Multiscale KF Algorithm for Strong Fractional Noise Interference...

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2011, Article ID 356421, 9 pagesdoi:10.1155/2011/356421

Research ArticleMultiscale KF Algorithm for Strong FractionalNoise Interference Suppression in Discrete-TimeUWB Systems

Liyun Su,1 Yuli Zhang,1 Yanju Ma,1 Jiaojun Li,2 and Fenglan Li3

1 School of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, China2 School of Electronic Information and Automation, Chongqing University of Technology,Chongqing 400054, China

3 Institute of Library, Chongqing University of Technology, Chongqing 400054, China

Correspondence should be addressed to Liyun Su, [email protected]

Received 26 July 2011; Accepted 4 September 2011

Academic Editor: Daniele Fournier-Prunaret

Copyright q 2011 Liyun Su et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

In order to suppress the interference of the strong fractional noise signal in discrete-time ultra-wideband (UWB) systems, this paper presents a new UWB multi-scale Kalman filter (KF)algorithm for the interference suppression. This approach solves the problem of the narrowbandinterference (NBI) as nonstationary fractional signal in UWB communication, which does notneed to estimate any channel parameter. In this paper, the received sampled signal is transformedthroughmultiscale wavelet to obtain a state transition equation and an observation equation basedon the stationarity theory of wavelet coefficients in time domain. Then through the Kalman filtermethod, fractional signal of arbitrary scale is easily figured out. Finally, fractional noise interferenceis subtracted from the received signal. Performance analysis and computer simulations reveal thatthis algorithm is effective to reduce the strong fractional noise when the sampling rate is low.

1. Introduction

Ultrawideband (UWB) radio is an emerging technology for future short-rang high-speedwireless communications. In recent years, the Federal Communication Commission (FCC)confined the working bandwidth of UWB systems in the 3.1–10.6GHz frequency band rangand the power spectral density (PSD) in less than −41.3 dBm/MHz [1, 2]. UWB signal isdefined as any wireless signal that has a relative bandwidth more than 25% or an absolutebandwidth in excess of 500MHz, satisfying PSD required by FCC. Compared with theexisting radio communication technologies, the spread spectrum technique of UWB systems

2 Discrete Dynamics in Nature and Society

bears many advantages, for instance, low power consumption, low probability of intercept,strong ability of penetrating materials, and large space capacity.

Since a conventional UWB system transfers data by using ultrashort pulses, itpossesses considerably large bandwidths. Usually, it causes mutual interference with themost existing electronic narrowband systems over the same frequency band inevitably, suchas radio navigation or detection system, wireless local area network and global positioningsystem (GPS) [3]. Therefore, to guarantee the successful deployment of UWB, it is relatedto not only the deployment of efficient multiple access technologies but also narrow-bandinterference suppression technologies.

Recently a number of the narrowband interference (NBI) suppression technologieshave been published thoroughly in many references. Some NBI cancellation solutions[4–11] have been utilized in UWB systems, for example, pulse waveform designingtechnology, minimum mean square error (MMSE) rake receiver technology, templatewaveform estimation technology, frequency domain technology, notch filter techniques, andso forth. These methods all selected simple and stable model as NBI model [12, 13] andused a large number of estimated channel parameters to eliminate interference. Moreover,they studied less about the nonstationary narrowband interference model and the reductionof channel parameter estimation. Hence, this paper proposes a new multi-scale Kalmanfilter algorithm for strong fractional noise interference suppression in time hopping (TH)UWB systems. As the nonstationary fractional signal is very strong and UWB signal mixedwith white noise is very weak, the received signal in this paper is modeled as narrowbandsignal which concentrated mainly fractional signal energy. Then it performs the multi-scalewavelet analysis of the received signal and establishes the state space model of Kalmanfilter estimation [14–17]. The state space model includes a time domain state equationand observation equation. They are obtained by the stationarity of fractional waveletcoefficients. Lastly, we can estimate the fractional Brownianmotion (FBM), and a novel multi-scale Kalman filter for UWB systems is proposed. Its performance analysis and computersimulations are also discussed in Section 5.

This paper is organized as follows. In Section 2, the discrete-timemodel of the receivedsignal is defined. In Section 3, we introduce the FBM. A novel multi-scale Kalman filteralgorithm is proposed in Section 4. The performance of this multi-scale Kalman filter isanalyzed, and simulation results are given in Section 5. The conclusions are drawn inSection 6.

2. Discrete-Time UWB System Model

Consider a multiple-access TH-UWB system with single user. The transmitted UWB signalemploying binary pulse amplitude modulation (2-PAM) is provided by

st(t) =+∞∑

j=−∞b�j/Ns�ω

(t − jTf − CjTc

), (2.1)

where ω(t) is a single link transmitted pulse waveform. Tf is the pulse repetition period. Tcis the TH unit time. Ns is the number of repeated pulses over one symbol period. b�j/Ns� ∈{+1,−1} represents the user’s information during the jth frame. If a symbol period durationis Tb = NsTf , the transmitted rate is Rs = 1/(NsTf). {Cj} is the pseudorandom TH sequencewhich adds an additional time shift Tc to each pulse, and 0 ≤ Cj ≤ Nh − 1.

Discrete Dynamics in Nature and Society 3

Assume dn stands for the nth symbol of single user. Then, (2.1) can be expressed as

st(t) =N0∑

n=1

bnv(t − nTb), (2.2)

where v(t) =∑Ns−1

j=0 ω(t − jTf − CjTc).N0 is the number of transmitted symbols.Assume the nonstationary FBM model is used as NBI model. Then, while the

transmitted UWB signal propagates through a flat channel and is corrupted by fractionalnoise signal BH(t) and additive white Gaussian noise (AWGN) N(t), the observation signalis given by

r(t) = S(t) + BH(t) +N(t), (2.3)

where S(t) = α·st(t). α is the channel gain.N(t) ∼ N(0, σ2Nt). S(t)+N(t) denotes the wideband

component of the received signal.However, in practical situation, the received signal must be discrete. Suppose r(t) is

sampled at the rate of 1/Ts and an output r(n) = r(t)|t=nTs is received in discrete-time form.Then, the following received signal becomes

r(n) = BH(t) + S(n) +N(n), (2.4)

where S(n) = α · ∑N0m=1 bnv(n − mM). M is the number of sampling points in every symbol

period.Thus, in the nth symbol period, (2.4) can be changed into

rn = Bn + Sn +Nn, (2.5)

where Sn = X · η and X = [v(0), v(1), . . . , v(M − 1)]T . Here η = αbn is a constant.In this paper, the estimated scheme is proposed to stress against the fractional

narrowband interferer. The whole basic idea is to realize the prediction of FBM signal, so thatit can be removed from the received signal to reduce the fractional interference. In theory,this paper preforms an analysis of FBM. However, in practice, the estimation of the pureFBM is difficult to achieve because the FBM is often unknown. And as the narrowbandsignal occupies most of the received signal energy, the prediction of r(t) is used to replace theestimation of FBM. On the other hand, the error signal r(t)−r̂(t), where r̂(t) is an estimation ofr(t), is very important to eliminate the FBM. Kalman filter technique is also cited to estimatethe FBM in this paper. It is shown in Figure 1.

3. Fractional Brownian Motion

Brownian motion is a kind of random, irregular motion. In the 1960s, a new theory offractional Brownian motion, commonly referred to as the FBM, is advanced by Mandelbrotand Van Ness [18].

Denote

BH(t) =1

Γ(H + 1/2)·{∫ t

0(t − λ)H−1/2dB(λ) +

∫0

−∞(t − λ)H−1/2 − (−λ)H−1/2dB(λ)

}, (3.1)


−

Kalmanfilter

Widebandsignal

A state spacemodel

Receivedsignal Transformation

Multiscalewavelet

Figure 1: Estimated scheme of fractional noise suppression in UWB systems.

where Γ(·) is Gamma function andH (0 < H < 1) is a Hurst index. For the general Brownianmotion, when H = 1/2, it is a standard Brownian motion. Obviously, FBM, in a continuousGaussian process with a zero mean, has the nonstationarity, self-similarity, and so forth [18,19].

Certainly, there are many methods of generating Brownian motion. For example,Weierstrass-Mandelbrot random function method, under certain conditions, emerges FBMthrough an almost everywhere nondifferentiable continuous function; in fractional Gaussiannoise method, it is that the fractional Gaussian noise is represented as the incrementalderivative of FBM; in covariance matrix transform method, Gaussian white noise is turnedinto a sequence with the covariance matrix through the Cholesky decomposition technologyto get FBM; wavelet transform method, that is to say, in terms of adjacent times frequencyconversion rate, makes Gaussian white noise into FBM by the inverse wavelet transform.However, this paper uses Weierstrass-Mandelbrot random function to obtain FBM.

4. Multiscale Kalman Filter Algorithm on Fractional Noise Model

With the purpose of NBI suppression to be successfully realized, we consider the discreteorthogonal wavelet decomposition of BH(t). Then, (3.1) can be expressed in the followingform:

BH(t) =+∞∑

n=−∞cJ[n]ΦJ,n(t) +

+∞∑

j=−∞

+∞∑

n=−∞dj[n]Ψj,n(t), (4.1)

where aJ[n] = 2−J/2∫∞−∞ I(t)Φ(2−J t − n)dt, dj[n] = 2−j/2

∫∞−∞ I(t)Φ(2−j t − n)dt. ΦJ,n(t) =

2−J/2Φ(2−J t − n) is a scale function of the wavelet transform, and the wavelet function iswritten by ΨJ,n(t) = 2−j/2Ψ(2−j t − n) [20, 21].

If (2.4) is converted applying the wavelet decomposition theory, we have the followingequation:

rj[n] = Sj[n] + dj[n] +Nj[n] = dj[n] + uj[n], (4.2)

where j = 1, 2, . . . , J . uj[n] = Sj[n]+Nj[n] has the mean μj = Sj[n] and variance Var(uj[n]) =σ2uj. {dj[n]} is stable [22]. Then, due to the stationarity of wavelet coefficients {dj[n]}, this

AR(p) model can be found in this study:

dj[m] =p∑

i=1

aidj[m − i] + εj[m], (4.3)


where εj[m] with a zero mean and variance σ2εj , independent of S[m] and BH[m], is AWGN.

ai, i = 1, 2, . . . , p are parameter coefficients.Define

Fj =

[O(M/2j−1)×1 EM/2j−1

O1×(M/2j−p) BTj

],

Bj =[ap, ap−1, . . . , a1]

T , G =[O1×(M/2j−1) 1

]]T,

d(M/2j )j,m =

[dj

[m − M

2j+ 1

], . . . , dj[m − 1], dj[m]

]T.

(4.4)

For (4.3), we have

d(M/2j )j,m+1 = Fjd

(M/2j )j,m +Gε(m + 1). (4.5)

Equation (4.5) is one-step prediction from d(M/2j )j,m to d

(M/2j )j,m+1 , so there M/2j iterative

steps of dj,n from dj,n+1, namely,

dj,n+1 = FM/2jj dj,n +GM/2j εn+1, (4.6)

where dj,n = d(M/2j )j,(n+1)M/2j−1, and

GM/2j =[F(M/2j−1)j G, F

(M/2j−2)j G, . . . , FjG,G

],

εn+1 =[ε

((n + 1)M

2j

), . . . , ε

((n + 2)M

2j− 1

)]T.

(4.7)

What is more, to facilitate the study below, we need to transform formula (2.5) into anobservation equation. That is,

rj,n = dj,n + Sj,n +Nj,n. (4.8)

Thus, if (4.6) is seen as a state transition equation in a dynamic space, combined with(4.8), we can obtain a state space model for BH(t):

dj,n+1 = FM/2jj dj,n +GM/2j εn+1,

rj,n = dj,n + uj,n,(4.9)

where

uj,n = Sj,n +Nj,n. (4.10)

To estimate the state space model (4.9) in the third part, the new Kalman filteralgorithm is structured as follows.


−4

−2

0

2

4

Amplitud

e(V

)0 10 20 30 40 50 60 70 80 90 100

t(s)

Figure 2: Fractional noise signal generated by Weierstrass-Mandelbrot function.

(1) Initialization: at n = 0, initialize dj,0 and P0 are given.

(2) State update:

d̂−j,n+1 = FM/2j

j · d̂j,n,

P−j,n+1 = FM/2j

j,n · Pj,n ·(FM/2jj,n

)T+ σ2

εj ·GM/2j (GM/2j )T ,

Kj,n+1 = P−j,n+1

{P−j,n+1 + σ2

uj(n)E(M/2j )×(M/2j )

}†,

d̂j,n+1 = d̂−j,n+1 +Kn+1

(rj,n+1 − d̂−

j,n+1

),

Pj,n+1 =(I −Kj,n+1

)P−j,n+1.

(4.11)

Therefore, this algorithm can predict all the high-frequency coefficients of the waveleton any scale according to the KF iterative algorithm. It means that FBM can be calculatedthrough the inverse wavelet transform to receive the interference suppression in UWBsystems.

5. Simulation

In this section, the performance of our proposed NBI suppression method is surveyed. It issimulated by theMatlab7.1 software, and the pulse waveformω(t) from the TH-UWB systemis adopted as follows:

ω(t) =

(1 − 4π

(t

τ

)2)

· exp(−2π

(t

τ

)2), (5.1)

where τ denotes the pulse forming factor. Suppose the parameters are Tf = 8 ns, Ns = 2,Nh = 5,M = 160, and J = 3, and the sampled time interval is 0.05 ns, which is low. Moreover,the Hausdorff-Besicovitch (fractional) dimension of the graph of Weierstrass-Mandelbrotrandom function is D = 1.5. Therefore, there are some measured results for this simulationin the following portion. In Figure 2, the original fractional noise signal is shown. Figures 3and 4 describe the estimated fractional signal and the noisy error signal, respectively. That is,the proposed new algorithm can effectively estimate the strong fractional noise from theseresults of Figures 1–3.


−4

−2

0

2

4

(t/s)Amplitud

e(V

)0 10 20 30 40 50 60 70 80 90 100

Figure 3: The estimated fractional signal based on multiscale wavelet transform.

0

Amplitud

e(V

)

0 10 20 30 40 50 60 70 80 90 100−0.5

0.5

t(s)

Figure 4: Error signal being wideband component of the observation signal.

Generally, the signal-to-interference ratio (SINR) can detect the performance of aalgorithm very well. In order to preferably evaluate the performance of the new fractionalnoise suppression algorithm from another angle, SINR is improved in this paper:

SINRimprovement =SINRout

SINRin=

E{|r(n) − S(n)|2

}

E{|e(n) − S(n)|2

}

=Var(BH(n)) + σ2

N

Var(e(n)) + σ2N

,

(5.2)

where e(n) = BH(n) − B̂H(n), which is the estimated error signal of BH(n). σ2N is the variance

of the white noise.Thus, 100 independent tests are done, where the lower sampling rate is 0.1 ns and

D = 1.2 is employed. From this simulation, it is indicated that the actual SINR has greatlyimproved in Figure 5. On the other hand, it is shown that this new method is effective insuppressing the nonstationary BH(t) for UWB systems, which is a strong NBI.

6. Conclusion

In this paper, the multi-scale KF solution is suggested to suppress the fractional noiseinterference in discrete-time UWB systems. Particularly, after a discrete-time received signalis modeled in the receiver and the fractional signal is transformed based on the waveletanalysis, the fractional signal can be estimated by Kalman filter, which is supposed as a strongnonstationary NBI process. From the simulation above, the algorithm estimates the fractionalnoise signal under the weak UWB signals by combining the theory of the multi-scale wavelettransform and the Kalman filter. So that it achieves the separation of the narrowband signal


10

15

20

25

30

35

40

45

Input SINR (dB)

SINRim

prov

emen

t(dB)

−20 −15 −10 −5

IdealReality

Figure 5: Comparison of the performance of the ideal and proposed multi-scale KF method.

and wideband signals successfully. One of the advantages of this paper is not estimatingany channel parameter. Simulation results show that this method is an effective narrowbandinterference suppression method of UWB systems.

Acknowledgments

This work was supported by Natural Science Foundation Projects of CQ CSTC of China(CSTC2010BB2310), and Chongqing CMEC Foundations of China (KJ080614, KJ100810,KJ100818).

References

[1] L. Y. Su and W. Pang, “Ultra-wideband wireless technology—the forefront technology of short-rangwireless communication,” Technology and Market, no. 5, pp. 35–36, 2005.

[2] Z. Z. Zhang, X. J. Sha, and Q. Y. Zhang, Ultra-Wideband Communication Systems, Electronic IndustryPress, 2010.

[3] G. Zhu, Ultra-Wideband (UWB) Principles and Interference, Tsinghua University Press, 2009.[4] N. Boubaker and K. B. Letaief, “A low complexity MMSE-RAKE receiver in a realistic UWB channel

and in the presence of NBI,” in Proceedings of the IEEE Wireless Communications and NetworkingConference (WCNC ’09), vol. 1, no. 3, pp. 233–237, 2009.

[5] B. Parr, B. L. Cho, K. Wallace, and Z. Ding, “A novel ultra-wideband pulse design algorithm,” IEEECommunications Letters, vol. 7, no. 5, pp. 219–221, 2003.

[6] T. Ikegami and K. Ohno, “An interference mitigation study for coexistence of DS-UWB and OFDM,”in Proceedings of the 18th Annual IEEE International Symposium on Personal, Indoor and Mobile RadioCommunications, (PIMRC’07), vol. 1, no. 9, pp. 583–587, September 2007.

[7] J. Z. Wang and T. T. Wong, “Narrowband interference suppression in time-hopping impulse radioultra-wideband communications,” IEEE Transactions on Communications, vol. 54, no. 6, pp. 1057–1067,2006.

[8] A. E. C. Tan and K. Rambabu, “Modeling the effects of interference suppression filters on ultra-wideband pulses,” IEEE Transactions on Microwave Theory and Techniques, vol. 59, no. 1, pp. 93–98,2010.

[9] K. Ohno and T. Ikegami, “Interference mitigation study for UWB radio using template waveformprocessing,” IEEE Transactions on Microwave Theory and Techniques, vol. 54, no. 4, pp. 1782–1792, 2006.


[10] R. D. Weaver, “Frequency domain processing of ultra-wideband signals,” in Proceedings of the 37thAsilomar Conference on Signals, Systems and Computers, vol. 2, no. 11, pp. 1221–1224, Pacific. Grove,Calif, USA, November 2003.

[11] Y. Dhibi and T. Kaiser, “Impulsive noise in UWB systems and its suppression,” Mobile Networks andApplications, vol. 11, no. 4, pp. 441–449, 2006.

[12] X. Chu and R. D. Murch, “The effect of NBI on UWB time-hopping systems,” IEEE Transactions onWireless Communications, vol. 3, no. 5, pp. 1431–1436, 2004.

[13] Y. H. Zhu, Study of Narrowband Interference Suppression in UWB Systems, Yangzhou University, Jiangsu,China, 2009.

[14] M. S. Grewal and A. P. Andrews, Kalman Filtering: Theory and Practice Using MATLAB, John Wiley &Sons, New York, NY, USA, 2nd edition, 2008.

[15] J. Peng and L. Y. Su, “Performance analysis of blind multi-user detector with fuzzy Kalman filter inIR-UWB,” International Journal of Distributed Sensor Networks, vol. 5, no. 1, p. 47, 2009.

[16] Z. F. Ye, Statistical Signal Processing, University Press of Science and Technology, 2008.[17] M. A. Badamchizadeh, I. Hassanzadeh, and M. A. Fallah, “Extended and unscented Kalman filtering

applied to a flexible-joint robot with jerk estimation,” Discrete Dynamics in Nature and Society, ArticleID 482972, 14 pages, 2010.

[18] B. B. Mandelbrot and J. W. Van Ness, “Fractional Brownian motions, fractional noises andapplications,” SIAM Review, vol. 10, pp. 422–437, 1968.

[19] X. J. Li, D. X. Nie, and L. G. Yuan, “Computer research with fractional brown motion,” FujianComputer, no. 1, pp. 1–2, 2007.

[20] L. Y. Su, H. Ma, and S. F. Tang, “Weak signal detection of strong fractional noise using chaos oscillatorand kalman filtering,” Journal of Sichuan University (Engineering Science Edition), vol. 39, no. 3, pp.149–154, 2007.

[21] L. Su and F. Li, “Deconvolution of defocused imagewithmultivariate local polynomial regression anditerative wiener filtering in DWT domain,” Mathematical Problems in Engineering, vol. 2010, Article ID605241, 14 pages, 2010.

[22] P. Flandrin, “Wavelet analysis and synthesis of fractional Brownian motion,” Institute of Electrical andElectronics Engineers. Transactions on Information Theory, vol. 38, no. 2, pp. 910–917, 1992.

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