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→Scale factor: Included GIF of smoothly adjusted scale factor (in Morlet wavelet) to better convey properties of the CWT. |
→Scale factor: Fixed inconsistency (changed "scale factor" to "\sigma parameter") and added context (interpolation between time-series and Fourier transform). |
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[[File:Continuous wavelet transform.gif|thumb|300px|right]]
The scale factor <math>a</math> either dilates or compresses a signal. When the scale factor is relatively low, the signal is more contracted which in turn results in a more detailed resulting graph. However, the drawback is that low scale factor does not last for the entire duration of the signal. On the other hand, when the scale factor is high, the signal is stretched out which means that the resulting graph will be presented in less detail. Nevertheless, it usually lasts the entire duration of the signal.
[[File:Wavelet scale sweep for FM signal.gif|thumb|300px|Visualizing the effect of changing a wavelet's scale factor. Here, a [[Frequency modulation|frequency-modulated]] tone (plus noise) is analyzed with a [[Morlet wavelet]]; the scale is adjusted from <math>a=1</math> to <math>a=200</math> in steps of unity.]]▼
==Continuous wavelet transform properties==
In definition, the continuous wavelet transform is a [[convolution]] of the input data sequence with a set of functions generated by the mother wavelet. The convolution can be computed by using a [[fast Fourier transform]] (FFT) algorithm. Normally, the output <math>X_w(a,b)</math> is a real valued function except when the mother wavelet is complex. A complex mother wavelet will convert the continuous wavelet transform to a complex valued function. The power spectrum of the continuous wavelet transform can be represented by <math>|X_w(a,b)|^2</math> .
▲[[File:Wavelet scale sweep for FM signal.gif|thumb|300px|Visualizing the effect of changing a [[Morlet wavelet|Morlet wavelet's]]
==Applications of the wavelet transform==
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