TIME-FREQUENCY ANALYSIS - INAF - OA-Brerabelloni/ASTROSAT/Home_files/Time-Frequency.pdf · TIME-FREQUENCY ANALYSIS ... COHEN’S KERNEL ALL time-frequency representations come from:

TIME-FREQUENCY ANALYSIST.M. Belloni (INAF-Osservatorio Astronomico di Brera)

Friday, January 27, 12

TIME-FREQUENCY ANALYSIST.M. Belloni (INAF-Osservatorio Astronomico di Brera)


NON-STATIONARY SIGNALS

• A stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space.

• A non-stationary process is a process which is not stationary

•Most signals in real life are non-stationary

•Most analysis methods are for stationary signals


OBVIOUS EXAMPLES

•Musical instruments

• Stock market time series

• Speech and animal sounds

• Sound of gunshots

• Sound of flushing toilets


RELEVANT EXAMPLES

•Drifting Quasi-Periodic Oscillations

• Changing noise components

• X-ray burst oscillations

•Orbital modulation of pulsations


THE UNCERTAINTY PRINCIPLE

• You cannot beat it

• It’s a big limitation

Time

Freq

uenc

y

T 2 = �2t =

Z(t� < t >)2|s(t)|2dt

B2 = �2! =

Z(!� < ! >)2|S(!)|2d!

TB � 1

2

�t

�!

Duration

Bandwidth


THE EASY WAY OUT• Spectrogram (from short-term Fourier Transform)

• Sliding window to select time (window can be chosen)

•Obtain a time-frequency image

st(⌧) = s(⌧)h(⌧ � t)

P (t,!) =

��1p(2⇡)

Re(�i!⌧)s(⌧)h(⌧ � t)d⌧

��2


AN EXAMPLE• Type-B QPO from GX 339-4 T = 128 s






NON-OVERLAPPING• Sliding window to select time st(⌧) = s(⌧)h(⌧ � t)

t = ⌧


NON-OVERLAPPING• Sliding window to select time st(⌧) = s(⌧)h(⌧ � t)

t < ⌧


LINEAR SHIFT AND ADD• Good to recover features at a constant distance in ν


MULTIPLICATIVE SHIFT& ADD•What is you expect something at 2.5 times your ν?

• Two approaches:

• Linear shift, but concentrate only on feature

•Multiplicative shift: technically how?

• Step 1: multiply

• Step 2: sort

• Step 3: rebin (logartithmic rebin helps here)


SOME REAL-WORLD CASES

X-ray bursts


SOME REAL-WORLD CASESkHz QPOs

Méndez et al. 1998

T. Strohmayer


SOME REAL-WORLD CASESTransient features

Belloni et al. 2004

Altamirano et al. 2008


SOME REAL-WORLD CASESAccreting millisecond pulsar Swift J1749.4-2807 (518 Hz)

1 2 3 4 5 6 7 8 9x 104

1035.2

1035.4

1035.6

1035.8

1036

1036.2

1036.4

1036.6

1036.8

Orbital modulation






SOME REAL-WORLD CASESBowhead whale


SOME REAL-WORLD CASES

Human voice


ALTERNATIVE TECHNIQUES• The Wigner Distribution

• Signal in the past by the signal in the future!

• Problem: only for Gaussian chirps W is everywhere positive

• You can beat the uncertainty principle, but at the cost..

• ... of generatin additional monstruosities

W (t,!) = 12⇡

Rs⇤(t� 1

2⌧)s(t+12⌧)e

�i⌧!d⌧


THE WIGNER DITRIBUTION

Signal: sum of two chirps


THE WIGNER DITRIBUTIONComparison with the spectrogram


COHEN’S KERNELALL time-frequency representations come from:

C(t,!) = 14⇡2

R R Rs⇤(u� 1

2⌧)s(u+ 12⌧)�(✓, ⌧)e

�i✓t�i⌧!+i✓udu d⌧ d✓

Where φ(θ,τ) is the kernel

Changing the kernel you change the representation

The properties of the representation depend on the proprties of the kernel



Bibliography

L. Cohen: “Time-frequency Analysis,” Prentice-Hall PTR, 1995

T. Butz: “Fourier Transformation for Pedestrians,” Springer, 2006


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TIME-FREQUENCY ANALYSIS - INAF - OA-Brerabelloni/ASTROSAT/Home_files/Time-Frequency.pdf · TIME-FREQUENCY ANALYSIS ... COHEN’S KERNEL ALL time-frequency representations come from: