## 4 Comparison of the Modified Godard Algorithm to the Gardner Algorithm  
In this section, the timing estimation algorithms are extended to include the Gardner algorithms. For this, we first comment on how timing jitter can be determined, and then compare the different versions of the Godard algorithm from above with the two versions of the Gardner algorithm. Timing jitter can be calculated from the statistics of the zero crossing of the s-curve (see Figure 7) [7; 18]. This can be achieved in four steps. First, we resample the signal to several sets with a different timing offset. Second, we evaluate the timing estimate with a feedback algorithm. Third, we fit a sinusoidal to the estimates. Fourth, we determine the zero crossing of the sinusoidal. This zero crossing will have zero mean, and its variance gives a quantitative estimate of the timing jitter. The timing jitter can be calculated with Equation (10):  
\[Jitter_{\text{dB}}=10\log_{10}(\text{var}(\hat{\tau}_{\epsilon})). \tag{10}\]  
The Gardner algorithm can be used in two ways. It can be applied to the signal amplitude [1] as well as the signal power [18]. Figure 8 shows the jitter over roll-off factor of six different timing estimation algorithms for a 16 QAM signal with an SNR of 16.5 dB. The six different algorithms consist of different versions of the Godard algorithm as well as the two Gardner algorithms. It can be seen that the Gardner algorithm with the signal amplitude lies in the range of the Godard and modified Godard algorithms. The Gardner algorithm applied to the signal power provides the lowest jitter for signals with roll-off factors smaller than 0.05. Figure 6: Simulation results for the jitter over roll-off factor for Godard and modified Godard and the multiplier-free version. (Signals with an oversampling of two, 16 QAM, and SNR = 16.5 dB were simulated). Our simulations show that the result of Figure 8 applies to a wide range of modulation formats and SNRs, and will only result in an increased timing jitter for the Gardner algorithm applied to the signal power when switching to more advanced modulation formats or decreasing the SNR. Next, we will determine the BER for a certain timing jitter. To simulate the timing jitter, we used a normal distribution with zero mean and a variance according to the desired timing jitter. This distribution was used to resample the signal and measure the resulting BER afterwards. Figure 9 shows the resulting BER for different timing jitter values. A 16 QAM signal with a roll-off factor of 1, 0.5, and 0.01 is simulated. The signals are evaluated with an SNR of 100 dB and an SNR of 16.5 dB. It can be seen that a signal with a roll-off factor of 1 is most resilient to timing jitter. When operating at a BER of \(10^{-3}\), the signal quality below a timing jitter of \(-20\) dB is dominated by the SNR. Therefore, a timing jitter below \(-20\) dB does not bring benefit to the signal quality. Figure 8: Jitter over roll-off factor for a signal with 16 QAM with an SNR of 16.5 dB (BER = \(10^{-3}\)). BER: bit-error ratio  
Figure 7: S-curve for a signal with 16 QAM, SNR = 16.5 dB, and roll-off factor of 0.1, 6 million symbols were simulated. The s-curve shows the response of a feedback timing estimation algorithm (Gardner algorithm in this case) to a signal with a timing error varying between \(-0.5\) and 0.5. The color indicates the probability of occurrence. The jitter corresponds to the distribution of the zero crossing. The timing jitter has a different impact on signals with different modulation formats. The higher the order of the modulation format, the stronger the impact of the timing jitter. Figure 10 shows the BER over timing jitter for a signal with roll-off factor of 0.1 and quadrature phase-shift keying (QPSK), 16 QAM, and 64 QAM as modulation formats. As expected, it can be seen that QPSK is more robust to timing jitter than 16 QAM and 64 QAM. The values in Figures 9 and 10 were obtained without averaging the timing jitter.