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or
or
where for [link]
The corresponding relationships for the wavelet coefficients are
or
where
These are very important properties of the DWT of a periodic signal, especially one artificially constructed from a nonperiodic signal in orderto use a block algorithm. They explain not only the aliasing effects of having a periodic signal but how to calculate the DWT of a periodicsignal.
If $f\left(t\right)$ is essentially infinite in length, then the DWT can be calculated as an ongoing or continuous process in time. In other words,as samples of $f\left(t\right)$ come in at a high enough rate to be considered equal to ${c}_{J1}\left(k\right)$ , scaling function and wavelet coefficients at lower resolutions continuously come out of the filter bank. This is best seenfrom the simple two-stage analysis filter bank in Section: Three-Stage Two-Band Analysis Tree . If samples come in at what is called scale ${J}_{1}=5$ , wavelet coefficients at scale $j=4$ come out the lower bank at half the input rate. Wavelet coefficients at $j=3$ come out the next lower bank at one quarter the input rate and scaling function coefficients at $j=3$ come out the upper bank also at one quarter the input rate. It is easy to imagine morestages giving lower resolution wavelet coefficients at a lower and lower rate depending on the number of stages. The last one will always be thescaling function coefficients at the lowest rate.
For a continuous process, the number of stages and, therefore, the level of resolution at the coarsest scale is arbitrary. It is chosen to be thenature of the slowest features of the signals being processed. It is important to remember that the lower resolution scales correspond to aslower sampling rate and a larger translation step in the expansion terms at that scale. This is why the wavelet analysis system gives good timelocalization (but poor frequency localization) at high resolution scales and good frequency localization (but poor time localization) at low orcoarse scales.
For finite length signals or block wavelet processing, the input samples can be considered as a finite dimensional input vector, the DWT as asquare matrix, and the wavelet expansion coefficients as an output vector. The conventional organization of the output of the DWT places the outputof the first wavelet filter bank in the lower half of the output vector. The output of the next wavelet filter bank is put just above that block.If the length of the signal is two to a power, the wavelet decomposition can be carried until there is just one wavelet coefficient and one scalingfunction coefficient. That scale corresponds to the translation step size being the length of the signal. Remember that the decomposition does nothave to carried to that level. It can be stopped at any scale and is still considered a DWT, and it can be inverted using the appropriatesynthesis filter bank (or a matrix inverse).
The one-sided tree structure of Mallet's algorithm generates the basic DWT. From the filter bank in Section: Three-Stage Two-Band Analysis Tree , one can imagine putting a pair of filters and downsamplers at the output of the lower wavelet bankjust as is done on the output of the upper scaling function bank. This can be continued to any level to create a balanced tree filter bank. Theresulting outputs are “wavelet packets" and are an alternative to the regular wavelet decomposition. Indeed, this “growing" of the filter banktree is usually done adaptively using some criterion at each node to decide whether to add another branch or not.
Still another generalization of the basic wavelet system can be created by using a scale factor other than two. The multiplicity-M scaling equationis
and the resulting filter bank tree structure has one scaling function branch and $M-1$ wavelet branches at each stage with each followed by a downsampler by $M$ . The resulting structure is called an $M$ -band filter bank, and it too is an alternative to the regular wavelet decomposition. This is developed in Section: Multiplicity-M (M-band) Scaling Functions and Wavelets .
In many applications, it is the continuous wavelet transform (CWT) that is wanted. This can be calculated by using numerical integration to evaluatethe inner products in [link] and [link] but that is very slow. An alternative is to use the DWT to approximate samples of the CWT much asthe DFT can be used to approximate the Fourier series or integral [link] , [link] , [link] .
As you can see from this discussion, the ideas behind wavelet analysis and synthesis are basically the same as those behind filter bank theory.Indeed, filter banks can be used calculate discrete wavelet transforms using Mallat's algorithm, and certain modifications and generalizationscan be more easily seen or interpreted in terms of filter banks than in terms of the wavelet expansion. The topic of filter banks in developed in Chapter: Filter Banks and the Discrete Wavelet Transform and in more detail in Chapter: Filter Banks and Transmultiplexers .
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