Support vector regression -based robust frequency estimation algorithm by instantaneous phase

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Published in IET CommunicationsReceived on 7th July 2013Revised on 19th August 2013Accepted on 26th August 2013doi: 10.1049/iet-com.2013.0589

50The Institution of Engineering and Technology 2014

ISSN 1751-8628

Support vector regression-based robust frequencyestimation algorithm by instantaneous phaseXueqian Liu, Hongyi Yu

Wireless Communication Department, Zhengzhou Information Science and Technology Institute, No. 7 JianXue Street,

Zhengzhou 450001, People’s Republic of China

E-mail: [email protected]

Abstract: There are two important factors which impact the performance of phase-based frequency estimation algorithmsremarkably: the approximations of noise phase model and imperfections in phase unwrapping process. Support vectorregression (SVR) exits excellent capabilities for learning unknown data and forecasting the future ones, especially under thesmall sample condition. Therefore, the authors introduce it to predict the variation trend of instantaneous phase and unwrapphases efficiently. Even though with that being the case, errors still exist in phase unwrapping process because of itsambiguous phase characteristic. Furthermore, a SVR-based frequency estimation algorithm is proposed and makes it immuneto these error phases by means of setting the SVR’s parameters properly. The results show that, compared with other phased-based algorithms, not only does the proposed one maintain a wide estimation range and quality capabilities at lowfrequencies, but also improves accuracy at high frequencies and decreases the impact with the initial phase. The proposedalgorithm is fit for not only linear phase signal but also polynomial phase signal, under both the Gaussian and non-Gaussiancondition.

1 Introduction

Estimating frequency of a single sinusoid has attractedconsiderable attentions for many decades, and undoubtedly,phase-based algorithm is an important part of it. Tretter [1]was the first person to propose a phase-based approach byintroducing an approximate and linear model forinstantaneous signal phase. Subsequently, a great deal ofimprovements have erupted mainly in following three parts:taking differences over one or more delays, which iswell-known as Kay and generalised Kay estimators [2–7];introducing autocorrelations and their difference functions,such as Fitz, L&R and M&M estimators [8–12]; andpreprocessing by means of lowpass filter, blocking averageand filter banks to increase signal-to-noise ratio (SNR) [13–16]. All of these methods must make compromises inestimation range, estimation accuracy and computationalcomplexity. In other words, increasing any one of them isat the expense of degrading the others.It is well known that threshold effects have existed in

frequency estimation all the while. Namely, when SNR ishigher than a value, the mean-square error (MSE) curve isobserved lying very close to Cramer–Rao lower bound(CRLB), or it decreases rapidly. There are two reasons ofthis [17]: one is the approximations of phase-based modeland the other is imperfections in phase unwrapping.The work of [18] and [19] improved them and obtained a

better performance. However, they are still based on the

previous approximate model. Fu and Kam [20–24] haveaddressed a novel phase noise model and phase unwrappingprocess. Moreover, they supposed a priori probabilitydensity function (pdf) of frequency and initial phase andderived a series of algorithms, e.g. time-domain maximumlikelihood (ML) and maximum a posteriori probability(MAP) ones. However, the a priori assumption consists oftwo sides: if frequency is consistent with that, estimationperformance is greatly improved; or otherwise, it is severelydeteriorated.Statistical learning theory (SLT) and structure risk

minimisation (SRM) principle are specialising in theresearch of small sample condition. As their concreteimplement, Support vector regression (SVR) overcomes theoverfitting and local minimum problems currently existingin artificial neural network (ANN). For the first time,therefore, this paper employs SVR to explore the linearrelationship between absolute signal phase and time series,unwrap phases and estimate frequency recursively. Weavoid taking the rationality of phase noise model intoaccount by not making any approximations in it. At onetime, we reduce the estimation performance’s dependenceon phase unwrapping process with a proper choice ofSVR’s parameters. It has been shown that except for theproperty of wide estimation range, low sensitivity to initialphase and closely approaching the estimation accuracy ofMAP algorithm at low frequencies, the proposed algorithmimproves its estimation performance at high frequencies.

IET Commun., 2014, Vol. 8, Iss. 2, pp. 250–257doi: 10.1049/iet-com.2013.0589

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2 Proposed algorithm
2.1 Signal model

The sinusoid signal polluted by noise is modelled as

rn = anAej 2pf0nTs+u0( ) + wn, n = 0, . . . , N − 1 (1)

Here, an ∈ {exp( j2πm/M );m = 0, …, M− 1} is anindependent symbol, M is the modulation order; A > 0,f0∈ [− 0.5, 0.5), θ0∈ [− π, π) are the amplitude,deterministic but unknown frequency and initial phase,respectively; TS is the sampling period, N is the samplesize; wn is an independent complex additive white Gaussiannoise (AWGN) with zero-mean and variance σ2. For thesake of simplicity, we set Ts = 1 and rearrange (1)

rn = Aej2pf0n+u0 + wna∗n = ej2pf0n+u0

A+ wna∗ne

−j 2pf0n+u0( )( )= ej2pf0n+u0 A+ w′

n

( ), n = 0, . . . , N − 1

(2)

where (·)* is a conjugate operator, w′n = wna

∗ne

−j 2pf0n+u0( ) isstill an independent complex AWGN with zero-mean andvariance σ2. Absolute phase of rn is presented as

/rn = 2pf0n+ u0 +/ A+ w′n

( ), n = 0, . . . , N − 1 (3)

When SNR = (A2/σ2) is high enough, the approximate modelof previous phase-based algorithms is expressed as

/ A+ w′n

( )≃ arctanIm w′

n

( )A+ Re w′

n

( )≃ arctanIm w′

n

( )A

≃ Im w′n

( )A

, n = 0, L, N − 1 (4)

A more reasonable model proposed by Fu and Kam is given[21]

/ A+ w′n

( )≃ arcsinIm w′

n

( )rn∣∣ ∣∣

≃ Im w′n

( )rn∣∣ ∣∣ , n = 0, L, N − 1 (5)

SVR-based Phase Unwrapping Process and FrequencyEstimation AlgorithmIn (5), there are two approximations still existing [21].

Based on the linear relationship in (3), this paper utilisesSVR’s excellent capability for learning unknown models tounwrap phases and estimate frequency. At time point k(1≤k≤ N− 1), we yield a training set Sk = xki , y

ki

( )i =|{

1, . . . , k.}, xki = i− 1, yki = /ri−1, where /ri−1 denotes theestimation value of /ri−1 and /r0 = arg[u0 +/(A+ w0)].At first, we define a line fk(x) = (wk·φ(x)) + bk and its

ε-insensitive loss function as (see (6))

L yki , fk xki( )( ) = yki − fk xki

( )∣∣ ∣∣1=

{


where εk is the insensitive loss coefficient at time point k.Next, we assume that fk(x) insofar as for εk can completely

fit all elements of Sk, where (·) is an inner product operatorand φ(·) is a non-linear mapping from low tohigh-dimensional feature space.We denote dki as thedistance from point xki , y

ki

( )[ Sk to fk(x)

dki = wk · f xki( )( )+ bk − yki

∣∣ ∣∣��1+ wk

∥∥ ∥∥2√≤ 1k��

1+ wk

∥∥ ∥∥2√ , i = 1, L, k (7)

According to (7), we optimise fk(x) through maximising

1k/

��1+ wk

∥∥ ∥∥2√, that is, minimising ||wk||

2. Thereby, SVR is

presented as

min J wk , bk( ) = 1

2wk

∥∥ ∥∥2s.t. wk · f xki

( )( )+ bk − yki∣∣ ∣∣ ≤ 1k ,

i = 1, L, k

(8)

In fact, fitting errors larger than εk always exist. By

introducing slack variables jki , j∗( )k

i≥ 0, i = 1, . . . , k

and penalty factor Ck at time point k, (8) is converted into

min J wk , bk , jk , j∗k

( ) = 1

2wk

∥∥ ∥∥2+ Ck

∑ki=1

jki + j∗( )k

i

[ ]

s.t.yki − wk ·f xki

( )( )− bk ≤ 1k + jki

wk · f wki

( )( )+ bk − yki ≤ 1k + j∗( )k

i

{, i = 1, L, k

(9)

where jk= jk1, jk2, L, j

kk

[ ]T, j∗k = j∗

( )k1, j∗( )k

2, L, j∗( )k

k

[ ]T,

[·]T is a transpose operator, C is a positive constant to takecompromise in SVR’s generalisation capability and fittingerrors, which are denoted by the first and second items of J(wk, bk), respectively.Equation (9) is a strict convex quadratic programming (QP)

problem in optimal theories. Then, using Lagrange multipliermethod

wk =∑ki=1

xki aki − a∗( )k

i

[ ](10)

0, yki − fk xki( )∣∣ ∣∣ ≤ 1k

yki − fk xki( )∣∣ ∣∣− 1k , else

(6)

251& The Institution of Engineering and Technology 2014

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∑ki=1
a∗( )ki−ak

i

[ ]= 0 (11)

Ck − aki − gki = 0, i = 1, . . . , k (12)

Ck − a∗( )ki− g∗( )k

i= 0, i = 1, . . . , k (13)

where aki , a∗( )k

i , gki , g∗( )k

i are the ith Lagrange multipliers ofyi − wk · f xi

( )( )− bk − 1k − ji ≤ 0, wk · f xi( )( )+ bk−

yi − 1k − j∗( )k

i≤ 0, ji ≥ 0, j∗

( )ki≥ 0, respectively.

Substituting (10)–(13) into (9), replacing(f(xki) · f(xkj ))

with linear kernel function K(xki , x

kj

) = xki xkj in this study

and deriving the wolf dual problem of (9)

max W ak , a∗k

( )= − 1

2

∑ki,j=1

aki − a∗( )k

i

[ ]akj − a∗( )k

j

[ ]xki x

kj

+∑ki=1

yki aki − a∗( )k

i

[ ]− 1k

∑ki=1

aki + a∗( )k

i

[ ]

s.t.

∑ki=1

aki − a∗( )k

i

[ ]= 0

0 ≤ aki , a∗( )k

i≤ Ck , i = 1, . . . , k

(14)

where ak = ak1, a

k2, . . . , a

kk

[ ]T, a∗

k = a∗( )k1, a∗( )k

2,

[. . . , a∗( )k

k]T.

We obtain ak , a∗k through solving (14). Ultimately,

we obtain the best approximation of (3) by using

Fig. 1 Recursive implementation of SVR-based algorithm


Karush–Kuhn–Tucker (KKT) conditions

fk(x) = wk · f x( )( )+ bk =∑ki=1

aki − a∗( )k

i

[ ]xki x+ bk (15)

where

bk = yki −∑kj=1

akj − a∗( )k

j

[ ]xki x

kj

− 1k , aki [ (0, Ck) or

bk = yki −∑kj=1

akj − a∗( )k

j

[ ]xki x

kj

+ 1k , a∗( )ki[ (0, Ck )

are bias at time point k.Not only does (15) guarantee less fitting errors, but it also

possesses a good generalisation capability. It means that afterlearning previous unwrapped phases y0, …, yk−1, (15) canpredict the variation trend of phase efficiently for the current

time point k and derive /SVRrk =∑k

i=1 aki − a∗( )k

i

[ ]xki k+

bk . The phase /rk is then unwrapped within a 2π-intervalcentred around ∠SVRrk, by adding multiples of ± 2π to theprincipal value of /rk when the absolute difference between∠SVRrk and the principal value of /rk is greater than π.Namely, we select a proper m to satisfy /rk = arg rk

( )+2pm [ /SVRrk − p, /SVRrk + p

[ ). Till time point k =N,

we take f N0 = ∑Ni=1 aN

i −[(

a∗( )Ni]xNi /2p) and u N

0 = bN asthe ultimate estimation value. This algorithm can be


Fig. 2 Graphs showing

a Mean with θ0 = 0, N = 32, SNR = 0 dBb Mean with θ0 = 0, N = 32, SNR = 8 dB

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implemented recursively in time, as shown schematically inFig. 1.
2.2 SVR’s parameter settings

Setting SVR’s parameters, including insensitive losscoefficient ε, penalty factor C and so on, is an ever presentdifficulty in corresponding fields all the time and has nocomplete theoretical basis or explicit closed-form. In spiteof that, it has a pronounced impact on SVR’s performance.Until recently, cross validation was a common, but complexand time-consuming method. We formulate properparameter values by understanding SVR’s theory andintegrating a large number of references and experiments inthis study.When k is small, imperfect phase unwrapping at a

particular time point can have an impact on the variationtrend of f (x) easily. In order to avoid this discrepancy, anew model must be made with the aptitude of bettergeneralisation capability. As k increases, its necessitydecreases contrarily, which is because of the degradingimpact of improperly unwrapped phase. Simply speaking,the model’s generalisation capability is inverselyproportional to the size of the set S.Intuitively, insensitive loss coefficient ε is the vertical

height of ε-tube. The larger ε is, the less support vectorsthere are, and the better generalisation capability exists.Nevertheless, too large ε will cause unfixable b. Noted that,if the predicted phase is in the vicinity of π’s odd times, theestimation performance is deteriorated rapidly for itsambiguous phase characteristic. Selecting a proper 1 canreduce this impact. At the same time, ε is directlyproportional to SNR, so insensitive loss coefficient at timepoint k is given by [25]

1k = tA

s

��ln k

k

√, k = 1, . . . , N − 1 (16)

where t is a positive constant, and t = 1 is the set in this study,(s/A) = ��

SNR√

is assumed to be known.Penalty factor C controls the penalty degree of vectors

outside the ε-tube, and determines SVR’s generalisationcapability. C is directly proportional to the sample size, andinversely proportional to SNR. As (15) is a line in atwo-dimensional plane, the first item of target functionduring solving SVR’s QP problem is directly proportionalto its slope, so a very small C will result a horizontal andunderfitting fk(x). Inspired by [25] and [26], penalty factorat time point k is given by

Ck = ds

A

��k3

√max gk − lsk

∣∣ ∣∣, gk + lsk

∣∣ ∣∣( ),

k = 1, . . . , N − 1(17)

where

gk =1

k

∑k−1

n=0

yn∣∣ ∣∣2, sk =

��1

k

∑k−1

n=0

yn∣∣ ∣∣2−gk

( )2√√√√ ,

yn = /rn−1

is a section of the element of training set Sk; as (16),(s/A) = ��

SNR√

is assumed to be known; δ, λ are positiveconstants, and δ = 0.1, λ = 0.5 are set in this study.


3 Results and analyses

We have compared the proposed algorithm entitled as SVRestimator with other fours: the time-domain ML and MAPestimators proposed in [21]; the least-squares phaseunwrapping estimator (LSPUE) proposed in [18]; thegeneralised weighted linear predictor (GWLP) proposed in[5].

3.1 Mean performance

Fig. 2 illustrates the mean of these five estimators, whileSNR is 0 and 8 dB, respectively. The number of MonteCarlo experiments is 10 000, and θ0 = 0, N = 32. Here, f0and θ0 are assumed to be random variables with α = β = 50,Ω =Θ = 0 Tikhonov pdfs in MAP estimator.


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From Fig. 2, all of these five estimators exhibit narrow
unbiased estimation range in the condition of low SNR.As SNR increases, the range increases correspondingly.In addition, unbiased performance of SVR estimator isalways the best one.

3.2 Phase unwrapping process

Everything is as in Fig. 2 except that SNR is 8 dB, becauseGWLP does not need the phase unwrapping process, Fig. 3illustrates the arbitrary phase unwrapping processes of theother four while f0 = 0 and f0 = 0.3. It is shown that SVRestimator can unwrap phase accurately whether f0 = 0 orf0 = 0.3.


a Arbitrary phase unwrapping processes with f0 = 0, θ0 = 0, N = 32,SNR = 8 dBb Arbitrary phase unwrapping processes with f0 = 0.3, θ0 = 0, N = 32,SNR = 8 dB


3.3 Frequency Estimation performance

Thirdly, the estimation performance is worked out. As inFig. 2, Figs. 4 and 5 illustrate the MSE curves against SNRand f0, respectively, where MSE is defined as

E[f0 − f0( )2]

, f0 is the estimation value of f0.It is shown that when f0 = 0, SVR estimator is only not as

good as the MAP one; their MSE performances all decreaseas f0 = 0 increases, where SVR estimator is the second lesssensitive to it ( just only behind the LSPUE one) and hashad a great advantage already when f0 = 0.3. The reason isthat when SNR is low, MAP estimator depends on thefollowing three approximations

sin /rn − 2pf0 + u0( )[ ]≃/rn − 2pf0 + u0

( )(18)


a MSE of frequency estimation against SNR with f0 = 0, θ0 = 0, N = 32b MSE of frequency estimation against SNR with f0 = 0.3, θ0 = 0, N = 32



a MSE of frequency estimation against f0 with θ0 = 0, N = 32, SNR = 0 dBb MSE of frequency estimation against f0 with θ0 = 0, N = 32, SNR = 8 dB

Fig. 6 Arbitrary frequency estimation processes with f0 = 0,θ0 = 0, N = 32, SNR = 8 dB

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sin 2pf0 −V( )≃2pf0 −V (19)

sin u0 −Q( )≃u0 −Q (20)

We can know that, satisfying (18) is well related to phaseunwrapping process perfectly; satisfying (19) and (20) aredependent of the relationship between the a priori mean Ω,Θ and the actual value 2πf0, 2πf0, θ0. Bearing Ω = 0 inmind, (19) is satisfied only when f0 = 0, which means, MAPestimator now can acquire reliable knowledge from the apriori assumption; otherwise, their unreliability will cause anegative impact on the MSE performance. However, SVRestimator predicts phase for the next time point, merely bylearning the unwrapped phases at previous time points, sothat phenomenon will not occur.


Besides, we note that when f0 = 0, the MSE curve of MAPestimator is lower than CRLB at first, but then returns to theabove of it again and nears to it closely as SNR increases. Thereason is that, if SNR is higher than the threshold, MAPestimator can attain Bayesian CRLB (BCRLB) proposed in[21]. Also, when f0 = 0, BCRLB is lower than CRLB forthe reason of a priori assumption of f0 and θ0, and thedifference is directly proportional to α, β. On the otherhand, BCRLB reduces to CRLB at comparatively highSNR. In addition, the MSE of MAP estimator decreasesrapidly as f0 increases, and the speed is inverselyproportional to α, β. Emphasised that, SVR estimatorderives frequency absolutely through utilising SVR to learnunknown phase noise model, so that its MSE will not belower than CRLB.Impact of sample size NEverything is as in Fig. 3 other than f0 = 0, because LSPUE

and GWLP are not sample-to-sample iterative estimators,arbitrary frequency acquisition processes and the MSEcurves of the other three against N are plotted in Figs. 6and 7, correspondingly.It is clear that ML estimator is the most fluctuant one during

the initial part of frequency acquisition process, and itsthreshold is highest too. The reasoning, its frequencyacquisition process starts with the first two phases that arealready known, and their accuracy can impact MSEdirectly. At the same time, each estimator’s MSE is directlyproportional to N if its SNR is higher than the threshold;otherwise, it will keep the value unchanged after somemeasurements, for example, the ML estimator in Fig. 7.Contemporarily, it is to note in Fig. 7, because the initialestimation value of SVR estimator is fixed, its MSE islower than CRLB before convergence and directlyproportional to f0. Accordingly, any estimation value beforeconvergence is ineffective and meaningless.Moreover, when these three estimators’ SNRs are above the

threshold, SVR estimator has the slowest convergence speed.Upon that, the MSE curves of them with f0 = 0.3, θ0 = 0,N = 16 are plotted against SNR in Fig. 8. From Fig. 8, SVRestimator would not come close to CRLB, even if SNR


Fig. 8 MSE of frequency estimation against SNR with f0 = 0.3,θ0 = 0, N = 16

Fig. 7 Impact of N on MSE with f0 = 0, θ0 = 0, SNR = 8 dB


a MSE of frequency estimation against θ0 with f0 = 0, N = 32, SNR = 8 dBb MSE of frequency estimation against θ0 with f0 = 0.3, N = 32, SNR = 8 dB

Table 1 Consuming time with different N (ms)

Algorithm N = 8 N = 16 N = 32 N = 64

ML 15 31 62 93MAP 16 31 63 141LSPUE 187 374 1358 3611GWLP 10 19 39 80

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goes to infinity, which is different from the other three.Consequently, N must be large enough to ensure thevalidity of SVR estimator; and the suitable value is about25 after numerous experiments.

proposed 219 499 1622 5320

3.4 Impact of initial phase θ0

Everything is as in Fig. 3, the MSE curves are plotted againstθ0 in Fig. 9. We can see that, when SNR is under thethreshold, all of these five estimators’ MSE curves varywith θ0 irregularly, however SVR estimator is the most


robust one; when SNR is above the threshold, SVRestimator is immune to θ0, but MAP one is not. This isachieved because (20) is satisfied only when θ0 = 0, whichmeans, MAP estimator now can acquire reliable knowledge


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from the a priori assumption; otherwise, their unreliabilitywill cause a negative impact on the MSE performance.
3.5 Computational complexity

Because SVR is translated into QP problem and need tosearch the minimums during the process, we cannot derivethe explicit form of the computational complexity of SVRestimator. However, the consuming time can be derived. Soeverything is as in Fig. 2 except that the number of MonteCarlo experiments is 100, SNR is 8 dB and f0 = 0, theconsuming times are list in Table 1 while N is 8, 16, 32and 64, respectively. The running comuter isASUS-PC1111 having Intel(R) Pentium 2.13 GHz CPU and2.00 GB RAM.

4 Conclusions

Phase unwrapping process is a key point in phase-basedfrequency estimation. MAP algorithm proposed in [21] is anovel and effective one. However, we have found in thisstudy that, on one side, it can improve the estimationaccuracy of frequency adjacent to the a priori mean; on theother side, it can degrade the estimation accuracy offrequency far away from the a priori mean. Therefore, aSVR-based robust frequency estimation algorithm ispresented. First, we adopt SVR to learn the unwrappedphases at previous time points, predict the variation trend ofphase efficiently and derive the estimation value for thecurrent time point. Once acquired, in terms of relationshipbetween instantaneous signal phase and frequency, weaddress a simple and effective frequency estimationalgorithm. Comparing with MAP algorithm and other threeclassical ones, the proposed algorithm completely exhibitsits advantages of wider estimation range, higher estimationaccuracy, lower sensitivity of frequency and initial phase,by sacrificing higher demands of the sample size and moreconsuming times.Furthermore, we can extend to estimate the frequency of

polynomial phase signal (PPS), just through replacing thelinear kernel function by a polynomial one; and also,because SVR predicts the curve’s variation trend merely interms of training set which consists of previous points’values, we even can estimate frequency under thenon-Gaussian condition by the same way.Stressed that the proposed algorithm learns training set and

obtains the approximate values of SVR’s insensitive losscoefficient ε and penalty factor C. As a next step, therefore,improving SVR’s parameter setting is an important researchpoint.

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Support vector regression -based robust frequency estimation algorithm by instantaneous phase