# Multiscale KF Algorithm for Strong Fractional Noise Interference Suppression in Discrete-Time UWB Systems

Authors: Liyun Su; Yuli Zhang; Yanju Ma; Jiaojun Li; Fenglan Li
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

In order to suppress the interference of the strong fractional noise signal in discrete-time ultrawideband (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 narrowband interference (NBI) as nonstationary fractional signal in UWB communication, which does not need to estimate any channel parameter. In this paper, the received sampled signal is transformed through multiscale wavelet to obtain a state transition equation and an observation equation based on the stationarity theory of wavelet coefficients in time domain. Then through the Kalman filter method, fractional signal of arbitrary scale is easily figured out. Finally, fractional noise interference is subtracted from the received signal. Performance analysis and computer simulations reveal that this algorithm is effective to reduce the strong fractional noise when the sampling rate is low.


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## Body

## 1. Introduction

Ultrawideband (UWB) radio is an emerging technology for future short-rang high-speed wireless communications. In recent years, the Federal Communication Commission (FCC) confined the working bandwidth of UWB systems in the 3.1–10.6 GHz frequency band rang and the power spectral density (PSD) in less than −41.3 dBm/MHz [1, 2]. UWB signal is defined as any wireless signal that has a relative bandwidth more than 25% or an absolute bandwidth in excess of 500 MHz, satisfying PSD required by FCC. Compared with the existing radio communication technologies, the spread spectrum technique of UWB systems 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, it possesses considerably large bandwidths. Usually, it causes mutual interference with the most existing electronic narrowband systems over the same frequency band inevitably, such as radio navigation or detection system, wireless local area network and global positioning system (GPS) [3]. Therefore, to guarantee the successful deployment of UWB, it is related to not only the deployment of efficient multiple access technologies but also narrow-band interference suppression technologies.Recently a number of the narrowband interference (NBI) suppression technologies have been published thoroughly in many references. Some NBI cancellation solutions [4–11] have been utilized in UWB systems, for example, pulse waveform designing technology, minimum mean square error (MMSE) rake receiver technology, template waveform estimation technology, frequency domain technology, notch filter techniques, and so forth. These methods all selected simple and stable model as NBI model [12, 13] and used a large number of estimated channel parameters to eliminate interference. Moreover, they studied less about the nonstationary narrowband interference model and the reduction of channel parameter estimation. Hence, this paper proposes a new multi-scale Kalman filter algorithm for strong fractional noise interference suppression in time hopping (TH) UWB systems. As the nonstationary fractional signal is very strong and UWB signal mixed with white noise is very weak, the received signal in this paper is modeled as narrowband signal which concentrated mainly fractional signal energy. Then it performs the multi-scale wavelet analysis of the received signal and establishes the state space model of Kalman filter estimation [14–17]. The state space model includes a time domain state equation and observation equation. They are obtained by the stationarity of fractional wavelet coefficients. Lastly, we can estimate the fractional Brownian motion (FBM), and a novel multi-scale Kalman filter for UWB systems is proposed. Its performance analysis and computer simulations are also discussed in Section 5.This paper is organized as follows. In Section2, the discrete-time model of the received signal is defined. In Section 3, we introduce the FBM. A novel multi-scale Kalman filter algorithm is proposed in Section 4. The performance of this multi-scale Kalman filter is analyzed, and simulation results are given in Section 5. The conclusions are drawn in Section 6.


## 2. Discrete-Time UWB System Model

Consider a multiple-access TH-UWB system with single user. The transmitted UWB signal employing binary pulse amplitude modulation (2-PAM) is provided by(2.1)st(t)=∑j=-∞+∞b⌊j/Ns⌋ω(t-jTf-CjTc),
where ω(t) is a single link transmitted pulse waveform. Tf is the pulse repetition period. Tc is 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 duration is  Tb=NsTf, the transmitted rate is Rs=1/(NsTf). {Cj} is the pseudorandom TH sequence which adds an additional time shift Tc to each pulse, and 0≤Cj≤Nh-1.Assumedn stands for the nth symbol of single user. Then, (2.1) can be expressed as
(2.2)st(t)=∑n=1N0bnv(t-nTb),
where v(t)=∑j=0Ns-1ω(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 fractional noise signalBH(t) and additive white Gaussian noise (AWGN) N(t), the observation signal is given by
(2.3)r(t)=S(t)+BH(t)+N(t),
where  S(t)=α·st(t).  α is the channel gain. N(t)~N(0,σNt2).  S(t)+N(t) denotes the wideband component of the received signal.However, in practical situation, the received signal must be discrete. Supposer(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
(2.4)r(n)=BH(t)+S(n)+N(n),
where  S(n)=α·∑m=1N0bnv(n-mM).  M is the number of sampling points in every symbol period.Thus, in thenth symbol period, (2.4) can be changed into
(2.5)rn=Bn+Sn+Nn,
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 that it 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 pure FBM is difficult to achieve because the FBM is often unknown. And as the narrowband signal occupies most of the received signal energy, the prediction ofr(t) is used to replace the estimation of FBM. On the other hand, the error signal r(t)-r̂(t), where r̂(t) is an estimation of r(t), is very important to eliminate the FBM. Kalman filter technique is also cited to estimate the FBM in this paper. It is shown in Figure 1.Figure 1
Estimated scheme of fractional noise suppression in UWB systems.


## 3. Fractional Brownian Motion

Brownian motion is a kind of random, irregular motion. In the 1960s, a new theory of fractional Brownian motion, commonly referred to as the FBM, is advanced by Mandelbrot and Van Ness [18].Denote(3.1)BH(t)=1Γ(H+1/2)⋅{∫0t(t-λ)H-1/2dB(λ)+∫-∞0(t-λ)H-1/2-(-λ)H-1/2dB(λ)},
where Γ(·) is Gamma function and H (0<H<1) is a Hurst index. For the general Brownian motion, when  H=1/2, it is a standard Brownian motion. Obviously, FBM, in a continuous Gaussian 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 FBM through an almost everywhere nondifferentiable continuous function; in fractional Gaussian noise method, it is that the fractional Gaussian noise is represented as the incremental derivative of FBM; in covariance matrix transform method, Gaussian white noise is turned into a sequence with the covariance matrix through the Cholesky decomposition technology to get FBM; wavelet transform method, that is to say, in terms of adjacent times frequency conversion 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 discrete orthogonal wavelet decomposition ofBH(t). Then, (3.1) can be expressed in the following form:
(4.1)BH(t)=∑n=-∞+∞cJ[n]ΦJ,n(t)+∑j=-∞+∞∑n=-∞+∞dj[n]Ψj,n(t),
where aJ[n]=2-J/2∫-∞∞I(t)Φ(2-Jt-n)dt,  dj[n]=2-j/2∫-∞∞I(t)Φ(2-jt-n)dt.  ΦJ,n(t)=2-J/2Φ(2-Jt-n) is a scale function of the wavelet transform, and the wavelet function is written by ΨJ,n(t)=2-j/2Ψ(2-jt-n) [20, 21].If (2.4) is converted applying the wavelet decomposition theory, we have the following equation:
(4.2)rj[n]=Sj[n]+dj[n]+Nj[n]=dj[n]+uj[n],
where j=1,2,…,J.  uj[n]=Sj[n]+Nj[n] has the mean μj=Sj[n] and variance Var(uj[n])=σuj2.  {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:
(4.3)dj[m]=∑i=1paidj[m-i]+εj[m],
where εj[m] with a zero mean and variance σεj2, independent of S[m] and BH[m], is AWGN. ai, i=1,2,…,p are parameter coefficients.Define(4.4)Fj=[O(M/2j-1)×1EM/2j-1O1×(M/2j-p)BjT],Bj=[ap,ap-1,…,a1]T,G=[O1×(M/2j-1)1]T,dj,m(M/2j)=[dj[m-M2j+1],…,dj[m-1],dj[m]]T.
For (4.3), we have
(4.5)dj,m+1(M/2j)=Fjdj,m(M/2j)+Gε(m+1).Equation (4.5) is one-step prediction from dj,m(M/2j) to dj,m+1(M/2j), so there M/2j iterative steps of dj,n from dj,n+1, namely, (4.6)dj,n+1=FjM/2jdj,n+GM/2jεn+1,
where dj,n=dj,(n+1)M/2j-1(M/2j), and
(4.7)GM/2j=[Fj(M/2j-1)G,Fj(M/2j-2)G,…,FjG,G],εn+1=[ε((n+1)M2j),…,ε((n+2)M2j-1)]T.What is more, to facilitate the study below, we need to transform formula (2.5) into an observation equation. That is,
(4.8)rj,n=dj,n+Sj,n+Nj,n.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):(4.9)dj,n+1=FjM/2jdj,n+GM/2jεn+1,rj,n=dj,n+uj,n,
where
(4.10)uj,n=Sj,n+Nj,n.To estimate the state space model (4.9) in the third part, the new Kalman filter algorithm is structured as follows.(1)
Initialization: atn = 0, initialize dj,0 and P0 are given.(2)
State update:(4.11)d̂j,n+1-=FjM/2j⋅d̂j,n,Pj,n+1-=Fj,nM/2j⋅Pj,n⋅(Fj,nM/2j)T+σεj2⋅GM/2j(GM/2j)T,Kj,n+1=Pj,n+1-{Pj,n+1-+σuj2(n)E(M/2j)×(M/2j)}†,d̂j,n+1=d̂j,n+1-+Kn+1(rj,n+1-d̂j,n+1-),Pj,n+1=(I-Kj,n+1)Pj,n+1-.Therefore, this algorithm can predict all the high-frequency coefficients of the wavelet on any scale according to the KF iterative algorithm. It means that FBM can be calculated through the inverse wavelet transform to receive the interference suppression in UWB systems.


## 5. Simulation

In this section, the performance of our proposed NBI suppression method is surveyed. It is simulated by the  Matlab7.1 software, and the pulse waveformω(t) from the TH-UWB system is adopted as follows:
(5.1)ω(t)=(1-4π(tτ)2)⋅exp⁡(-2π(tτ)2),
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-Mandelbrot random function is  D=1.5. Therefore, there are some measured results for this simulation in the following portion. In Figure 2, the original fractional noise signal is shown. Figures 3 and 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 these results of Figures 1–3.Figure 2
Fractional noise signal generated by Weierstrass-Mandelbrot function.Figure 3
The estimated fractional signal based on multiscale wavelet transform.Figure 4
Error signal being wideband component of the observation signal.Generally, the signal-to-interference ratio (SINR) can detect the performance of a algorithm very well. In order to preferably evaluate the performance of the new fractional noise suppression algorithm from another angle, SINR is improved in this paper:(5.2)SINRimprovement=SINRoutSINRin=E{|r(n)-S(n)|2}E{|e(n)-S(n)|2}=Var⁡(BH(n))+σN2Var⁡(e(n))+σN2,
where e(n)=BH(n)-B̂H(n), which is the estimated error signal of BH(n). σN2 is the variance of the white noise.Thus, 100 independent tests are done, where the lower sampling rate is 0.1 ns andD=1.2 is employed. From this simulation, it is indicated that the actual SINR has greatly improved in Figure 5. On the other hand, it is shown that this new method is effective in suppressing the nonstationary BH(t) for UWB systems, which is a strong NBI.Figure 5
Comparison of the performance of the ideal and proposed multi-scale KF method.


## 6. Conclusion

In this paper, the multi-scale KF solution is suggested to suppress the fractional noise interference in discrete-time UWB systems. Particularly, after a discrete-time received signal is modeled in the receiver and the fractional signal is transformed based on the wavelet analysis, the fractional signal can be estimated by Kalman filter, which is supposed as a strong nonstationary NBI process. From the simulation above, the algorithm estimates the fractional noise signal under the weak UWB signals by combining the theory of the multi-scale wavelet transform and the Kalman filter. So that it achieves the separation of the narrowband signal and wideband signals successfully. One of the advantages of this paper is not estimating any channel parameter. Simulation results show that this method is an effective narrowband interference suppression method of UWB systems.



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*Source: 356421-2011-11-15.xml*