Impulsive noise mitigation in OFDM systems using sparse bayesian learning Jing Lin, Marcel Nassar and Brian L. Evans Department of Electrical and Computer Engineering The University of Texas at Austin

Impulsive Noise at Wireless Receivers Background | Prior Work | System Model | Estimation Algorithms | Simulation Results Impulsive Noise at Wireless Receivers Antennas Non-Communication Sources Electromagnetic radiations Wireless Communication Sources Uncoordinated transmissions Today’s wireless receivers are exposed to a variety of interference sources. Uncoordinated transmissions in the wireless network causes co-channel interference. Non-communication resources such as power lines and microwave oven are also generating radio frequency interference in the wireless transmission band. Also there is in-platform interference from clocks, buses and processors, etc. These interferences are generally impulsive in time domain, and therefore cannot be modeled by AWGN. Baseband Processor Computational Platform Clocks, busses, processors Other embedded transceivers

In-Platform Interference Background | Prior Work | System Model | Estimation Algorithms | Simulation Results In-Platform Interference May severely degrade communication performance Impact of LCD noise on throughput for IEEE 802.11g embedded wireless receiver ([Shi2006]) As a result, traditional receivers, which are designed under the assumption of AWGN, may encounter severe degradation in communication performance. For example, the presence of LCD noise significantly decreases the throughput of an embedded wireless receiver working on particular channels.

Noise Trace for Platform Noise Background | Prior Work | System Model | Estimation Algorithms | Simulation Results Noise Trace for Platform Noise The impact of impulsive noise is also seen in low-voltage powerline communications. The switching transients of home appliances could introduce random impulsive noise with the power of 20-40dB above the background noise.

OFDM System in Impulsive Noise Background | Prior Work | System Model | Estimation Algorithms | Simulation Results OFDM System in Impulsive Noise FFT spreads out impulsive energy across all tones SNR of each tone is decreased Receiver performance degrades Noise in each tone is asymptotically Gaussian (as 𝑁 𝐷𝐹𝑇 →∞) In this work, we are going to look at the effect of impulsive noise in OFDM systems. At the OFDM receiver, taking the discrete Fourier transform of the received signal spreads out the impulsive energy in time domain over all subcarriers (or tones). As the DFT size increases, noise in each tone is asymptotically Gaussian by CLT. However, the impulsive noise may significantly decreases the SNR in each tone, causing dramatic performance degradation.

Prior Work Parametric vs. non-parametric methods (Noise Statistics) Background | Prior Work | System Model | Estimation Algorithms | Simulation Results Prior Work Parametric vs. non-parametric methods (Noise Statistics) Impulsive noise mitigation in OFDM * Semi-non-parametric since threshold tuning is needed ** MMSE: Minimum mean squared error; LS: least squares Param. Non-Param. Assume parameterized noise statistics Yes No Performance degradation due to model mismatch Training needed Parametric? Technique Optimality Complexity [Nassar09] Yes Pre-filtering [Haring02] MMSE** estimate [Haring03] Iterative decoder [Caire08] No* Compressed sensing & LS** Various parametric and non-parametric methods have been proposed in the literature to mitigate impulsive noise in OFDM systems. Parametric methods assume a parameterized form of noise statistics and estimates the parameters through training. Not only does the training stage introduce extra overhead, the performance of parametric methods can suffer due to model mismatch, or when the noise statistics is fast varying. On the other hand, non-parametric methods

𝑦=𝐹𝑒+𝐹𝐻 𝐹 ∗ 𝑥+𝐹𝑛=𝐹𝑒+𝑣, 𝑣~𝐶𝑁(𝛬𝑥, 𝜎 2 𝐼) Background | Prior Work | System Model | Estimation Algorithms | Simulation Results System Model A linear system with Gaussian disturbance 𝑦=𝐹𝑒+𝐹𝐻 𝐹 ∗ 𝑥+𝐹𝑛=𝐹𝑒+𝑣, 𝑣~𝐶𝑁(𝛬𝑥, 𝜎 2 𝐼) Estimate the impulsive noise and remove it from the received signal 𝑦 =𝑦−𝐹 𝑒 =𝛬𝑥+𝑔+𝐹( 𝑒 −𝑒 𝑒 ≈𝑒 𝛬𝑥+𝑔 Apply standard OFDM decoder as if only Gaussian noise were present Goal: Non-parametric impulsive noise estimator 𝛬 𝑔 𝑣 Let’s first take a look at the system model of the OFDM receiver. The received signal after taking the DFT can be expressed as the summation of three terms: the transmitted signal shaped by the channel frequency response, the impulsive noise in frequency domain, and the AWGN in the frequency domain, which is also AWGN. Combining the signal with the Gaussian noise into a single term v, we have a linear system y=Fe+v, where y is the observation, F is the system matrix, e is the input, and v is the disturbance. Notice that conditioned on \Gamma x, the disturbance v is Gaussian distributed. Given the observed signal y, if we could obtain an estimate of e by linear regression, we can then remove it from the received signal to get

Estimation Using Null Tones Background | Prior Work | System Model | Estimation Algorithms | Simulation Results Estimation Using Null Tones Underdetermined linear regression 𝐹 𝐽 : over-complete dictionary e: sparse weight vector 𝑔~𝐶𝑁(0, 𝜎 2 𝐼 with 𝜎 2 unknown Sparse Bayesian learning (SBL) Prior: 𝑒|𝛾 ~ 𝐶𝛮 0,𝛤 ,𝛤≜diag 𝛾 Likelihood: 𝑦 𝐽 |𝛾, 𝜎 2 ~ 𝐶𝑁(0,𝐹𝛤 𝐹 ∗ + 𝜎 2 𝐼) Posterior: 𝑒| 𝑦 𝐽 ;𝛾, 𝜎 2 ~ 𝐶𝑁(𝜇, 𝛴 𝑒 ) Step 1: Maximum likelihood estimate of hyper-parameters: 𝛾 , 𝜎 2 = argmax 𝛾, 𝜎 2 𝑝 𝑦𝐽;𝛾, 𝜎 2 Treat e as latent variables and solve by expectation maximization (EM) 𝛾 and 𝜎 2 are inter-dependent and updated iteratively Step 2: Estimate e from posterior mean: 𝑒 =𝐸 𝑒 𝑦 𝐽 ; 𝛾 , 𝜎 2 = 𝜇 Guaranteed to converge to a sparse solution. 𝐽: Index set of null tones ( 𝒙 𝐽 =𝟎)

Estimation Using All Tones Background | Prior Work | System Model | Estimation Algorithms | Simulation Results Estimation Using All Tones Joint estimation of data and noise 𝑦=𝐹𝑒+𝑣 𝑣~𝐶𝑁(𝛬𝑥, 𝜎 2 𝐼) Similar SBL approach with additional hyper-parameters of the data Step 1 involves ML optimization over 3 sets of hyper-parameters: 𝛾, 𝜎 2 , 𝛬𝑥 𝐽 𝑥 𝐽 is relaxed to be continuous variables to insure a tractable M-step Estimate of 𝛬𝑥 𝐽 is sent to standard OFDM channel equalizer and MAP detector Increase complexity from O(N2M) to O(N3) per EM iteration 𝑧≜𝛬𝑥 𝐽 : Index set of data tones

Background | Prior Work | System Model | Estimation Algorithms | Simulation Results Symbol error rate (SER) performance in different noise scenarios ~7dB ~4dB Symmetric alpha stable model ~6dB ~8dB ~4dB ~10dB Middleton’s Class A model Gaussian mixture model

Background | Prior Work | System Model | Estimation Algorithms | Simulation Results Performance of the first algorithm vs. the number of null tones SNR = 0dB 256 tones Middleton Class A noise In both algorithms, EM converges after a few iterations

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