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Signal AnalysisOne of the goals of signal analysis is determining how well the data matches a given function or model.
Fourier AnalysisGiven certain assumptions, one can decompose a ( -periodic) signal in terms of complex exponentials (e.g., sines and cosines). 1-D Fourier series:
( ) i tf t c e ωω
ω
∞
=−∞
= ∑2
0
1 ( )2
i tc f t e dtπ
ωω π
−= ∫
2π
Fourier Analysis, cont’dFourier analysis allows for the study of the frequency characteristics of the data.A key tool for doing that is the power spectra:
2( ) cωωΦ =
Fourier Analysis, cont’dAnother tool in Fourier analysis is the Fourier transform:
( ) ( ) i tf f t e dtωω+∞
−∞
= ∫
Problems with Fourier AnalysisThe signal is assumed to be periodic on the given interval – which is rarely true. Other basis functions can also be used to expand the signal – and these are often chosen to reflect expected characteristics of the data.
Problems with Fourier Analysis, cont’d
A key drawback to these approaches is that the signal is decomposed in terms of a single parameter, the frequency. So, for a non-stationary signal, any temporal information is lost (smeared).
Problems with Fourier Analysis, cont’dTo accommodate non-stationarity, one could use short-time windowed Fourier transforms.But this results in an equal partition in time/freq.
Frequency
Time
Finally…Wavelets!We want fairly general, local functions that can be used to analyze signals.But we also want functions that are scalable, so that they have simultaneous frequency-resolving and location-resolving capabilities.
“Scale”
Time
Wavelets: DefinitionSimilar to the Fourier series, we have a wavelet series and wavelet transform:
Wavelet Series:
, ,,
( ) ( )j k j kj k
f t c tψ+∞
=−∞
= ∑ , ( )(2 ,2 )j jj kc Wf k− −=
1/ 2 *( )( , ) ( ) ( )t bWf a b a f t dta
ψ+∞
−∞
−= ∫Wavelet Transform:
Wavelet RequirementsIn order to reconstruct the signal from it’s transform without loss of information, it must be admissible. This fact also implies that
( ) 0t dtψ+∞
−∞
=∫
This implies that the function must be oscillatory,i.e., wave-like.
ψ
Wavelet Requirements, cont’dThe other condition imposed on wavelets is regularity: the wavelet function should have some smoothness and concentration in both time and frequency domains.So, we have a localized, oscillatory function: A Wavelet!
Wavelet Application, cont’d
Raw Data -Single Radial
Wavelet-Filtered“Plume Region”
Wavelet-Filtered“Noise Region”