High-Resolution Group Quantization Phase Processing Method in Radio Frequency Measurement Range

Aiming at the more complex frequency translation, the longer response time and the limited measurement precision in the traditional phase processing, a high-resolution phase processing method by group quantization higher than 100 fs level is proposed in radio frequency measurement range. First, the phase quantization is used as a step value to quantize every phase difference in a group by using the fixed phase relationships between different frequencies signals. The group quantization is formed by the results of the quantized phase difference. In the light of frequency drift mainly caused by phase noise of measurement device, a regular phase shift of the group quantization is produced, which results in the phase coincidence of two comparing signals which obtain high-resolution measurement. Second, in order to achieve the best coincidences pulse, a subtle delay is initiatively used to reduce the width of the coincidences fuzzy area according to the transmission characteristics of the coincidences in the specific medium. Third, a series of feature coincidences pulses of fuzzy area can be captured by logic gate to achieve the best phase coincidences information for the improvement of the measurement precision. The method provides a novel way to precise time and frequency measurement.


Results and Discussion
In the following experiment, the reference signal and the measured signal are from the same frequency source.
The reference signal f 1 is produced by 5071A Cs atomic clock. The measured signal f 2 is from precise frequency synthesizer by the reference signal caused by 5071A Cs atomic clock.
Frequency standard comparison with simple phase relationships. The experimental scheme of the frequency standard comparison is shown in Fig. 1. Here, the frequency of the reference signal f 1 is 10 MHz, and the frequency of the measured signal f 2 is 8 MHz.
First, the reference signal f 1 and the measured signal f 2 are converted to the same frequency pulse with a suitable pulse width by pulse generator, as shown in Fig. 2. Second, the same frequency pulse of f 1 and f 2 is sent to the phase coincidence detector in the same time. The result of the phase coincidence leads to the phase coincidences fuzzy area, as shown in Fig. 3.
The quantized phase resolution T maxc denotes here measurement resolution, and is given by T maxc = (1/10, 1/8) = 25 ns. However, the detection resolution of the phase coincidence circuit is usually 2 ns. Obviously, every phase difference obtained by phase comparison between f 1 and f 2 can be identified by the detector. The fuzzy area is in fact a series of the phase coincidences pulse. These phase coincidences pulses are strict phase synchronization by the T minc . That is to say, where T 1 and T 2 are their periods, N 1 is counter value of the reference signal f 1 from counter I, N 2 is counter value of the measured signal f 2 from counter II. The obtained measurement data is processed by the signal processing circuit. The experimental result shows that the frequency stability in Fig. 1 can be reached E-13/s order of magnitude.
Frequency measurement with the complex phase relationships. The experimental scheme of frequency measurement is shown in Fig. 4.
Here the frequency of the reference signal f 1 is 10 MHz, and the frequency of measured signal f 2 is 10.23 MHz. First, the reference signal f 1 and the measured signal f 2 are converted to the same frequency pulse with a suitable pulse width by pulse generator, as shown in Fig. 5.
Second, the same frequency pulse of f 1 and f 2 is sent to the phase coincidence circuit in the same time. The result of the phase coincidence generates the phase coincidences fuzzy area, as shown in Fig. 6.
The quantized phase resolution is T maxc = (1/10, 1/10.23) ≈ 10 ps. Every phase difference less than the detection resolution 2 ns in phase comparison result f out can not be obviously identified by the phase coincidence detection circuit. The fuzzy area occurs and is made of a series of the phase coincidences pulse. These fuzzy areas are very stable and strict synchronization by the group period T gp . That is to say,   That is to say, it is very difficult to achieve frequency stability of E-13/s order of magnitude in the time. The phase coincidences fuzzy area is required to be further processed for high measurement precision and authenticity. Third, the phase coincidences fuzzy area is precisely shifted in phase, and there is an exclusive OR processing with the original fuzzy region in order to obtain the phase coincidences pulse of fuzzy region edge. The processing result f out1 is shown in Fig. 7.
In order to obtain single fuzzy area edge pulse, the fuzzy area is further processed by precise phase shift, non logic, and logic, etc. The further processing result f out2 is shown in Fig. 8.
The single fuzzy area edge pulse is here used as the open signal and stop signal of measurement gate. The reference signal and the measured signal are counted in gate time. The obtained counter value is processed by the signal processor, and the frequency stability data of the measured signal can be easily obtained in this experiment.  The experimental result demonstrates that the frequency stability in Fig. 4 can be also reached E-13/s order of magnitude.
The above experiments show that the principle of the group quantization and its synchronization is scientific and reasonable, the quantization phase resolution is the measurement resolution, and high measurement resolution does not necessarily achieve the high measurement precision due to the limited resolution of detection circuit. The key is whether the measurement resolution is stable. The stability of the measurement resolution depends on the stability of frequency relationships between the reference signal and the measured signal. However, the stability of the fuzzy area depends on the stability of the detection resolution of the detector. If the measurement resolution and the detection resolution are stable in a measurement system, the system hardware error caused by the inconsistence and mismatching of detection device can be reduced or eliminated, which greatly improves the authenticity and precision of the measurement to ensure the success of the scientific research.

Conclusion
According to the analysis of the high-resolution phase processing in the international, and by combining the inherent phase or frequency relations between different frequency comparison signals, a high-resolution group quantization phase processing method in radio frequency range is presented. The method depends on the quantization of phase difference to achieve the higher measurement resolution, and relies on the stability of the fuzzy area to obtain the higher measurement precision. It has a significant meaning in precise measurement. Compared to high-resolution dual mixer time difference method which is widely used in the international, there are the advantages of simple circuit structure, low development cost, small phase noise, quick response time , high authenticity and stability.

Methods
As a key step in phase measurement, the resolution conflicts with the stability of detection equipment. Resolution which directly leads to measurement error is considered to be the decisive factor affecting the measurement precision, especially in the condition of super-high resolution quantization measurement. Thus, high-resolution phase processing method is an important way for improving measurement precision in time and frequency measurement. For the same frequency signals, there is a high measurement resolution in the traditional phase processing. However, for the different frequency signals, some frequencies synthesis and transformation, such as mixing and multiple frequency, etc., are first used for frequency normalization of comparison signals. There are some additional errors caused by frequency conversion circuit with the inherent noise characteristics, which partly affects the improvement of measurement precision. Even if the switch phase comparison with the higher phase comparison precision is used in phase comparison, it is also confined to further improve the measurement precision due to the "dead zone" phenomenon and "nonlinearity" problem of phase comparison. Hence, the research of novel measurement principle and the exploration of frequency relationships of comparison signals and its variation law which is used in precise measurement fields for the improvement of measurement precision and resolution have a realistic significance. Frequency signal as an object of phase comparison and processing is one of periodic motion phenomenon in nature, and it follows the law of periodic motion in phase comparison. Hence, the essence of the law can be described by some new concepts such as the greatest common factor period, group quantization, phase quantization, group quantization synchronization and group quantization period, etc.
Suppose f 1 and f 2 are two stable frequencies of phase comparison signals with the initial phase difference from the same frequency source, whereT 1 and T 2 denote their periods.
Let T 1 = m 1 T maxc and T 2 = m 2 T maxc , where m 1 and m 2 are two positive integers without common factor and m 1 > m 2 . The T maxc is here called the greatest common factor period between f 1 and f 2 frequencies signals, and can be calculated by equation (5), c max 1 2 Every phase difference in phase comparison results f out is quantized by T maxc , as shown in Fig. 9.
With the time, there is a series of phase coincidences caused by the stability of electromagnetic wave signal transmission in the specific medium and the differences of two signals periods T 1 and T 2 in phase comparison. The obtained ideal coincidences are a minimum of phase comparison results f out . The coincidences with a stable transmission can parallel shift with the time. When the shifting time is just multiples of the least common multiple period T minc , the coincidences occur again. The least common multiple period can be calculated by equation (6), As shown in Fig. 10. So the continuity and periodicity of the coincidences such as A, B, C, D, E, etc. for the interval of the least common multiple period is the key characteristics of different frequency phase comparison.
The least common multiple period directly reflects the mutual relationships of phase between two frequencies signals. The relationships can be obtained by equation (7), Hence it is not difficult to conclude that the phase comparison for the interval of one signal period (T 1 or T 2 ) as reference phase can not be implemented in the different frequency phase processing. This is the main reason It is demonstrated by experiments that there is a strict phase coincidence for the interval of the least common multiple period. However, the obtained ideal coincidence is not a single pulse, but a fuzzy area including a series of the coincidences pulses, as shown in Fig. 11.
The decisive factor of the actual measurement precision is the stability of the fuzzy area. Thus, reducing coincidence pulses number and width of fuzzy area to improve measurement precision is an important aspect that we should work hard continuously.
The frequency f 1 is used as a reference signal in Fig. 10. For every positive pulse of f 1 , the phase comparison result is the first positive pulse phase difference of the measured frequency f 2 such as a, b, c, etc. Then a phase difference group here called "group" is formed in the neighboring phase coincidences. So there are the following three important characteristics for all the phase differences in a group.
First, it is not monotonous by sequence of the phase differences. Second, it is not continuous for the neighboring phase difference. Third, it is repeatable for the interval of the least common multiple period.
For every positive pulse of f 2 in Fig. 9, phase comparison results are Δ p 1 , Δ p 2 , …, Δ p m2 , where m 2 is a number of T 1 in a group, Δ p m2 is phase difference of the phase coincidence. The phase comparison results can be calculated by equation (8),  T maxc called phase quantization is the least phase difference in all phase differences, and it is used as a basic quantization unit of the phase differences. The detection circuit can distinguish T maxc , while its resolution is only better than T maxc . The T maxc is usually used as a reference standard of detection circuit resolution in the different frequency phase processing. Hence, the T maxc is also called the quantized phase resolution. It is found that the quantization step of T maxc is linear by rearranging the phase differences from small to large in a group, as shown in Fig. 12, where A, B, C, D are the coincidences. From the aspect of the quantized phase difference, all phase differences in a group can be calculated by equation (9), where P gq called group quantization. TheP gq is repeatable for the interval of the least common multiple period, so the least common multiple period is also called group quantization period T gqp in different frequency phase processing. That is to say, T gap = T minc . Eventually it is the nominal frequency of two comparison signals that determines the group quantization, which is not affected by phase noise or frequency drift. The quantized rate of the phase differences in a group depends on the f pq which can be calculated by equation (10), The f pq is here called phase quantization frequency. It is shown by equation (4) that any phase difference Δ p in a group is the integer multiple of the T maxc , so the quantized situation of the Δ p is shown in Fig. 13.
The T maxc is the least phase difference in phase comparison results, so the concept of the f pq is based on the analysis of phase by equation (10). However, the change situation of power caused by the T maxc can be observed, only while the variation of phase which is not a physical quantity is converted into the change of voltage by phase comparison circuit. So the frequency spectrum of the f pq cannot be observed in frequency analyzer.
There is a strict phase synchronization of group quantization for the interval of group in phase comparison, which generates the phase coincidences in the least common multiple period. The phase coincidences fuzzy area can be obtained by different frequency phase coincidence detection circuit. It is concluded by the concept of group quantization that different frequency phase comparison is essentially the periodic accumulation of the T maxc for the interval of group with the time. The monotonous situation of the accumulation is determined by the frequency relationships of two comparison signals. While the T maxc changes to the P gq or the opposite, there is a strict phase coincidence between two frequencies signals. The T maxc is the least unit of phase difference which is used as the step value for the monotonous variation of any phase difference in a group. The step variation of the T maxc is repeatable, periodic and synchronous for the interval of the least common multiple period. It is a fact that  the initial value of the T maxc is linearly converted into group quantization P gq or the opposite, which is the step law of the quantized phase, as shown in Fig. 12 above.
It can be seen by the analysis above that the greatest phase difference in a group is the T 1 , and T 1 = m 1 T maxc . Hence, the time that theT 1 is quantized by the T maxc is just T minc . That is to say, while a single phase difference accumulated by phase quantization T maxc in a group is increased to theT 1 , there is a strict phase coincidence between two comparison signals.
Similarly, the formation of fuzzy area width of the phase coincidences caused by step of the phase quantization is similar to the formation of group quantization caused by the accumulation of the T maxc . The fuzzy area is made of a number of very narrow coincidence pulses less than the detection circuit resolution in the form of Gauss distribution. That is to say, the very narrow coincidence pulses less than the detection resolution are also constantly accumulated in the process of the formation of group quantization. Their distribution is only random in the form of fuzzy area. The number of them is continuously increased until the formation of group quantization. Experimental results show that it is a variation law of periodic movement from quantitative changes to qualitative changes in precise measurement fields. The results of the qualitative changes with certain physical phenomenon which is periodically repeated finally return to a new starting point. If the width of fuzzy area is reduced by the periodic variation law of the quantized phase, the precision of measured quantity can be improved two, three or more orders of magnitude. The super-high resolution better than ps even 100 fs level is also obtained by further reducing the value of the T maxc .
The influence of the errors caused by system hardware is mainly reflected in the volatility of group quantization phase Δt. By establishing the model of the Δ t, the frequency of the measured signal can be easily obtained, as shown in Fig. 14.
Suppose f ′ 1 and f ′ 2 are two frequencies of phase comparison signals with a relative frequency difference Δ f caused by system hardware or phase noise, and f ′ 1 = f 1 + Δ f, f ′ 2 = f 2 . The f 1 and f 2 are nominal frequencies of f ′ 1 and f ′ 2 , and their periods are T 1 = m 1 T maxc and T 2 = m 2 T maxc , respectively, where m 1 and m 2 are two positive integers without common factor, and m 1 < m 2 . So the periods of two comparison signals are as follows. Fig. 14, the Δ t is a phase drift quantity generated by the Δ f for the interval of T minc . Hence the mathematical model of the Δ t can be easily established by equation (7), equation (10), equation (11) and equation (12), which is followed by Here, the T gp is group period 29 between the frequencies signals f ′ 1 and f ′ 2 , it reflects the state of phase coincidence between two comparison signals in frequency standard comparison and can be calculated by reverse cycle of Lissajous figure obtained by measurement on the oscilloscope.
From the equation (11), theΔ T′ is the phase jitter of frequency signals f 1 , it reflects a pollution level of the f 1 caused by phase noise, and can be precisely calculated by the following equation (14)  T  f  f  m T  T  f   T  T f  m T   T  mT  T f m T  T f  T  mT   m T  f  m T  f  T  f   m T  T  The phase quantization error is also an important factor that affects the accuracy of measurement. It affects the measurement resolution T maxc of detection system and the minimum phase difference in a T minc , that determines the size of phase quantization error. The smaller the T maxc , the higher the resolution of measurement and the smaller the phase quantization error.
In the practical application circuit based on group quantization phase processing, the precision of measurement is mainly determined by the measurement resolution and its stability, and the reliability of the system hardware. The resolution of measurement can be improved by possibly reducing the periods of two comparison signals and increasing the T minc . The stability of measurement resolution is mainly determined by the stability of frequencies relationships between two comparison signals that can be easily achieved by precise frequency linking method based phase group synchronization 32 . The variation law of the group phase differences can be therefore revealed by the relationships between the group quantization phase Δ t and the measurement resolution T maxc .
Similar to equation (8), for every positive pulse of f 2 in Fig. 14, all phase differences in the Nth T minc are Δp′ 1 , Δp′ 2 , ..., Δp′ m1 , where m 1 is a number of T 2 in a group, Δ p′ m1 is phase difference of the phase coincidence. The phase comparison results can be calculated by equation (15)   where Y 1 ,Y 2 ,Y 3 , …,Y m2 are nonnegative integers. The Δ T′ in the equation (15) is the result of phase jitter due to non-fixed phase relationships between two comparison signals. The (N − 1)Δ t caused by system hardware and phase noise is the phase of group quantization. If the phase relationships between two comparison signals are fixed, that is to say, the phase comparison is generated between two nominal frequency signals, both Δ T′ and (N − 1)Δ t are equal to zero. The equation (15) will be converted into the equation (8). The measurement precision is mainly determined by the quantization error T maxc in the condition of the nominal frequency of two comparison signals. The phase locked-loop (PLL) technology from different frequency signals and the homologous noise canceling technology are therefore used in group quantization phase processing method. It can be seen by equation (15) that if the Y 1 , Y 2 , Y 3 , …, Y m2 are equal to 0, 1, 2, …, m 2 − 1 respectively, m 2 phase differences in the first T minc can be obtained by the equation (16) The equation (16) denotes that every phase difference in the Nth T minc will vary periodically while the group quantization phase (N − 1)Δ t is T maxc , 2T maxc , 3T maxc , …, (m 2 − 1)T maxc , T 2 . While (N − 1)Δ t = T maxc in equation (16), m 2 phase differences in Nth T minc are as follows While (N − 1)Δ t = 2T maxc in equation (16), m 2 phase differences in Nth T minc can be obtained by the following equation According to the law above, while (N − 1)Δ t = T 2 in equation (16), m 2 phase differences in Nth T minc can be just calculated by the equation (  The equation (16) and the equation (18) is obviously the same. That is, while (N − 1)Δ t = T 2 , m 2 phase differences in Nth T minc have a variation of full cycle. The full cycle is called group period T gp that is equal to the (N − 1)T minc .
Besides, the sensitivity of detection is also an important aspect that improves the performance index of system. It depends on the sensitivity of phase comparison in group quantization phase processing. The measurement precision of the system is determined by the sensitivity of phase comparison. The called sensitivity of phase comparison is an output voltage corresponding to the unit phase difference. Suppose f 1 = 5 MHz and f 2 = 4 MHz are two stable frequencies of phase comparison signals, V PPS = 10 V is the peak to peak of output voltage of phase comparison with the same frequency, then the sensitivity of phase comparison can be calculated as follows where the V′ PPD is peak to peak of output voltage of phase comparison after the T 1 and the T 2 are converted into the T maxc . If the generation of phase comparison with different frequency is direct, the V′ PPD is changed into theV PPD . Similar to the equation (24), the V PPD = 10 V can be easily obtained by calculation.
From the above analysis, if the generation of phase comparison is direct, the sensitivity of phase comparison S′ D is as shown in equation (25 It can be seen that the measurement precision with 20 times is easily improved in group quantization phase processing.