0. 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

1. 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

2. 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 g 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.

3. 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.

4. 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.

5. 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

6. 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

7. 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)

8. 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

9. 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

10. 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

11. Thank you for your attention!

12. References
D. Middleton, “Non-Gaussian noise models in signal processing for telecommunications: new methods and results for Class a and Class b noise models,” IEEE Trans. on Info. Theory, vol. 45, no. 4, pp. 1129–1149, 1999.
M. Nassar, K. Gulati, M. DeYoung, B. Evans, and K. Tinsley, “Mitigating near-field interference in laptop embedded wireless transceivers,” Journal of Signal Proc. Sys., pp. 1–12, 2009.
J. Haring, Error Tolerant Communication over the Compound Channel, Shaker-Verlag, Aachen, 2002
J. Haring and A. Vinck, “Iterative decoding of codes over complex numbers for impulsive noise channels,” IEEE Trans. Info. Theory, vol. 49, no. 5, pp. 1251–1260, 2003.
G. Caire, T. Al-Naffouri, and A. Narayanan, “Impulse noise cancellation in OFDM: an application of compressed sensing,” Proc. IEEE Int. Sym. on Info. Theory, 2008, pp. 1293–1297.
D. Wipf and B. Rao, “Sparse Bayesian learning for basis selection,” IEEE Trans. Signal Proc., vol. 52, no. 8, pp. 2153–2164, 2004.