## 3 Multiplier-Free Modified Godard Timing Recovery
### Modified Godard Algorithm for Simplified Clock Tone Calculation  
The clock tone calculation can be simplified. To illustrate the necessary steps, we start with Figure 2 that shows a graphical representation of the DFT coefficients used to calculate the Godard estimate. The Godard algorithm multiplies the positive frequency components (upper sideband, red part in Figure 2) with the complex conjugate of the negative frequency components (lower sideband, green part in Figure 2). Note that the DFT spectrum is \(N\) periodic, so that the coefficients \(\widetilde{r}_{k+N/2}\) and \(\widetilde{r}_{k-N/2}\) are equal. Figure 2 displays a spectrum from \(-N\) to \(N\). Commonly, a spectrum is displayed from \(-N/2\) to \(N/2\). In the following derivations, we will display the spectrum from \(0\) to \(N\) for explanatory reasons. Figure 2: Illustration of the magnitude of typical DFT coefficients needed to calculate the Godard timing error estimate. The DFT spectrum is \(N\) periodic. The upper sideband is colored in red and the lower sideband is colored in green. The Godard algorithm can be improved in two steps, resulting in the modified Godard technique. In the first step, terms that do not contribute to the timing estimate are neglected. In the second step, the algorithm is adapted to arbitrary oversampling, which allows for operation with less than two samples per symbol. For the first step, we simplify Equation (3) by dropping terms in the summation that only give negligible contributions. To illustrate this process, we show it graphically in Figure 3. Here, the magnitude of \(\widetilde{r}_{k}\) and \(\widetilde{r}^{*}{}_{k+N/2}\) is displayed in Figure 3a,b, respectively. Figure 3c shows the overlap of both spectral parts and their multiplication according to Equation (3) in blue. This illustrates that only some elements provide meaningful contributions to the timing estimate. The other elements are zero in magnitude and only contaminate the estimate with additional noise. Moreover, it can be seen that it is sufficient to calculate the sum from \(0\) to \(N/2-1\), because the sum from \(N/2\) to \(N-1\) would only add a factor of two. With the first adaption of neglecting the terms without information, the estimation of the timing error changes to Equation (4). \[\tau_{\mathrm{z}}=\sum_{ \begin{subarray}{c}k=\frac{1-\beta}{2}\frac{N}{2}-1 \end{subarray} }^{\frac{1+\beta}{2}\frac{N}{2}-1}\mathrm{Im}\left\{\widetilde{r }_{k}\widetilde{r}_{k+N/2}^{*}\right\} \tag{4}\]  
Consequently, the size of the area (see the blue in Figure 3c) depends on the roll-off factor and therefore the excess bandwidth. It is interesting to note that [PERSON] has already discussed band-pass filtering of the signal sidebands for a practical implementation. Such a band-pass filtering corresponds to a reduction of the summation limits as elaborated above. Limiting the summation by the excess bandwidth was also discussed in [15]. To evaluate the influence of the adaption on performance, we use the variance \(e^{2}\) of the timing estimation error in logarithmic scale. This so-called jitter is calculated from the fluctuations of the estimate around the actual value. The calculated jitter is displayed in Figure 4. The red curve shows the jitter over roll-off factor for the original [PERSON] technique, and the green curve shows that for the modified Godard technique. It can be seen that the modified Godard technique has a reduced jitter, especially for smaller roll-off factors. For larger roll-off factors, the modified Godard algorithm converges to the original Godard algorithm. Therefore, it can be summarized that neglecting the outer elements reduces the amount of computational steps, but nevertheless improves the performance of the estimate. Figure 3: First adaption towards the modified Godard algorithm by neglecting unnecessary terms. (**a**) Shows the magnitude of the coefficients \(\widetilde{r}_{k}\) of a twofold oversampled, raised-cosine spectrum with roll-off factor \(1/3\). (**b**) Shows the spectrum shifted by N/2.