Blind Interference Reduction in Wigner Distributions using the Fractional Fourier Transform

1
Blind Interference Reduction in Wigner
Distributions using the Fractional Fourier
Transform

• S. Qazi L.K. Stergioulas
• School of Information Systems, Computing and
  Mathematics, Brunel University, UK

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Overview

• PART I
• Introduction Background to TF methods and WDF
• PART II
• The Idea the generic method
• The proposed step-by-step method for Interference
  Reduction using fracFT
• PART III
• Application and results
• Discussion

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Introduction

• Time-frequency (TF) analysis a powerful signal
  processing method
• TF representations (TFRs) are usually classified
  into linear and quadratic methods
• Quadratic TFRs are preferable (for analysis)
• Approximate TF energy density functions
• Increased TF concentration

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Cohens class distributions

• All bilinear T-F representations that satisfy
  the time and frequency shift invariance belong to
  a general class of distributions introduced by L.
  Cohen (1989)
• where t, u, and ? are time and frequency lags
• F(?, t) is a 2-D kernel function, which
  determines the specific distribution and its
  properties.

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The Wigner distribution (WD)

• A prominent member of Cohens class of quadratic
  TFRs
• Satisfies a large number of desirable
  mathematical properties
• Exhibits excellent concentration in the TF plane
• Straightforward Calculation.
• Highest achievable resolution.
• Resolution related to the intrinsic resolution of
  the signal.
• However, the WD possesses spurious components
  (cross-terms)
• Cross-terms reduce the readability of the WD in
  practical applications

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Definition of the WDF
or

• Advantages
• dynamics of change of the frequency components in
  time
• instantaneous signal energy and power spectrum
• instantaneous frequency (average frequency over
  time)
• group delay (average time over frequency)
• Disadvantages
• negative values
• computationally expensive
• Interference

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Important properties of the WDF
Marginals
Total energy
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Reducing the interference

• Considerable amount of research has been carried
  out to develop modified WDs
• Current methods are based on smoothing the WD
• Direct smoothing in the TF plane
• Filtering in the Ambiguity domain
• However, smoothing in the WD results in a
  trade-off between cross-term reduction and
  auto-term concentration

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This work

• We present a simple method to eliminate
  cross-terms
• Our approach is based on two steps
• Cross-term detection
• Cross-term suppression
• The method does not require any prior knowledge
  of the signal (blind)

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Preliminaries

• The fracFT of s(t) is
• It produces a rotation of s(t) in T-F. The
• T-F map of a fracFT-rotated signal shows
  significant variation of amplitude only in the
  cross-term regions. This observation led to the
  following algorithm

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T-F maps of fracFTed signals
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The algorithm (1-4)

• fracFT signal at various angles
• Step-wise computation of the WDs of the fracFTed
  versions (Standard WD, Pseudo WD or RID Hanning)
• Alignment of the WD maps back to original T-F
  axes
• .Variance calculation across T-F maps

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The algorithm (5-6)

• 5. Thresholding (soft or hard) of variance maps
• 6. The resulting mask M(t,f) yields the
  Cross-Term Free Distribution (CFD), which is the
  desired outcome

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APPLICATION EXAMPLES

• Two Gaussians
• Four Gaussians
• Challenge
• From WDF to derive a CFD, which is as close as
  possible to ITFD (Ideal T-F Distribution)
• FracFT rotation ? Engineering flaw to an Advantage

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Two Gaussians
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Four Gaussians
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Conclusions

• A novel method to detect cross terms in the
  Wigner distribution using the FracFT.
• The algorithmic steps of the method have been
  presented, along with two tutorial examples
• The proposed algorithm
• is consistent (good SNR)
• does not distort the auto-terms
• Yields a CFD close to desired ITFD.
• The proposed method increases the readability of
  the WD.