Summary: This module introduces continuous wavelet transform.

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 (Figure 1):

The Continuous Wavelet Transform (CWT) accomplishes the above
multi-resolution tiling by time-scaling and time-shifting a
prototype function

The Morlet wavelet is a classic example of the
CWT. It employs a windowed complex exponential as the mother
wavelet:

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.