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

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.

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 Section 2, 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 by𝑠_𝑡(𝑡)=^+∞_𝑗=−∞𝑏_{⌊𝑗/𝑁𝑠⌋}𝜔𝑡−𝑗𝑇_𝑓−𝐶_𝑗𝑇_𝑐, (2.1) where 𝜔(𝑡) is a single link transmitted pulse waveform. 𝑇_𝑓 is the pulse repetition period. 𝑇_𝑐 is the TH unit time. 𝑁_𝑠 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 𝑇_𝑏=𝑁_𝑠𝑇_𝑓, the transmitted rate is 𝑅_𝑠=1/(𝑁_𝑠𝑇_𝑓). {𝐶_𝑗} is the pseudorandom TH sequence which adds an additional time shift 𝑇_𝑐 to each pulse, and 0≤𝐶_𝑗≤𝑁_ℎ−1.

Assume 𝑑_𝑛 stands for the 𝑛th symbol of single user. Then, (2.1) can be expressed as 𝑠_𝑡(𝑡)=^𝑁0_𝑛=1𝑏_𝑛𝑣𝑡−𝑛𝑇_𝑏, (2.2) where 𝑣(𝑡)=∑^𝑁𝑠−1_𝑗=0𝜔(𝑡−𝑗𝑇_𝑓−𝐶_𝑗𝑇_𝑐). 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 signal 𝐵_𝐻(𝑡) and additive white Gaussian noise (AWGN) 𝑁(𝑡), the observation signal is given by 𝑟(𝑡)=𝑆(𝑡)+𝐵_𝐻(𝑡)+𝑁(𝑡), (2.3) where 𝑆(𝑡)=𝛼⋅𝑠_𝑡(𝑡). 𝛼 is the channel gain. 𝑁(𝑡)∼𝑁(0,𝜎²_𝑁𝑡). 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/𝑇_𝑠 and an output 𝑟(𝑛)=𝑟(𝑡)|_{𝑡=𝑛𝑇𝑠} is received in discrete-time form. Then, the following received signal becomes 𝑟(𝑛)=𝐵_𝐻(𝑡)+𝑆(𝑛)+𝑁(𝑛), (2.4) where 𝑆(𝑛)=𝛼⋅∑^𝑁0_𝑚=1𝑏_𝑛𝑣(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 𝑟_𝑛=𝐵_𝑛+𝑆_𝑛+𝑁_𝑛, (2.5) where 𝑆_𝑛=𝑋⋅𝜂 and 𝑋=[𝑣(0),𝑣(1),…,𝑣(𝑀−1)]^𝑇. Here 𝜂=𝛼𝑏_𝑛 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 of 𝑟(𝑡) is used to replace the estimation of FBM. On the other hand, the error signal 𝑟(𝑡)−𝑟(̂𝑡), where 𝑟(̂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 𝐵_𝐻(𝑡)=1Γ()⋅𝐻+1/2^𝑡₀(𝑡−𝜆)^𝐻−1/2𝑑𝐵(𝜆)+⁰_−∞(𝑡−𝜆)^𝐻−1/2−(−𝜆)^𝐻−1/2𝑑𝐵(𝜆), (3.1) 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 of 𝐵_𝐻(𝑡). Then, (3.1) can be expressed in the following form: 𝐵_𝐻(𝑡)=^+∞_𝑛=−∞𝑐_𝐽[𝑛]Φ_𝐽,𝑛(𝑡)+^+∞_𝑗=−∞^+∞_𝑛=−∞𝑑_𝑗[𝑛]Ψ_𝑗,𝑛(𝑡), (4.1) where 𝑎_𝐽[𝑛]=2^−𝐽/2∫^∞_−∞𝐼(𝑡)Φ(2^−𝐽𝑡−𝑛)𝑑𝑡, 𝑑_𝑗[𝑛]=2^−𝑗/2∫^∞_−∞𝐼(𝑡)Φ(2^−𝑗𝑡−𝑛)𝑑𝑡. Φ_𝐽,𝑛(𝑡)=2^−𝐽/2Φ(2^−𝐽𝑡−𝑛) is a scale function of the wavelet transform, and the wavelet function is written by Ψ_𝐽,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: 𝑟_𝑗[𝑛]=𝑆_𝑗[𝑛]+𝑑_𝑗[𝑛]+𝑁_𝑗[𝑛]=𝑑_𝑗[𝑛]+𝑢_𝑗[𝑛], (4.2) where 𝑗=1,2,…,𝐽. 𝑢_𝑗[𝑛]=𝑆_𝑗[𝑛]+𝑁_𝑗[𝑛] has the mean 𝜇_𝑗=𝑆_𝑗[𝑛] and variance Var(𝑢_𝑗[𝑛])=𝜎²_𝑢𝑗. {𝑑_𝑗[𝑛]} is stable [22]. Then, due to the stationarity of wavelet coefficients {𝑑_𝑗[𝑛]}, this AR(p) model can be found in this study: 𝑑_𝑗[𝑚]=^𝑝_𝑖=1𝑎_𝑖𝑑_𝑗[𝑚−𝑖]+𝜀_𝑗[𝑚], (4.3) where 𝜀_𝑗[𝑚] with a zero mean and variance 𝜎²_𝜀𝑗, independent of 𝑆[𝑚] and 𝐵_𝐻[𝑚], is AWGN. 𝑎_𝑖, 𝑖=1,2,…,𝑝 are parameter coefficients.

Define 𝐹_𝑗=𝑂_(𝑀/2^𝑗_−1)×1𝐸_𝑀/2^𝑗₋₁𝑂_1×(𝑀/2^𝑗_−𝑝)𝐵^𝑇_𝑗,𝐵_𝑗=𝑎_𝑝,𝑎_𝑝−1,…,𝑎₁]^𝑇,𝐺=𝑂_1×(𝑀/2^𝑗₋₁₎1^𝑇,𝑑^(𝑀/2𝑗)_𝑗,𝑚=𝑑_𝑗𝑀𝑚−2^𝑗+1,…,𝑑_𝑗[𝑚−1],𝑑_𝑗[𝑚]^𝑇. (4.4) For (4.3), we have 𝑑^(𝑀/2𝑗)_𝑗,𝑚+1=𝐹_𝑗𝑑^(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/2^𝑗 iterative steps of 𝑑_𝑗,𝑛 from 𝑑_𝑗,𝑛+1, namely, 𝑑_𝑗,𝑛+1=𝐹^𝑀/2𝑗_𝑗𝑑_𝑗,𝑛+𝐺_𝑀/2^𝑗𝜀_𝑛+1, (4.6) where 𝑑_𝑗,𝑛=𝑑^(𝑀/2𝑗)_{𝑗,(𝑛+1)𝑀/2𝑗−1}, and 𝐺_𝑀/2^𝑗=𝐹^(𝑀/2𝑗_𝑗⁻¹⁾𝐺,𝐹^(𝑀/2𝑗_𝑗⁻²⁾𝐺,…,𝐹_𝑗,𝜀𝐺,𝐺_𝑛+1=𝜀(𝑛+1)𝑀2^𝑗()𝑀,…,𝜀𝑛+22^𝑗−1^𝑇. (4.7)

What is more, to facilitate the study below, we need to transform formula (2.5) into an observation equation. That is, 𝑟_𝑗,𝑛=𝑑_𝑗,𝑛+𝑆_𝑗,𝑛+𝑁_𝑗,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 𝐵_𝐻(𝑡):𝑑_𝑗,𝑛+1=𝐹^𝑀/2𝑗_𝑗𝑑_𝑗,𝑛+𝐺_𝑀/2^𝑗𝜀_𝑛+1,𝑟_𝑗,𝑛=𝑑_𝑗,𝑛+𝑢_𝑗,𝑛, (4.9) where 𝑢_𝑗,𝑛=𝑆_𝑗,𝑛+𝑁_𝑗,𝑛. (4.10)

To estimate the state space model (4.9) in the third part, the new Kalman filter algorithm is structured as follows.(1)Initialization: at n = 0, initialize 𝑑_𝑗,0 and 𝑃₀ are given.(2)State update: 𝑑⁻_𝑗,𝑛+1=𝐹^𝑀/2𝑗_𝑗⋅𝑑_𝑗,𝑛,𝑃⁻_𝑗,𝑛+1=𝐹^𝑀/2𝑗_𝑗,𝑛⋅𝑃_𝑗,𝑛⋅𝐹^𝑀/2𝑗_𝑗,𝑛^𝑇+𝜎²_𝜀𝑗⋅𝐺_𝑀/2^𝑗𝐺_𝑀/2^𝑗𝑇,𝐾_𝑗,𝑛+1=𝑃⁻_𝑗,𝑛+1𝑃⁻_𝑗,𝑛+1+𝜎²_𝑢𝑗(𝑛)𝐸_(𝑀/2^𝑗_)×(𝑀/2^𝑗₎^†,𝑑_𝑗,𝑛+1=𝑑⁻_𝑗,𝑛+1+𝐾_𝑛+1𝑟_𝑗,𝑛+1−𝑑⁻_𝑗,𝑛+1,𝑃_𝑗,𝑛+1=𝐼−𝐾_𝑗,𝑛+1𝑃⁻_𝑗,𝑛+1. (4.11)

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 waveform 𝜔(𝑡) from the TH-UWB system is adopted as follows: 𝜔(𝑡)=𝑡1−4𝜋𝜏²𝑡⋅exp−2𝜋𝜏², (5.1) where 𝜏 denotes the pulse forming factor. Suppose the parameters are 𝑇_𝑓=8 ns, 𝑁_𝑠=2, 𝑁_ℎ=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:SINR_improvement=SINR_outSINR_in=𝐸|𝑟(𝑛)−𝑆(𝑛)|²𝐸|𝑒(𝑛)−𝑆(𝑛)|²=Var𝐵_𝐻(𝑛)+𝜎²_𝑁Var(𝑒(𝑛))+𝜎²_𝑁, (5.2) where 𝑒(𝑛)=𝐵_𝐻(𝑛)−𝐵_𝐻(𝑛), which is the estimated error signal of 𝐵_𝐻(𝑛). 𝜎²_𝑁 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 greatly improved in Figure 5. On the other hand, it is shown that this new method is effective in suppressing the nonstationary 𝐵_𝐻(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.

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).