CHIN. PHYS. LETT. Vol. 26, No. 7 (2009) 070602
A Group-Period Phase Comparison Method Based on Equivalent Phase
Comparison Frequency *
DU Bao-Qiang(杜保强)1,2** , ZHOU Wei(周渭)1 , DONG Shao-Feng(董绍锋)1 , ZHOU Hai-Niu(周海牛)1
1
2
Department of Measurement and Instrument, Xidian University, Xi’an 710071
Department of Information Engineering, Henan Vocational and Technical College, Zhengzhou 450046
(Received 11 March 2009)
Based on the principle of equivalent phase comparison frequency, we propose a group-period phase comparison
method. This method can be used to reveal the inherent relations between periodic signals and the change laws
of the phase difference. If these laws are applied in the processing of the mutual relations between frequency
signals, phase comparison can be accomplished without frequency normalization. Experimental results show that
the method can enhance the measurement resolution to 10−13 /s in the time domain.
PACS: 06. 30. Ft, 06. 20. Dk
At present, the phase comparison method is widely
used in the frequency standard comparison of high
precision. Because this method has very high measurement resolution, it can distinguish and process a
small phase difference variation.[1] However, the application of traditional phase comparison methods are
limited by the fact that the frequencies of comparison
signals must be the same or multiplied. In order to realize phase comparison among any signals, we propose
the method of group-period phase comparison based
on equivalent phase comparison frequency.
Besides the periodic changes of two frequency signals, it is the regular changes of phase differences that
have a significant impact on measurement, comparison
and control. We characterize the regular changes of
phase differences and the frequency relations of signals
in terms of the greatest common factor frequency, the
least common multiple period, quantized phase shift
resolution, equivalent phase comparison frequency and
so on. The greatest common factor frequency is defined as follows:[2] supposing that the periods of two
frequency signals 𝑓1 and 𝑓2 are 𝑇1 and 𝑇2 , respectively. If 𝑓1 = 𝐴𝑓max 𝑐 , 𝑓2 = 𝐵𝑓max 𝑐 , where 𝐴and 𝐵
are two positive integers and co-prime to each other
and 𝐴 > 𝐵. Then 𝑓max 𝑐 is called the greatest common factor frequency between 𝑓1 and 𝑓2 . The period
of the greatest common factor frequency 𝑇min 𝑐 is the
least common multiple period. The above relations
can be expressed as
frequency signals 𝑓1 and 𝑓2 (𝐴 and 𝐵 times, respectively). Their frequency relations are shown in Fig. 1.
There will be a phase difference variation between
𝑓1 and 𝑓2 because of their frequency difference. The
phase difference variation has certain regularity in
one 𝑇min 𝑐 and periodic variation appears in several
𝑇min 𝑐 . The variation law of phase difference status
is not continuous in one 𝑇min 𝑐 , and it is also not absolutely monotonic. Its monotonicity depends on the
frequency relationships between the two frequency signals.
Tmin c
f1
Tmin c
f2
fout
Fig. 1. The phase relationships between two frequency
signals.
In one 𝑇min 𝑐 , if 𝑓2 is used as a reference signal,
to each unique phase point, the phase differences between 𝑓1 and 𝑓2 are 𝑇1′ , 𝑇2′ , . . . , 𝑇𝐵′ , respectively. Then
we have the equation[3]
⎡ ′⎤
⎡
⎤
⎡ ⎤
𝑇1
𝑛1
1
⎢ 𝑇2′ ⎥
𝑛
⎢
⎥
⎢
2⎥
2
⎢
⎥
− 𝑇2 ⎢
.
(3)
.. ⎥
⎢ .. ⎥ = 𝑇1 ⎢
.. ⎥
⎣
⎦
⎣
⎣ . ⎦
. ⎦
.
𝑋
𝑛𝑋
𝑇𝐵′
In general,
𝑇min 𝑐 = 1/𝑓max 𝑐 = 𝐴/𝑓1 = 𝐴𝑇1 ,
(1)
𝑇𝐵′ = 𝑛𝑋 𝑇1 − 𝑋𝑇2 =
𝑇min 𝑐 = 1/𝑓max 𝑐 = 𝐵/𝑓2 = 𝐵𝑇2 .
(2)
From Eqs. (1) and (2) we know that the period 𝑇min 𝑐
is equal to the multiples of the period value of the two
𝑋
𝑛𝑋
−
𝑓1
𝑓2
𝑛𝑋
𝑋
−
𝐴𝑓max 𝑐
𝐵𝑓max 𝑐
𝑛𝑋 𝐵 − 𝑋𝐴
𝑛𝑋 𝐵 − 𝑋𝐴
= 𝑇1
.
=
𝐴𝐵𝑓max 𝑐
𝐵
=
* Supported by the National Natural Science Foundation of China under Grant Nos 60772135 and 10703004, and the CAST
Innovation Foundation of China under Grant No 20080403.
** Email: [email protected]
c 2009 Chinese Physical Society and IOP Publishing Ltd
○
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CHIN. PHYS. LETT. Vol. 26, No. 7 (2009) 070602
Let 𝑛𝑋 𝐵 −𝑋𝐴 = 𝑌𝐵 , then 𝑇𝐵′ = 𝑇1 𝑌𝐵 /𝐵, where 𝐴, 𝐵
and 𝑛𝐵 are all positive integers, and 𝑓1 > 𝑓2 , so 𝑌𝐵
can only be an integer equal to or greater than zero.
So Eq. (3) can be changed into
⎤
⎡
⎡ 𝐵𝑛1 −𝐴 ⎤
⎡ ′⎤
𝑌1 /𝐵
𝑇1
𝐵
⎢ 𝑌2 /𝐵 ⎥
⎢ 𝐵𝑛2 −2𝐴 ⎥
⎢ 𝑇2′ ⎥
𝐵
⎥
⎢
⎢
⎥
⎢
⎥
(4)
⎥ = 𝑇1 ⎢ .. ⎥ .
⎢ .. ⎥ = 𝑇1 ⎢
.
..
⎣ . ⎦
⎣
⎦
⎣ . ⎦
𝐵𝑛𝑋 −𝑋𝐴
𝑇𝐵′
𝑌𝐵 /𝐵
𝐵
From Eq. (4), in one 𝑇min 𝑐 , any phase difference 𝑇𝐵′
satisfies 0 ≤ 𝑇𝐵′ < 𝑇1 , and 𝑇1′ , 𝑇2′ , . . . , 𝑇𝐵′ are different
from each other, so 𝑌1 , 𝑌2 , . . . , 𝑌𝐵 (0 ≤ 𝑌𝐵 < 𝐵) are
different too. Thus, there are 𝐵 phase differences 0,
𝑇1 /𝐵, 2𝑇1 /𝐵, . . . , (𝐵 −1)𝑇1 /𝐵 between two signals. If
we arrange these values in order, we will find that the
phase difference ∆𝑇 between two frequency signals is
fixed. That is,
∆𝑇 =
𝑇1
𝑓max 𝑐
1
=
=
.
𝐵
𝑓1 𝑓2
𝐴𝐵𝑓max 𝑐
(5)
Let
𝑓equ = 𝐴𝐵𝑓max 𝑐 ,
two frequency signals, we should pay attention to the
greatest common factor frequency, the least common
multiple period, and equivalent phase comparison frequency. These phase relations connect different frequency signals together.
For signals of different nominal frequencies, it is
impossible to find out the regular variation of phase
difference in sequential periods. Thus, if 𝑇min 𝑐 is taken
as the period, we can combine all the phase differences
between two signals as a group, then all the phase differences will be different, and they will be in sequence
by the common difference ∆𝑇 .
If the interval is strictly 𝑇min 𝑐 , when the frequency
relations of two signals are fixed and there is no further
relative frequency variation, the sequence and values
of the phase difference are the same in each 𝑇min 𝑐 .
For example, each muster of the phase differences in
two neighboring 𝑇min 𝑐 ’s is used as a group, then the
phase differences in each 𝑇min 𝑐 are corresponding to
the group, as shown in Fig. 2. Figure 2 presents the results of the comparison between two signals of 7 MHz
and 3 MHz.
(6)
Tmin c
Tmin c
7 MHz
Then Eq. (5) is simplified to
(7)
3 MHz
where 𝑓equ is called the equivalent phase comparison
frequency (EPCF), ∆𝑇 is the phase difference in one
full variation period, and it is also the quantized phase
shift resolution. We often refer to ∆𝑇 as the period of the EPCF and call it the equivalent phase
period, which is expressed as 𝑇equ . Because ∆𝑇 reflects the phase shift between two neighboring states
of phase coincidences, EPCF is also called the samephase-point frequency. EPCF is an important feature of the mutual phase and frequency relations between two frequency signals. However, EPCF does
not truly exist but is a virtual frequency. We know
that the same-phase-point frequency is the frequency
corresponding to the period of relative phase shift reflected by the phase repetition of regular phenomenon
according to the changes of phase difference between
two frequency signals. Because the frequency is much
higher than any one of the two comparison frequencies,
we can obtain higher measurement resolution based on
it. Usually, the phase comparison based on the EPCF
is called the stagger phase comparison. It is a phase
comparison between different frequency signals.
As a periodic phenomenon in nature, frequency
signals have their own intrinsic frequencies. However,
because of many external disturbances,phase variation
and frequency shift must exist between two frequency
signals. Therefore, in practical phase comparison, frequency signals usually have different nominal frequencies. If we want to achieve phase comparison between
Results
∆𝑇 = 1/𝑓equ = 𝑇equ .
Tmin c
Group
Group
Group
Fig. 2. Strict group periods.
Tmin c
Tmin c
Tmin c
7 MHz+Df
3 MHz
Comparison α
results
β
Group
γ
α∋
β∋
Group
γ∋
α∀
β∀
γ∀
Group
Fig. 3. Group periods existing with a small frequency
difference.
When frequency relations of two signals are fixed
and there is a small frequency difference between
them, we collect the phase differences in each 𝑇min 𝑐
as a group. Then the parallel shift will occur in the
groups on the time scale, this is the frequency relations of signals that we often meet in nature, which is
shown in Fig. 3.
From Fig. 3, we know that the signal of 7 MHz has
a small frequency difference ∆𝑓 (∆𝑓 < 0), there are
less than 7 periods in one 𝑇min 𝑐 . Although the initial
phase difference is zero in the first 𝑇min 𝑐 , in the latter 𝑇min 𝑐 ’s all the corresponding phase differences will
change because of the existence of ∆𝑓 . In Fig. 3, 𝛼,
𝛼′ , 𝛼′′ , 𝛽, 𝛽 ′ , 𝛽 ′′ and 𝛾, 𝛾 ′ , 𝛾 ′′ are the corresponding
group phase differences in the three continuous 𝑇min 𝑐 .
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CHIN. PHYS. LETT. Vol. 26, No. 7 (2009) 070602
In Fig. 3, each phase difference is increasing, and there
is an upper limit for each phase difference. That is,
when it reaches the smaller period (upper phase difference), it will return to the phase coincidence state
(lower phase difference). Thus, in the comparison of
small frequency difference, there is a high probability
phase coincidence in the course of phase variation.
The continuity of the phase difference variation between two frequency signals does not appear in each
𝑇min 𝑐 , but appears between the groups of phase differences by the interval of 𝑇min 𝑐 , namely group correspondence. With a lapse of time, the groups of phase
differences will reflect the variation of phase difference
and additional relative frequency differences between
frequency signals by this specific continuity. From
Fig. 3 we know that although the sequences of phase
differences in each group are different, the variation of
the corresponding phase differences has the same regulation. For any phase difference in the group, only
the varying phase drift ∆𝑡 is caused by ∆𝑓 in a proper
sequence, where ∆𝑡 is the change of group phase difference. When the initial phase coincidence occurs,
we can easily obtain the equation
∆𝑡 = 𝐴𝑇1 − 𝐵𝑇2
𝐵
𝐴
=
−
𝐴𝑓max 𝑐 + ∆𝑓
𝐵𝑓max 𝑐
1
𝐴
−
=
.
𝐴𝑓max 𝑐 + ∆𝑓
𝑓max 𝑐
Dt
Tgp
Tequ
(8)
From Eq. (8), we find that the phase differences in
each group will be different because of different frequencies. When ∆𝑓 < 0 and ∆𝑡 > 0, each phase difference increases to the upper limit and then returns
to the lower limit, and vice versa when ∆𝑓 > 0 and
∆𝑡 < 0. When ∆𝑓 → 0 and ∆𝑡 → 0, phase differences
vary slightly in each group. With the accumulation of
time, the continuity of phase difference variation will
be clear. For each phase difference in one group from
the lower limit to the upper limit or from the upper
limit to the lower limit, or from a state to itself by the
above processes, the time cost is exactly the period
of the group phase difference and is called the group
period expressed as 𝑇𝑔𝑝 . The group period is exactly
corresponding to the time interval of the two phase coincidences and this time interval is integral multiples
of 𝑇min 𝑐 , that is, 𝑇𝑔𝑝 = 𝑛𝑇min 𝑐 , where 𝑛 is a positive
integer. In one group period, group phase shift 𝑇𝑔𝑠 is
equal to the quantized phase shift resolution ∆𝑇 . It
also exactly corresponds to an equivalent phase period
𝑇equ . Thus, we have equation,
𝑇𝑔𝑠 = ∆𝑇 = 𝑇equ .
upper limit is 𝑇equ , that is, ∆𝑡 is in the range 0–𝑇equ .
(i) Frequency stability experiment. We apply the
principle of the EPCF on an oscilloscope, and combine
the oscilloscope and frequency synthesizer, then a very
high precision frequency standard comparison can be
obtained by use of a Lissajous graph. At the same
time, we can know the frequency stability of signals.
This method will be very useful when the precision
of the two signals is very high and the ratio between
the frequencies of the two signals is an integer. Super
high EPCF can be obtained if a frequency synthesizer
is used properly.[4] For example, one signal is 5 MHz
and the other is 5.1 MHz, the EPCF will be 255 MHz.
If the frequency difference of the standard frequency
signals is fixed, the higher the EPCF is, the faster the
phase variation of the full period or the phase difference in the oscilloscope will be. Then we can observe
the rolling variation of the special meshes in the oscilloscope when the pulses 𝑋 and 𝑌 are strong enough.
The variations of 𝜋 and 2𝜋 are 𝑇equ /2 and 𝑇equ of the
EPCF, respectively. The principle diagram is shown
in Fig. 5.
(9)
Figure 4 is the variation of ∆𝑡 when ∆𝑓 > 0 and the
initial phase is zero. In Fig. 4, 𝑛 is an positive integer, and 𝐴, 𝐵 and 𝐶 are the phase coincidences. From
Fig. 4 we find that the variation of ∆𝑡 is linear and the
0
A
B
C
t
Fig. 4. Group phase differences.
Oscilloscope
HP8662A
High stability
Crystal oscillator
Fig. 5.
EPCF.
(a)
Frequency
meter
HP8662A
Measurement
crystal oscillator
Judgment of frequency stability based on the
(b)
(c)
Fig. 6. Lissajous graphs of different EPCEs.
Using the principle in Fig. 5, after the measured
frequency is processed by a HP8662A frequency synthesizer, a high precision comparison can be completed by the oscilloscope. Figure 6 shows the Lissajous graphs of 10 MHz frequency compared with
8 MHz, 11 MHz and 10.5 MHz frequencies respectively,
using the principle in Fig. 5, and the EPCFs are
40 MHz, 10 MHz and 210 MHz, respectively.
From Fig. 6, we can calculate the frequency accu-
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CHIN. PHYS. LETT. Vol. 26, No. 7 (2009) 070602
racy by
∆𝑓
𝑇equ
=
,
(10)
𝑓0
𝜏
where ∆𝑓 is the frequency difference between two
comparison signals, 𝑇equ is a fixed value called the
equivalent phase comparison period, 𝜏 is the comparison time of the graph rotation measured by counter,
𝑓0 is the standard frequency. By Eq. (9), we can also
calculate the frequency difference. This principle is
suitable when the EPCF is higher than 1 GHz.
In a word, in the experiments of measuring the
frequency stability based on the EPCF, the Lissajous
graph depends on the law of phase difference variation in one 𝑇min 𝑐 , and the period is 𝑇min 𝑐 . Howver,
the rotation period is equal to 𝑇equ . Many experiments show that when a signal is of high stability, the
Lissajous graph moves to the other end and is close to
a constant velocity. When the stability is lower, the
Lissajous graph will vacillate left and locate right at
an unfixed velocity, the curve is fuzzy. Thus, we can
judge the stability of the oscillator from the definition
of the graph, the thickness and the movement of the
graph.
(ii) Phase-locked loop (PLL) experiment. The
EPCF can be used in the PLL to improve the higher
standard frequency source.[5] The experimental principle is shown in Fig. 7.
fin
8607 OCXO 5 MHz
Standard source
fout
HP8662A
PLL
Frequency stability measurement
10 MHz VCO
Fig. 7. The experimental principle of frequency source
improvement.
Table 1. Experimental results of frequency source improvement.
𝑓in (MHz)
𝜎 (10−12 s−1 )
5
5.1
12.8
5.3
16.384
6.0
38.88
5.6
In the PPL experiments based on the EPCF, the
HP8662A frequency synthesizer, with the 5 MHz signal bearing second-stability 1.3 × 10−13 , which is output by the OCXO8607 as external frequency standard, generates 5 MHz, 12.8 MHz, 16.384 MHz and
38.88 MHz high-performance signals (whose secondstability is 1.1 × 10−12 ), which can be used as the
reference signals 𝑓in for the PPL to lock the 10 MHz
VCO OCXO (second-stability: 3.7×10−11 , tune range
±3 Hz). Two groups of OCXO8607 and HP8662A
should be used in this experiment, one is used to generate the input signals 𝑓in , and the other is to measure
frequency stability 𝜎. The experimental results of frequency sources improvement are listed in Table 1.
Table 1 shows that the second-stability of the crystal oscillator reaches the order of 10−12 s−1 . It can be
seen that the performance of the oscillator has been
improved with the method based on the EPCF. If the
noise of the electronic circuits can be reduced further
and a much faster device is used, then the two signals
can be compared by the higher EPCF to acquire a
higher performance and to make the performance of
output signals close to the input reference signals.
It can be seen from the experiments of the above
group-period phase comparison between two signals
that their low-frequency and high-frequency characteristics can be shown because of the different nominal
frequency values.[6] These two characteristics are related to each other. We consider high-frequency characters from the perspective of frequency, the signals
of different nominal values possess the high-frequency
EPCF. The low-frequency characteristic is considered
on the perspective of time and phase and is exhibited
by the change of phase difference and continuity of
phase differences.
The principle of equivalent phase comparison frequency and the theory of group phase difference are
proposed by the research of the frequency relationships between the signals and the law of periodic
change of phase difference. It is a new breakthrough in
the research of the relationships of frequency signals.
Phase comparison and processing could be accomplished without frequency normalization by adopting
the intrinsic relationships and the change law of the
periodic signals in nature, instead of the traditional
phase comparison which improves the accuracy of
measurement simply by the development of the microelectric devices or the circuit. Experiments show that
much higher measurement resolution can be obtained
by the present method.
References
[1] Li Z Q, Zhou W and Miao M 2008 Chin. Phys. Lett. 25
2820
[2] Zhou W 1992 IEEE Frequency Control Symp. 6 p270
[3] Zhou W 2008 IEEE Frequency Control Symp. 9 468
[4] Sun S X and Zhou W 2005 Mod. Electron. Technol. 10 10
(in Chinese)
[5] Chen F X 2008 IEEE Frequency Control Symp. 5 156
[6] Zhou H 2006 IEEE Frequency Control Symp. 8 267
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