## 3 Building Blocks of the Overall DFE Architecture
### FIR3 (Farrow Interpolator)  
FIR3 (Farrow interpolator) plays the part of converting the sample rate by a fractional or rational number. For LTE DFE, two decimation factors, 128/325 and 192/325, were employed. In Figure 8, the two signals far apart became near to each other after decimation by the Farrow interpolator, during which filtering was also carried out to prevent aliasing. Basically, to convert the sample rate by a rational factor N/M, an upsampling or interpolation by a factor of N was followed by a downsampling by M, leading to a significantly large number of filter coefficients. Farrow interpolator in LTE DFE enables fractional sample-rate conversion with a small number of coefficients by making use of polynomial operation explained in the following. In the process of sample-rate conversion by a rational number, interpolation [20] is needed in the Farrow interpolator to obtain new values that did not exist in the input, as illustrated in Figure 9 where the light green lines correspond to \(h_{I}\) in Equation (4) and red marks represent the interpolated values obtained from the sum over \(h_{I}\) values at that \(x\) coordinate. Equation (4) represents discrete-time interpolation [21; 22], where \(T_{i}\) is the input sample period and \(T_{o}\) is the output sample period. \[y(kT_{i})=\sum_{m}x(mT_{i})\ h_{I}(kT_{i}-mT_{o}) \tag{4}\]  
Figure 6: Input and output of FIR1 (CIC filter) in the frequency domain. Figure 7: Noble identity. Equation (4) is rearranged as Equations (5)-(7), subsequently. \[y(kT_{i})=\sum_{m=m_{k}-l_{2}}^{m_{l}-I_{1}}x(mT_{i})\ h_{I}(kT_{o}-mT_{i}) \tag{5}\]  
\[y(kT_{i})=\sum_{i=l_{1}}^{l_{2}}x((m_{k}-i)T_{i})\ h_{I}(kT_{o}-(m_{k}-i)T_{i}) \tag{6}\]  
\[y(kT_{i})=\sum_{i=l_{1}}^{l_{2}}x((m_{k}-i)T_{i})\ h_{I}\bigg{(}\bigg{(}\frac{ kT_{o}}{T_{i}}-m_{k}+i\bigg{)}T_{i}\bigg{)} \tag{7}\]  
where \(i=\text{int}[kT_{o}/T_{i}]-m_{k}\) and \(m_{k}=\text{int}[kT_{o}/T_{i}]\), where \(\text{int}[R]\) is the maximum integer, which is smaller than or equal to the real number \(R\). Further rearranged, Equation (7) is represented as Equation (8). \[ \begin{array}{ll}y(kT_{i})&=y[(m_{k}+\mu_{k})T_{i}]\\ &=\sum_{i=l_{1}}^{l_{2}}x((m_{k}-i)T_{i})\ h_{I}((i-\mu_{k})T_{i})\end{array}  \tag{8}\]  
where \(\mu_{k}=kT_{o}/T_{i}-m_{k}\) and \(0\leq\mu_{k}<1\). In FIR3 (Farrow interpolator) for LTE DFE, \(h_{I}\) (\((i-\mu_{k})T_{i}\)) was modeled as a cubic polynomial in \(\mu_{k}\), which was readily calculated. An interpolation filter corresponding to Farrow interpolator was formulated below in  
Figure 8: Input (**left**) and output (**right**) of FIR3 (Farrow interpolator) in the frequency domain. Figure 9: Illustration of interpolation for fractional sample-rate conversion. Matlab. The elements in farrow_matrix represent the cubic polynomial coefficients that are identically used for hardware implementation. \[ \begin{array}{l}\bullet\quad\text{frac\_delay}=1/325\text{:}1/325\text{:}1;\\ \bullet\quad\text{for}\ i=1\text{:}length(\text{frac\_delay}),\\ \bullet\quad\text{farrow\_matrix}=[-4\ 96\ -128\ 32;\\ \bullet\quad\text{3}-558\ 820\ -264;\\ \bullet\quad\text{839}-166\ -1340\ 664;\\ \bullet\quad\text{-3}\ 854\ 652\ -664;\\ \bullet\quad\text{1}-290\ 28\ 264;\\ \bullet\quad\text{-4}\ 64\ -32\ -32];\\ \bullet\quad\text{mu=frac\_delay}();\\ \bullet\quad\text{interp\_coefs}(i,:)=\text{flipud}(\text{farrow\_matrix}) *[mu.^(0:3)]^{\prime};\\ \bullet\quad\text{end};\\ \bullet\quad\text{hinterp=reshape}(\text{interp\_coefs},[],1);\\ \bullet\quad\text{plot}(0\text{:}1/325\text{:}6-1/325\text{,hinterp});\\ \bullet\quad\text{hold on,plot}(128/325\text{,hinterp}(129),\text{or}^{\prime});\\ \bullet\quad\text{xlabel}(^{\prime}\text{/}\text{/}\text{To}^{\prime});\\ \bullet\quad\text{ylabel}(^{\prime}\text{/}\text{interpolation filter value}^{\prime}).\\ \end{array} \]  
Figure 10 represents the continuous-time impulse response of the Farrow interpolator. Since the x coordinate denotes \(t/T_{o}\), the impulse response length is \(6T_{o}\).