y[n] equation, we have
\[y[n]=\frac{1}{6}[(x[n-2]-3x[n-1]+3x[n]-x[n+1])\tau^{3}+ \\
(3x[n-1]-6x[n]+3x[n+1])\tau^2+ \\
(-x[n-2]+6x[n-1]-3x[n]-2x[n+1])\tau+6x[n]] \]

Corresponding Farrow structure is shown below. It only needs three multipliers and \(\tau\) is input to all multipliers. This simplifies the hardware design. When polynomial is different, the coefficients of FIR filter change accordingly. We should also note that due to the difference between input and output rates, one input sample not necessarily generates one output sample. A control logic is needed to dictate when to generate outputs.