Research Article
Frequency estimator by combination
of phase difference method and
interpolation algorithm
Journal of Algorithms &
Computational Technology
Volume 13: 1–10
! The Author(s) 2019
Article reuse guidelines:
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DOI: 10.1177/1748301819833032
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Haitao Xu1 , Leng Han1, Qingqing Huang2, Song Feng1 and
Junhui Cao1
Abstract
In the traditional time-shifting based phase difference method, considerable errors may be introduced by wrapped phase
problem as long as translation coefficient tends to one even in a small-scale turbulence noise. In this paper, an improved
frequency estimator is proposed to overcome the problem of wrapped phase by combination of phase difference
method and interpolation algorithm. Compared with the traditional method of phase difference based on timeshifting, the improved algorithm can obtain an accurate estimate when translation coefficient exceeds one.
Comparative studies were done by means of root-mean-square error over Cramer–Rao lower bound. According to
the computer simulations, it is demonstrated that root-mean-square errors can cross Cramer–Rao lower bound if the
translation coefficient is properly selected even in the case of the low signal-to-noise ratio, which implies that
the proposed algorithm has a strong noise immunity. Finally, the advantage of the proposed algorithm is illustrated
for the simulation signal which contained strong local random noise.
Keywords
Phase difference, wrapped phase, translation coefficient, frequency estimation
Received 12 March 2018; Revised received 11 July 2018; accepted 28 October 2018
Introduction
Phase difference method and interpolation algorithm
are two of effective spectrum correction techniques to
reduce the bias caused by the non-integer period sampling.1 The interpolation-based method, in which the
frequency error can be corrected by the ratio of a certain number of known discrete Fourier transform
(DFT) spectral bins, has the advantage of low computational burden.2–9 Early in 1970s, the interpolation
algorithm, based on the modulus of two DFT bins,
was proposed by Rife and Vincent.2 In the following
decades, various improved algorithms were presented,
such as the multi-point interpolated DFT approach,3
the weighted interpolation approach.4–6 However, most
of the interpolation algorithms were established on the
specific window or a cluster of windows. In 2015,
Candan7 proposed a frequency estimator by calculating
the correcting factor of window function7 to break the
limit of window choice. At the same time, Luo et al.8,9
investigated the interpolation algorithms for classic
windows based on main-lobe fitting technique and
zero padding technique. In contrast, the phase difference method is feasible for all kinds of windows and
can be implemented conveniently without calculating
any parameters in advance. As a result, it is extensively
applied to various engineering fields, such as vibration
monitoring,10 fault diagnosis,11–13 coriolis mass flowmeter,14,15 power electronic parameter estimation16
and so on.
The phase interpolation estimator (PIE) method was
firstly proposed by McMahon et al.17 to estimate the
1
School of Advanced Manufacture Engineering, Chongqing University of
Posts and Telecommunications, Chongqing, China
2
School of Automation, Chongqing University of Posts and
Telecommunications, Chongqing, China
Corresponding author:
Haitao Xu, School of Advanced Manufacture Engineering, Chongqing
University of Posts and Telecommunications, Chongqing 400065, China.
Email: xuhaitao@zoho.com.cn
Creative Commons Non Commercial CC BY-NC: This article is distributed under the terms of the Creative Commons AttributionNonCommercial 4.0 License (http://www.creativecommons.org/licenses/by-nc/4.0/) which permits non-commercial use, reproduction and
distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.
sagepub.com/en-us/nam/open-access-at-sage).
2
Journal of Algorithms & Computational Technology
frequency. In 1994, the leakage-induced phase error
was researched through analyzing the windowing
signal.18 In 2002, an universal method of phase difference based on time-shifting (PDTS) was presented by
Ding and Zhong,19 in which the translation coefficient
was less than or equal to 1. The algorithm proposed by
Ding et al.20 was a special case of PDTS by employing
two continuous segment signals. In the light of the translation of the window center, Huang and Xu21 derived a
new phase difference algorithm. Meanwhile, the synthesized phase difference method based on time-domain
translation and the changing width of symmetrical
window were put forward by Ding et al. In 2007, the
estimation error of PDTS in the case of Gauss White
noise was analyzed.23 By considering the effect from
negative frequency, an algorithm for low-frequency
vibration signal was provided by Zhang and Tu.24
Recently, McKilliam et al.25 deduced a phase unwrapping estimator in the least squares sense. In view of utilizing the peak DFT bins of sub-segment from input
samples with N points, the phase difference method,
which is similar to the PDTS, was derived.26
Simultaneously, the calculation burden and statistical
properties were also investigated. What’s more, Luo
et al. made full use of the phase characteristics of asymmetric window and presented a new phase difference
method, which can gain precise frequency estimated
value only from one signal segment.1
The time-shifting based phase difference method has
many advantages. For example, as translation coefficient increases, the errors caused by random noise or
other interference would be reduced, consequently
enhancing the estimation accuracy. In particular,
the method is also advantageous for sampled signal containing strong local noise. However, once the translation
coefficient exceeds one, considerable estimation errors
would appear in the case of a large frequency deviation
because of wrapped phase problem. In this paper, we
focus on this problem and try to establish an improved
phase difference algorithm which can still work
when the translation coefficient is more than one. The
remaining structure of the paper is as follows, in the next
section, the theoretical background is illustrated. The
derivation of the improved phase difference based on
time shifting is then described. The simulations and
results can be found in the subsequent section. Finally,
some conclusions are summarized in the last section.
For simplicity but without losing the generality, we consider an exponential signal emerged in random white noise
f
2p f0s nþh0
þ zðnÞ
k0 ¼
f0
f0
¼
Df fs =N
(2)
At this stage, we ignore the noise term in the
following part. After multiplying the time-shifting
window function wN ðnÞ to the exponential signal in
equation (1), the weighted samples can be written as
xw0 ðnÞ ¼ x0 ðnÞwN ðnÞ
(3)
where wN ðnÞ can be achieved by
wN ðnÞ ¼ wðn N=2Þ
(4)
According to the convolution theorem, the DFT
coefficients at spectrum line k of equation (3) can be
calculated by
Xw0 ðkÞ ¼ AWN ðk k0 Þejh0
¼ AWðk k0 Þej½h0 þsðkÞðk0 kÞ
0kN1
(5)
where sðkÞ is the slope of phase for wN ðnÞ at k-bin, and
WN ðÞ, WðÞ denote the DFT coefficients of wN ðnÞ and
wðnÞ, respectively. In general, k0 lies between two DFT
bins due to non-integer sampling. Therefore, k0 can be
further described as
k0 ¼ m þ d
n ¼ 0; 1; . . . N 1
(1)
(6)
where m is the integer part which can be rapidly located
with its maximum peak of signal in frequency domain,
and d(0:5 d 0:5) is the fractional part of k0 .
Classic interpolation method
Subscribing equation (6) into equation (5) and replacing k by m, the highest amplitude can be given by
Xw0 ðmÞ ¼ AWðdÞ
Theoretical background
x0 ðnÞ ¼ Aej
where A, f0 , h0 , fs , N represent the amplitude, theoretical frequency, initial phase, sample rate and the
number of samples, respectively. The last term zðnÞ
denotes the Gauss White noise, in which the mean is
zero and variance is r2 . If the normalized frequency is
described as k0 , and frequency resolution is expressed
as Df, the relationship can be obtained
(7a)
Similarly, the second highest amplitude can be written as
Xw0 ðm1Þ ¼ AWðd1Þ
(7b)
Xu et al.
3
In two-point interpolation methods, the ratio of two
highest amplitude is defined as
a¼
jXw0 ðm1Þj jWðd1Þj
¼
jXw0 ðmÞj
jWðdÞj
(8)
d ¼ gw ðaÞ
(9)
It can be seen in equation (9) that frequency offset
can be solely determined once the window is selected.
For the maximum side-lobe decay (MSD) window, the
relationship can be simply written as5
Ha ðH 1Þ
aþ1
(10)
where H is the number of terms in the MSD window,
and the sign of d can be determined by the second
highest amplitude.28 For other windows, the polynomial approximation6 or the iterative approximation9
is suggested.
Compared with interpolation algorithm, the timeshifting based phase difference method requires two
signal frames. The signal with M points delay of equation (1) is expressed as
x1 ðnÞ ¼ Ae
f
f
n ¼ 0; 1; . . . N 1
(11)
Similar to equation (3) and equation (5), the DFT
coefficients of the weighted time-shifting signal can be
described as
k0
X1 ðkÞ ¼ AWðk k0 Þej½h0 þ2p N MþsðkÞðk0 kÞ
0kN1
(12)
After substituting m into equations (5) and (12), the
phases for the two signal frames can be calculated by
u0 ðmÞ ¼ h0 þ sðmÞ ðk0 mÞ
where {} denotes implementing 2p modulo operation.
Let us define D
u ¼ fDu 2pmSg. Since the phase
belongs to the range ðp; pÞ, D
u may exceed the
range ðp; pÞ. To guarantee the range of phase difference and frequency offset corrected by PDTS, the following operation is required
D
u¼
D
u 2p
D
u þ 2p
(13a)
and u1 ðmÞ ¼ h0 þ sðmÞ ðk0 mÞ þ 2pðm þ dÞS
(13b)
In equation (13b), S ¼ M=N is the translation coefficient. Since the period of phase is 2p, the phase
obtained through DFT is the main value of original
phase. If we define the phases calculated from DFT
D
u>p
D
u < p
(15)
For the traditional PDTS, the translation coefficient
is less than 1, equation (14) can be further simplified as
(see Appendix 1)
D
u ¼ 2pdS
(16)
As a result, the frequency offset can be expressed as
d¼
Phase difference method based on time-shifting
j 2p f0s nþ2p f0s Mþh0
Du ¼ /1 /0 ¼ u1 ðmÞ u0 ðmÞ ¼ f2pmS þ 2pdSg
(14)
In terms of equation (8), the frequency bias can be
described as a function of a related to window
d¼
as /0 and /1 , respectively, the phase difference Du
can be established as
D
u
2pS
(17)
Finally, the frequency estimate can be calculated by
f0 ¼ ðm þ dÞ Df
(18)
Derivation of the improved algorithm
It is known from ‘Phase difference method based on
time-shifting’ section that the traditional PDTS is built
on ignoring the phase ambiguity while translation coefficient is less than 1. However, phase ambiguity is supposed to be taken into account once translation
coefficient is equal to or beyond 1. Considering the
phase ambiguity and the noise interference on frequency estimator, frequency error corrected by PDTS
should be rewritten as (see Appendix 1)
^d ¼ Du þ 2pl 2pmS þ Zn ¼ Du þ 2pl 2pmS þ Zn
2pS
2pS
2pS
(19)
where l is the number of phase wrapping and Zn is the
noise term in frequency domain.
As illustrated in equation (19), the influence of noise
term would be reduced and would approach with the
increasing translation coefficient, and hence the estimation value would be more precise. However, the
number of wrapped phase cannot be known according
4
Journal of Algorithms & Computational Technology
to translation coefficient in advance, so the effectively
corrected range of traditional PDTS would be shrunk as
the increasing translation coefficient. To guarantee the
feasibility of PDTS using equation (16) while translation
coefficient is larger than one, it is necessary to add a
correction value of high accuracy which can ensure the
residual bias to meet the requirement in equation (32)
(Appendix 1). The initial correction value can be
achieved in the absence of noise and approximately
obtained as much as possible in the presence of noise
by the interpolation method (simplified as Selva) proposed in Selva.29 First of all, we employed, respectively,
Selva algorithm to estimate frequency from signal
xw0 ðnÞ; xw1 ðnÞ. Subsequently, two estimated values of
frequency can be obtained, and are defined as k1 and k2 .
There are a set of mean values in statistics, such as arithmetic mean value, geometric mean value, harmonic
mean value, and the quadratic mean value. In this
paper, we take the arithmetic mean operation to
reduce the random error of initial estimate and define
k as arithmetic mean. k can be given by equation (20).
k ¼ k1 þ k2
2
(20)
Replacing m with k in equation (14), and expressing
the bias addressed by the improved PDTS as d, the final
estimate of normalized frequency can be computed by
^k ¼ k þ d
(21)
At last, the improved PDTS can be summarized
as follows:
1. Two discrete signals x0 ðnÞ and x1 ðnÞ ¼ x0 ðn þ MÞ
with N samples are acquired.
2. Obtain the tappered sequences xw0 ðnÞ ¼ x0 ðnÞwN ðnÞ
and xw1 ðnÞ ¼ x1 ðnÞwN ðnÞ, where wN ðnÞ is the
adopted window function.
3. Estimate the coarse frequency estimates k1 and k2 ,
respectively, from xw0 ðnÞ and xw1 ðnÞ by Selva method.
4. Compute the arithmetic mean by k ¼ ðk1 þ k 2 Þ=2.
5. Calculate
the phase difference Du ¼ arg Y1 ðkÞ
arg Y0 ðkÞ , where Y1 ðÞ and Y0 ðÞ are the DFT
coefficients of xw1 ðnÞ and xw0 ðnÞ, respectively.
6. Return
the bias by d ¼ D
u =2pS, where D
u¼
Du 2pkS and S ¼ M=N.
7. The final frequency estimate is expressed
as f^ ¼ ðk þ d Þ Df.
Simulations and results
Performance of the improved algorithm
without noise
In this section, the comparative study between traditional PDTS and improved PDTS is investigated in the
case of noiselessness. For convenience but without
losing the generality, the sample rate and the number
of samples were set as 256, which implied the frequency
resolution was equal to 1. The theoretical frequency
and phase were varied from [63.5,64.5] with steps of
0.025 and ½p; p with steps of p=20, respectively, and
the translation coefficient was selected with 0.0625 interval in the range of [0.0625,4]. For different phases, the
maximum absolute error of frequency was selected.
Figure 1(a) and (b) depicts the curve of maximum absolute error corresponding to traditional PDTS and
improved PDTS. As described in Figure 1(a), the maximum error introduced by wrapped phase problem in
traditional PDTS is about 1 under noiseless situation,
and it also exams equation (16) that the effectively corrected range of traditional PDTS is decreased (blue district in Figure 1(a)) with the increasing translation
coefficient. Otherwise, Figure 1(b) reveals that the
improved method can be described as a frequency estimator of high accuracy, because the maximum absolute
errors are all below 1013 for different frequency deviations and translation coefficients.
Figure 1. Maximum absolute error of frequency versus d when Hanning window is used. (a) Traditional PDTS (b) Improved PDTS.
Xu et al.
5
Performance of the improved algorithm with random
Gaussian noise
To verify the feasibility and robustness of improved algorithm in the case of random noise, following simulations
are performed. The exponential signal corrupted by
Gauss White noise was generated by equation (1).
The ratio of root-mean-square error (RMSE) to
Cramér–Rao lower bound (CRLB) was investigated
to evaluate the behavior of the improved algorithm.
The expressions of RMSE1 and CRLB30 can be calculated by
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
Ntr
u 1 X
RMSE ¼ t
ðf i f 0 Þ2
Ntr i¼1
(22)
where Ntr denotes the number of independent instances
and fi denotes the estimation value of frequency for
each independent instance
8
>
>
>
>
<
6fs 2
h
i
2 SNR ðM þ NÞ ðM þ NÞ2 1
4p
CRLB ¼
>
>
6fs 2
>
>
: 2
8p SNR N ð4N2 1Þ
S<1
S1
(23)
In equation (23), the SNR is defined as
SNR ¼ 10log10 ðA2 =r2 Þ
(24)
Maximum translation coefficient. In this subsection, the
maximum translation coefficient in the case of different
SNRs (dB) is discussed. The theoretical frequency and
initial phase were selected randomly in [63.5,64.5] and
½p; p, respectively. The translation coefficient was
varied from 1 to 20 with steps of 1, and for each translation coefficient, a total of 10,000 instances were
tested. Figure 2 depicts the ratio of RMSE to CRLB
with different SNRs (dB) and window functions.
From the pictures, it should be pointed out that the
performance trend is identical with regard to different
SNRs when the translation coefficient is less than or
equal to 4. In addition, it is obvious for all selected
windows that the ratio curve initially declines as we
add translation coefficient, then minimum value and
starts to go up as the translation coefficient increases
except the situation that the SNR =10. The point with
maximum translation coefficient and minimum ratio
value, which reflects the excellent capability of antinoise, is different for variable SNR. The reason is
that the improved method can only compensate the
error caused by wrapped phase problem in certain
Figure 2. Ratio of RMSE to CRLB versus S. (a) Rectangle window (b) Hanning window (c) Hamming window (d) Blackman window.
6
extend. As the translation coefficient increases, the
interference from noise on PDTS is deceased, and the
accuracy is remarkably improved. For instance, the
minimum ratio can approximately attach to 0.3, while
SNR is equal to 5 dB. However, it should be stressed
that the interpolation algorithm in Step 3 would suffer
obstacle from noise disturbance if a low SNR is
encountered. The problem may, in turn, lead to a considerable residual error d1 , which is intended to be corrected by PDTS. Unfortunately, PDTS may not work
because it is possible that d1 S exceeds the effective
range in equation (32). As a result, we have to make
a careful balance between maximum translation coefficient and noise disturbance in practise.
However, it is complicated to establish the relationship by theoretical derivation, so we attempt to zoom
the simulation parameters to simulate the corresponding relationship between maximum translation coefficient and random noise. The parameters of theoretical
frequency and phase were same as above. The SNR was
varied from 5 dB to 15 dB with steps of 1 dB, and
translation coefficient was varied from 0.25 to 20
with steps of 0.25. Figure 3 describes the ratio of
RMSE to CRLB as a function of S and SNR for different windows.
It is shown in pictures that the relationship is almost
identical for different window techniques, because the
capability to resolve the problem of wrapped phase is
Journal of Algorithms & Computational Technology
dependent on the accuracy of the interpolation algorithm. The phenomenon implies that the relationship
between S and SNR can be obtained by curve fitting
technique through one of selected windows. The result
of curve fitting with rectangle window is depicted in
Figure 4, and the formula for the relationship is established in equation (25). In Figure 4, it can be seen that
the value of line-fitting is a little larger than data when
the SNR is in the range of [1 dB,6 dB]. To guarantee the
high accuracy with maximum translation coefficient
calculated by the fitted expression, the floor function,
where floor () denotes the nearest integers less than or
equal to (), should be operated. What’s more, it can
also be known that the maximum translation coefficient can be attached to 20, while SNR is larger than
10 dB in equation (25), and at this time, the RMSE of
improved method can be 0.07 times over CRLB.
Smax floorð7:338 e0:1114SNR Þ SNR < 10
20
SNR 10
(25)
Influence of frequency deviation. In order to assess the performance of the improved phase difference for changing deviations of frequency, we set theoretical
frequency to vary with 0.05 interval in the range of
[63.5,64.5] and SNR is equal to 5 dB. Initial phase
Figure 3. Ratio of RMSE to CRLB versus S and SNR (a) Rectangle window (b) Hanning window (c) Hamming window
(d) Blackman window.
Xu et al.
was randomly selected in ½p; p and translation coefficient was selected separately as S ¼ 1, S ¼ 2, S ¼ 3,
S ¼ 4. For different window functions, the performance
of the improved method for different deviations is
depicted in Figure 5.
From these pictures, it can be found that the trend
of ratio curves is almost a uniform flat in the whole
range of frequency deviations due to the first scheme
corrected by interpolation method.29 The interpolation
Figure 4. Line-fitting curve of S versus SNR.
7
method can eliminate the effect for wrong location
maximum spectral bins when frequency deviation is
close to 0:5. Moreover, with the increase of S, the
ratios are all reduced, and particularly, RMSEs can
cross CRLB when S is larger than or equal to 2 for
different window functions. The ratio of RMSE to
CRLB can approximately reach 0.58, 0.67, 0.7, 0.75
corresponding to rectangle window, Hamming
window, Hanning window and Blackman window supposing S ¼ 2, and approximated equal to 0.3, 0.33,
0.35, 0.37 supposing S ¼ 4. Thus, these features exam
that the capability of noise immunity of the improved
algorithm is independent on frequency deviations and
can be enhanced with the increase of translation coefficient as shown in equation (19).
Algorithm analysis under simulated signal interfered by strong
local random noise. In practical process of sampling, it is
possible that the signal may be corrupted by strong
local noise. In this subsection, the problem is considered to exam the advantages of the improved phase
difference algorithm. At first, the Gauss White noise
with zero mean was generated with SNR ¼ 15 dB
and 25 dB, respectively. In order to simulate well
the real environment noise, the Kaiser–Bessel window
(b ¼ 3:5) was used to weight the noise to change the
Figure 5. Ratio of RMSE and CRLB versus different deviations (a) Rectangle window (b) Hanning window (c) Hamming window
(d) Blackman window.
8
Journal of Algorithms & Computational Technology
Figure 6. Simulation sampled signal when the additive noise is weighted by Kaiser–Bessel window (b ¼ 3:5). (a)
SNR ¼ 15 dB (b) SNR ¼ 25 dB.
Figure 7. RMSEs of different estimation values versus deviations when the noise is weighted by Kaiser–Bessel window (b ¼ 3:5).
(a) SNR¼15 dB (b) SNR¼25 dB.
distribution of noise. For the sake of brevity, the basic
simulation parameters are set as follows. The number
of samples and sample rate was equal to 200, and the
frequency deviation was scanned with 0.05 interval in
[0.5,0.5]. A total of 5N signal was generated with
random phase. For example, the curve of simulated
signal (d ¼ 0:5) with different strong local noise can
be found in Figure 6. Here, we compared the RMSE
of the improved algorithm (rectangle window) and the
interpolation algorithm in Selva29 (simplified as Selva).
In Figure 7(a) and (b), FN and LN denote the first
segment signal and the last segment signal with N
points, respectively. The RMSEs of two methods and
arithmetic mean of estimation value are comprehensively compared. It can be seen that the RMSEs of
Selva method with N points are not related to the FN
or LN, and the operation, taking arithmetic mean, can
reduce the random error caused by noise. For the two
kinds of signal frames, the RMSE of improved algorithm is approximately 0.02 times over Selva (5N) in
Figure 7(a). It can be seen in Figure 7(b) that the
RMSE even approximately attach to 28, which vertifies
the Selva (5N) method has worst behavior due to the
wrong location of maximum spectral bin. In contrast,
the frequency corrected by the improved method
always keeps the privilege than other frequency estimates. Consequently, it can be concluded that the
improved PDTS can be a good strategy to achieve
the precise estimation of frequency when samples are
submerged by strong local random noise.
Conclusions
The paper illustrates a frequency estimator by combining the PDTS and interpolation algorithm to estimate
frequency. The extensive simulation results verify the
effectiveness of the proposed frequency estimator, and
the improved method can be applied for several kinds
of window functions. As the translation coefficient
increases, the capability against random noise becomes
stronger; in particular, the ratios of RMSE to CRLB
can even approximately attach to 0.35 for different
Xu et al.
9
window functions when translation coefficient is equal
to 4. For the sake of keeping the balance between maximum translation coefficient and random noise, curve
fitting technique is used to obtain the arithmetic relationship. It can be known in equation (25) that the
maximum translation coefficient can be 20 if SNR is
equal to or larger than 10 dB and high precise estimate
can be achieved. What’s more, while the stable samples
interfered by strong local noise, which may be generated by sampling environment, the improved algorithm
can still achieve high accuracy parameters estimation.
Acknowledgments
The authors would like to thank the anonymous reviewers for
their valuable comments and suggestions, and the help is of
great significance for improving the quality of the paper.
We also want to express our appreciation for the guidance
by Dr. J. Luo.
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of
this article.
Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this
article: This research was supported by the National Natural
Science Foundation of China (Grant Nos. 51705059 and
51605065), Chongqing Municipal Education Commission
(Grant No. KJ1600428), also partly supported by the found
of Chongqing Science and Technology Committee (No.
cstc2017jcyjAX0033).
ORCID iD
Haitao Xu
http://orcid.org/0000-0002-8629-6273
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Appendix 1
Du ¼ f2pmS þ 2pdSg
(26)
After some algebraic operation, equation (26) can be
transformed as
fDug f2pmS þ 2pdSg ¼ 0
(27)
because of Du 2 ð2p; 2pÞ. Given the property of
Modulo operation, equation (27) can be rewritten as
fDug f2pmS þ 2pdSg ¼ ðDu 2pmSÞ 2pdS
¼ D
u f2pdSg ¼ 0
(28)
where D
u ¼ fDu 2pmSg. To guarantee
D
u 2 ½p; p, the adjustment is required
D
u¼
D
u 2p
D
u þ 2p
D
u 2pdS ¼ 0
d¼
D
u
2p
D
u>p
D
u < p
(29)
(31)
As illustrated earlier, for different translation coefficients, the requirement, that equation (28) transforms
into equation (30), must meet p < 2pdS < p to guarantee 2p < D
u 2pdS < 2p.
Further, we can obtain
(32)
When S is less than one, equation (32) can be satisfied easily, and frequency error ð0:5 d 0:5Þ can be
addressed by equation (31). However, when S is equal
to or larger than one, equation (31) can only correct the
error d, which meets the requirement of equation (32).
It will lead to considerable error for other d. In such
cases, a supplementary term 2pl due to wrapped phase
has to be considered in the left part of equation (26).
It becomes
Du þ 2pl ¼ 2pmS þ 2pdS
(33)
The frequency bias can be rewritten as
d¼
the
(30)
The frequency bias can be corrected by
0:5 < dS < 0:5
Recall from equation (14)
It is easy to learn that D
u is in the range of ½p; p
after the adjustment, and 2pdS is in the range of
ðp; pÞ if S is less than one. As a result, D
u 2pdS
will be varied in the range of ð2p; 2pÞ. Accordingly,
equation (28) can be simplified as
Du þ 2pl 2pmS
2p
(34)