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Radial Kernel based Time-Frequency Distributions

with Applications to Atrial Fibrillation Analysis

Sandun KodituwakkuPhD Student

The Australian National UniversityCanberra, Australia.

Supervisors: A/Prof. Thushara Abhayapala

Prof. Rod Kennedy

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Outline

• Background – Time-Frequency Distributions (TFDs)

• Our work1) Multi-D Fourier Transform based framework for TFD kernel design2) Unified kernel formula for generalizing Wigner-Ville, Margenau-Hill, Born-Jordan and Bessel3) Applications to Atrial Fibrillation

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Motivation• Real world signals -- speech, radar, biological

etc. -- are non-stationary in nature.• Example: ECG Video• Non-stationary – Period, Amplitudes,

Morphology changes in time.• Limitations of Fourier Analysis – fails to locate

the time dependency of the spectrum.• This motivates joint Time-Frequency

representation of a signal.

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Historical background

• TFDs are a research topic for more than half a century

• Famous two

1. Short-time Fourier Transform

2. Wigner-Ville Distribution

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Classification

Time-Frequency Distributions

(TFDs)

Linear

•STFT•Wavelets

•Gabor

Quadratic

•“Cohen class”(Shift invariant)

•Affine class(Scale invariant)

Others

•Signal Dependent

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Linear vs. QuadraticLinear

Pros:• Linear superposition• No interference terms for

muti-component signals

Cons:• Trade off between time

and frequency resolutions

Heisenberg inequality

Quadratic

Pros:• Better time and frequency

resolutions than linear• Shows the energy

distribution

Cons:• Cross terms for multi-

component signals

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Cohen Generalization

• Breakthrough by L. Cohen in 1966

• All shift invariant TFDs are generalized to a one class (Cohen class)

• Kernel function uniquely specifies a distribution

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Prominent members of Cohen

• Wigner-Ville (1948)

• Page (1952)

• Margenau-Hill (1961)

• Spectrogram – Mod squared of STFT

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Prominent members of Cohen (cont.)

• Born-Jordan (1966)

• Choi-Williams (1989)

• Bessel (1994)

2-D time-frequency convolution of Wigner-Ville will result others

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Kernel Questions?

• Why so many?

• Which one is the best?

• How to generate them?

• What are the applications?

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Our work

• Multi-D Fourier Transform based framework for deriving Cohen kernels.

• Radial-δ kernel class generalizing Wigner-Ville, Margenau-Hill, Born-Jordan, and Bessel.

• Analysis of Atrial Fibrillation from surface ECG.

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Multi-D Fourier Framework

Let be a vector in n-D and f be a scalar-valued multivariate function satisfying following conditions.

C1: ie. Radially symmetric

C2: ie. Unit volume

C3: ie. Finite support

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Multi-D Fourier Framework (cont.)

• Consider n-D Fourier Transform of

• is radially symmetric as well.

Identify by to obtain the order-n radial kernel.

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Realization based on δ function

• n-D radial δ function:

• It is radially symmetric (C1)

• It is normalised to give unit volume (C2)

• It has finite support for α ≤ ½ (C3)

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Realization based on δ function (cont.)

• n-D Fourier transform of

• Thus order-n radial-δ kernel is given by,

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Lower dimensions simplified

Dimension n Kernel Name

1 , 1 Wigner-Ville

1 , Margenau-Hill

2 Our work

3 Born-Jordan

4 Bessel

5 Our work

6 Our work

and many more…..

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Kernel visualization

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TFD Properties

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TFD Properties (cont.)

• Realness

guaranteed by radial symmetry of

• Time and Frequency Shifting

guaranteed by independence of from t and ω

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TFD Properties (cont.)

• Time and Frequency marginals

guaranteed by unit volume condition

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TFD Properties (cont.)

• Instantaneous frequency and Group delay

guaranteed by radial symmetry of and unit volume condition together

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TFD Properties (cont.)

• Time and Frequency support

guaranteed by finite support condition

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Simulation of FM + Chirp signals

• Time-frequency analysis of the sum of FM and chirp signal.

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Simulation of FM + Chirp signals (cont.)

Born-Jordan Bessel

Order-5 Radial Order-6 Radial

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Simulation of FM + Chirp signals (cont.)

Order-7 Radial

Order-5 radial-δ kernel works best.

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Summary so far……..

• A unified kernel formula

which contains 4 of the famous kernels (Wigner-Ville, Margenau-Hill, Born-Jordan and Bessel).

• Formula derived from n-dimensional FT of a radially symmetric δ function.

• Superiority of high order radial-δ kernels.

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An application of novel TFDs

Atrial Fibrillation Analysis from surface ECG

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What is ECG?

• ECG – Electrocardiogram

• ECG is a time signal which shows the changes in body surface potentials due to the electrical activity of the heart.

• Gold standard for diagnosing cardiovascular disorders.

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Typical healthy ECG

Source: Wikipedia

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What is AF?

• AF – Atrial Fibrillation• Cardiac arrhythmia condition• Consistent P waves are replaced by rapid

oscillations.• Fibrillatory waves vary in amplitude, frequency

and shape.• Associates with an irregular ventricular

response.

healthy

AF

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Why AF important?

• AF is the most common sustained cardiac arrhythmia condition.

• Increases in prevalence with age.• Affects approx. 8% of the population over

age of 80.• Accounts for 1/3 of hospitalizations for

cardiac rhythm disturbances.• Associated with an increased risk of

stroke.

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Motivation

• Spectrum of Atrial activity of ECG under AF has a dominant peak (AF frequency ).

• AF frequency gives insight to spontaneous or drug induced termination of AF.

• Thus, importance of accurately tracking AF frequency in time.

• TFDs are a good tool for this task.

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Previous work• Stridh[01] used STFT and cross Wigner-

Ville distributions for estimating the AF frequency.

• Sandberg[08] used HMM based method for AF frequency tracking.

• We obtained better results using higher order radial-δ kernels.

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System model

• Atrial fibrillation is modelled by a sum of frequency modulated sinusoidals with time varying amplitudes, and its harmonics [Stridh & Sornmo 01]

where,

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Synthetic ECG with AF

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Objective

• AF frequency given by,

• Accurately estimate , especially when is higher compared to .

• Approximation to the real AF.

• Can be used to compare performance of different algorithms.

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Born-Jordan Bessel

Order-5 radial Order-6 radial

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Simulation Results (cont.)Order-7 radial

Order-6 radial-δ kernel works best.

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Performance measure

• Maximise ratio between auto term energy and interference term energy.

• Find the order (n) with maximum ratio

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Performance measure (cont.)

• Best results for the AF model obtained by order-6 radial-δ kernel

0 1 2 3 4 5 6 7 86.5

6.55

6.6

6.65

6.7

6.75

Kernel Order

(Aut

o te

rm)/

(Cro

ss te

rm)

dB

Bessel

Born-Jordan

Wigner-Ville

Margenau-Hill

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Comparison with Choi-Williams

• Less interference in order-6 radial-δ kernel.• Choi-Williams does not satisfy time and

frequency support properties.

Frequency (Hz)

Tim

e (s

)

Choi-Williams

0 2 4 6 8 10 12 14 16 18 20

5

10

15

20

25

30

35

40

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6th Order Radial

Frequency (Hz)

Tim

e (s

)

0 2 4 6 8 10 12 14 16 18 20

5

10

15

20

25

30

35

40

45

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PhysioBank data• AF termination challenge database- ECG

record n02

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Future directions

• Parameterizing TFD for paroxysmal and persistent AF conditions.

• Pharmacological therapy and DC cardioversion influence on TFD.

• Generalization for other supraventricular tachyarrhythmias – Atrial Flutter.

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Summary

• A unified kernel formula for Cohen class of TFDs

based on n-dimensional Fourier Transform of a radially symmetric δ function.

• Atrial Fibrillation cardiac arrhythmia condition analysis using TFDs with higher order radial-δ kernels.