## 3 Results
### Hardware Output  
Figure 10 illustrates the hardware output. An MSOX4104A (oscilloscope), which can measure a bandwidth of up to 1 GHz and maximum 5 GSa/s, was used to accurately measure the time and frequency of the waveform. Figure 10a-c correspond with Figures 7a, 8a and 9a. The left side of each figure is the I+ signal, and the right side is the baseband one-side bandwidth (0-250 MHz). The bandwidth measurements illustrate that compensation of phase error and pre-distortion do not  
Figure 8: Simulation results of MDDS with phase error compensation. (**a**) MDDS output signal; (**b**) phase value over time. Figure 9: Simulation results of MDDS with pre-distortion. (**a**) MDDS output signal; (**b**) phase value over time. affect the bandwidth quality because they only change the phase value, but do not affect the frequency. Each result is the input to the RF modulator, as depicted in Figure 6, up-converted to the X-band (9.75 GHz). Figure 11 is the X-band (9.5-10 GHz) bandwidth measurement using an N9020A (spectrum analyzer). Figure 11a-c correspond with Figure 10a-c. Figure 11a, which depicts the phase error, and Figure 11b, which depicts the compensated phase center, shows 1.318 and 1.347 dBm of ripple, respectively. Figure 11c, which depicts applying the pre-distortion, reveals a slightly improved ripple to 0.974 dBm (red dotted line). The 89600 VSA software was used to precisely measure the phase error time. The phase compensated waveform and pre-distorted waveform are depicted in Figure 12a,b. The bottom right of Figure 12a,b depict the phase error varying time (PET). Ranges for predistortion were selected by analysis of the PET (Figure 12a). The 89600 VSA was set to measure 95% of the bandwidth of the pulse. Start and end of the spectrum was excepted, so the ripple was not accurate in this software. Figure 10: Waveform generator output applying MDDS (**a**) before phase error compensation, (**b**) with phase error compensation, and (**c**) with pre-distortion. Although the pre-distortion can be applied with several sections, the greater the number of sections, the greater is the code complexity. Table 1 and Figure 13 are various ranges of the pre-distortion and analysis results. The minimum, maximum, and mean results can change with the trigger timing of the capture point. Variation and standard deviation were significantly improved in the pre-distortion result. Table 1(a) (Figure 13a) represents phase error before applying the pre-distortion shown as Figure 10b. Figure 11: The X-band (9.75 GHz) frequency spectrum (**a**) before phase error compensation, (**b**) with phase error compensation, and (**c**) with pre-distortion. Although Figure 10b is close to the ideal waveform at the baseband, it has a wide fluctuation in phase measurement at the X-band. We applied the pre-distortion to reduce this fluctuation. Since, the range of the pre-distortion is limited due to its implementation, we focused on decreasing the maximum phase error and increasing the minimum phase error. Table 1b-f (Figure 13b-f) are the results of phase error compensation according to the pre-distortion range. Figure 13b (Table 1f) shows that the range of 0-6 \(\upmu\)s was compensated. Since wide range was selected, it is effective to reduce the variation. Figure 13c (Table 1c) and Figure 13f (Table 1f) show the additional compensation for Figure 13b (Table 1b), i.e., the multiple range compensation. Compared with Figures 13d and 13e (a single range compensation), the number of range and the compensated value are not proportional. In particular, in Table 1, row f, the variation and standard deviation are decreased to 34.0% and 58.4%, respectively. Figure 12: Detailed measurement for phase error analysis. (**a**) Phase compensation waveform; (**b**) pre-distortion waveform in Figure 13f. Top left is the frequency spectrum at the X-band (9.75 GHz), middle left is the detailed pulse table, and bottom left illustrates acquisition time (pulse repetition interval was 576 \(\upmu\)s).