Farrow structure for cubic interpolation

Fractional Delay Filters Using the Farrow Structure

In the discussion on piecewise polynomial interpolation, we emphasized on the fact that the fractional interval $\mu_m$ needs to be updated for each symbol time $mT_M$ and hence the subscript $m$ in $\mu_m$. For this reason, the interpolation process becomes a two-step procedure.

  1. Update the filter coefficients $h_p[n]$.
  2. Perform the convolution between $z(nT_S)$ and $h_p[n]$.

This process can be simplified if the two steps above can be combined in such a way that $\mu_m$ update is weaved into the convolution operation. In other words, instead of a two-input hardware multiplication with two variable quantities, complexity can be reduced by restructuring the problem such that one input in the two-input hardware multiplication remains fixed. This is accomplished by separating the constant factors from the powers of $\mu_m$ in filter equations, which is discussed in detail below.

Cubic Interpolation


Odd degree polynomials approximate the interpolant by using an even number of samples that ensures the linear phase property. After $p=1$, the next odd degree polynomial is with $p=3$ and is known as cubic interpolation. Plugging $p=3$ in the interpolation equation,
\[
z(t) \approx c_p t^p + c_{p-1}t^{p-1} + \cdots + c_1 t + c_0
\]

we get
\begin{equation*}
z(t) \approx c_3t^3 + c_2 t^2 + c_1 t + c_0
\end{equation*}

Therefore,
\begin{align}
z(nT_S+\mu_m T_S) &\approx c_3 (nT_S+\mu_m T_S)^3 + c_2 (nT_S+\mu_m T_S)^2 + \nonumber \\
&\hspace{1.7in}c_1 (nT_S+\mu_m T_S) + c_0 \label{eqTimingSyncCubicInterpolation1}
\end{align}

Following the same derivation as in linear interpolation, the coefficients $c_3$, $c_2$, $c_1$ and $c_0$ need to be found every time $\mu_m$ changes. The four samples around the interpolant are given by
\begin{align*}
z((n-1)T_S) &= c_3((n-1)T_S)^3 + c_2((n-1)T_S)^2 + c_1(n-1)T_S + c_0 \\
z(nT_S) &= c_3(nT_S)^3 + c_2 (nT_S)^2 + c_1 nT_S+c_0 \\
z((n+1)T_S) &= c_3((n+1)T_S)^3 + c_2((n+1)T_S)^2 + c_1(n+1)T_S + c_0 \\
z((n+2)T_S) &= c_3((n+2)T_S)^3 + c_2((n+2)T_S)^2 + c_1(n+2)T_S + c_0
\end{align*}

These are four equations with four unknowns that can be easily solved to compute $c_3$, $c_2$, $c_1$ and $c_0$. Substituting them back in Eq~\eqref{eqTimingSyncCubicInterpolation1} yields

$$\begin{equation}
\begin{aligned}
z(nT_S+\mu_m T_S) &= \left(\frac{\mu_m^3}{6}- \frac{\mu_m}{6} \right)z((n+2)T_S) + \\
&\hspace{0.2in} \left(-\frac{\mu_m^3}{2}+\frac{\mu_m^2}{2}+\mu_m \right)z((n+1)T_S) + \\
&\hspace{0.2in} \left(\frac{\mu_m^3}{2}-\mu_m^2-\frac{\mu_m}{2}+1 \right) z(nT_S) + \\
&\hspace{0.2in} \left(-\frac{\mu_m^3}{6}+\frac{\mu_m^2}{2} – \frac{\mu_m}{3} \right) z((n-1)T_S)
\end{aligned}
\end{equation}\label{eqTimingSyncCubicInterpolation2}$$

As before, the desired interpolant can again be written as the output of a filter as
\begin{align}
z(nT_S+\mu_m T_S) &= h_3[-2] z((n+2)T_S) + h_3[-1] z((n+1)T_S) + \nonumber\\
&\hspace{1.3in}h_3[0] z(nT_S) + h_3[1] z((n-1)T_S) \nonumber\\
&= \sum \limits _{i=-2} ^1 h_3[i] z((n-i)T_S) \label{eqTimingSyncCubicFiltering}
\end{align}
where the filter coefficients are given by
$$ \begin{equation}
\begin{aligned}
h_3[-2] &= \frac{\mu_m^3}{6}- \frac{\mu_m}{6} \\
h_3[-1] &= -\frac{\mu_m^3}{2}+\frac{\mu_m^2}{2}+\mu_m\\
h_3[0] &= \frac{\mu_m^3}{2}-\mu_m^2-\frac{\mu_m}{2}+1\\
h_3[1] &= -\frac{\mu_m^3}{6}+\frac{\mu_m^2}{2} – \frac{\mu_m}{3}
\end{aligned}
\end{equation}\label{eqTimingSyncCubicFilter}$$

In the above equations, the subscript $3$ denotes cubic interpolation as $p=3$. An example of cubic interpolation is drawn in the figure below.

New sample value is computed by cubic interpolation using four neighbouring samples

Observe how the curve has been smoothed out as compared to linear interpolation. This gain comes at a price of higher computational complexity paid for cubic interpolation.

Simplification through Farrow Technique


Begin with filter coefficients from cubic interpolation in Eq (\ref{eqTimingSyncCubicFilter}) and rewrite it as
\begin{align*}
h_3[-2] &= \frac{1}{6}\mu_m^3 + 0\cdot\mu_m^2 – \frac{1}{6}\mu_m + 0 \\
&= b_3[-2]\mu_m^3 + b_2[-2]\mu_m^2 +b_1[-2]\mu_m^1 + b_0[-2]\mu_m^0 = \sum \limits_{l=0}^{3} b_l[-2]\mu_m^l
\end{align*}

where $b_3[-2] = 1/6$, $b_2[-2]=0$, etc. Similarly, for other coefficients, we can write
\begin{align*}
h_3[-1] &= -\frac{1}{2}\mu_m^3+\frac{1}{2}\mu_m^2+1\cdot\mu_m + 0 = \sum \limits_{l=0}^{3} b_l[-1]\mu_m^l \\
h_3[0] &= \frac{1}{2}\mu_m^3-1\cdot\mu_m^2-\frac{1}{2}\mu_m+1 = \sum \limits_{l=0}^{3} b_l[0]\mu_m^l \\
h_3[1] &= -\frac{1}{6}\mu_m^3+\frac{1}{2}\mu_m^2 – \frac{1}{3}\mu_m+0 = \sum \limits_{l=0}^{3} b_l[1]\mu_m^l
\end{align*}

In general,
\begin{equation*}
h_3[i] = \sum \limits_{l=0}^{3} b_l[i]\mu_m^l
\end{equation*}

Observe that $b_l[i]$ are fixed coefficients independent of $\mu_m$ in the above expression and are given in the table below.

Farrow coefficients for cubic interpolator

With this understanding in place, the filtering process from Eq (\ref{eqTimingSyncCubicFiltering}) can be rearranged such that convolution of the matched filter output is performed with the fixed coefficients $b_l[i]$.
\begin{align*}
z(nT_S+\mu_m T_S) &= \sum \limits _{i=-2} ^1 h_3[i] z((n-i)T_S)\\
&= \sum \limits _{i=-2} ^1 \cdot \sum \limits_{l=0}^{3} b_l[i]\mu_m^l \cdot z((n-i)T_S)\\
&= \sum \limits_{l=0}^{3}\mu_m^l \underbrace{\sum \limits _{i=-2} ^1 b_l[i] \cdot z((n-i)T_S)}_{y(l)}\\
&= \sum \limits_{l=0}^{3} y(l) \mu_m^l
\end{align*}

To obtain a hardware efficient filter structure known as Farrow structure, the above expression can be opened as
\begin{align*}
z(nT_S+\mu_m T_S) &= y(3)\mu_m^3 + y(2)\mu_m^2 + y(1)\mu_m^1 + y(0)\mu_m^0\\
&= \Big\{y(3)\mu_m^2 + y(2)\mu_m + y(1)\Big\}\mu_m + y(0)
\end{align*}

One more such step produces the Farrow structure for a cubic interpolator.

\begin{equation}\label{eqTimingSyncFarrowStructure}
z(nT_S+\mu_m T_S) = \bigg\{\Big\{y(3)\mu_m +y(2)\Big\}\mu_m + y(1)\bigg\}\mu_m + y(0)
\end{equation}

The Farrow structure for cubic interpolation is drawn in the figure below where $T_S$ is a sample delay. Notice the fixed coefficients $b_l[i]$ operating on the matched filter output samples while a changing $\mu_m$ operates on the subsequent output $y(l)$. Although its block diagram looks complicated, the expression above is quite simple. Click on the figure to see the enlarged image.


Farrow structure for cubic interpolation

A similar strategy can be applied to devise Farrow structures for polynomial interpolation for the cases when $p \neq 3$. For example, for $p=2$, it is given by
\begin{equation*}
z(nT_S+\mu_m T_S) = \Big\{y(2)\mu_m +y(1)\Big\}\mu_m + y(0)
\end{equation*}

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