Continuous Wavelet Transformm10418Continuous Wavelet Transform2.142001/12/302009/05/31 18:11:21.954 GMT-5PhilSchniterPhil Schniterschniter@ee.eng.ohio-state.eduPhilSchniterPhil Schniterschniter@ee.eng.ohio-state.eduElizabethChanElizabeth Chanlizychan@rice.eduPhilSchniterPhil Schniterschniter@ee.eng.ohio-state.eduCWTtime-frequency analysisuncertainty principleMathematics and Statistics Science and TechnologyThis module introduces continuous wavelet transform.en
The STFT provided a means of (joint) time-frequency analysis
with the property that spectral/temporal widths (or resolutions)
were the same for all basis elements. Let's now take a closer
look at the implications of uniform resolution.
Consider two signals composed of sinusoids with frequency 1 Hz
and 1.001 Hz, respectively. It may be difficult to distinguish
between these two signals in the presence of background noise
unless many cycles are observed, implying the need for a
many-second observation. Now consider two signals with pure
frequencies of 1000 Hz and 1001 Hz-again, a 0.1%
difference. Here it should be possible to distinguish the two
signals in an interval of much less than one second. In other
words, good frequency resolution requires longer observation
times as frequency decreases. Thus, it might be more convenient
to construct a basis whose elements have larger temporal width
at low frequencies.
The previous example motivates a multi-resolution time-frequency
tiling of the form ():
The Continuous Wavelet Transform (CWT) accomplishes the above
multi-resolution tiling by time-scaling and time-shifting a
prototype function
ψt, often called the mother wavelet. The
a-scaled and
τ-shifted basis elements is
given by
ψa,τt1aψtτa
where
aτtψt0CψΩψΩ2Ω
The conditions above imply that
ψt is bandpass and sufficiently smooth. Assuming that
ψt1, the definition above ensures that
ψa,τt1 for all a and
τ. The CWT is then defined by
the transform pair
XCWTaτtxtψa,τtxt1CψaτXCWTaτψa,τta2
In basis terms, the CWT says that a waveform can be decomposed
into a collection of shifted and stretched versions of the
mother wavelet
ψt. As such, it is usually said that wavelets perform a
"time-scale" analysis rather than a time-frequency analysis.
The Morlet wavelet is a classic example of the
CWT. It employs a windowed complex exponential as the mother
wavelet:
ψt12Ω0tt22ΨΩΩΩ022
where it is typical to select
Ω022. (See illustration.) While this wavelet does not
exactly satisfy the conditions established earlier, since
Ψ07-70, it can be corrected, though in practice the
correction is negligible and usually ignored.
While the CWT discussed above is an interesting theoretical and
pedagogical tool, the discrete wavelet transform (DWT) is much
more practical. Before shifting our focus to the DWT, we take a
step back and review some of the basic concepts from the branch
of mathematics known as Hilbert Space theory (Vector Space, Normed Vector Space, Inner Product Space, Hilbert Space, Projection Theorem). These
concepts will be essential in our development of the DWT.