0.3 Filter banks and the discrete wavelet transform

In many applications, one never has to deal directly with the scaling functions or wavelets. Only the coefficients $h\left(n\right)$ and $h_1\left(n\right)$ in the defining equations [link] and [link] and $c\left(k\right)$ and $d_j\left(k\right)$ in the expansions [link] , [link] , and [link] need be considered, and they can be viewed as digital filters and digital signals respectively [link] , [link] . While it is possible to develop most of the results of wavelet theory using only filter banks, we feel that both the signalexpansion point of view and the filter bank point of view are necessary for a real understanding of this new tool.

Analysis – from fine scale to coarse scale

In order to work directly with the wavelet transform coefficients, we will derive the relationship between the expansion coefficients at a lowerscale level in terms of those at a higher scale. Starting with the basic recursion equation from [link]

and assuming a unique solution exists, we scale and translate the time variable to give

which, after changing variables $m=2k+n$ , becomes

If we denote $V_j$ as

then

is expressible at a scale of $j+1$ with scaling functions only and no wavelets. At one scale lower resolution, wavelets are necessary forthe “detail" not available at a scale of $j$ . We have

where the $2^{j/2}$ terms maintain the unity norm of the basis functions at various scales. If ${\phi }_{j,k}\left(t\right)$ and ${\psi }_{j,k}\left(t\right)$ are orthonormal or a tight frame, the $j$ level scaling coefficients are found by taking the inner product

which, by using [link] and interchanging the sum and integral, can be written as

but the integral is the inner product with the scaling function at a scale of $j+1$ giving

The corresponding relationship for the wavelet coefficients is

Filtering and down-sampling or decimating

In the discipline of digital signal processing, the “filtering" of a sequence of numbers (the input signal) is achieved by convolvingthe sequence with another set of numbers called the filter coefficients,taps, weights, or impulse response. This makes intuitive sense if you think of a moving average with the coefficients being the weights.For an input sequence $x\left(n\right)$ and filter coefficients $h\left(n\right)$ , the output sequence $y\left(n\right)$ is given by

There is a large literature on digital filters and how to design them [link] , [link] . If the number of filter coefficients $N$ is finite, the filter is called a Finite Impulse Response (FIR) filter. If the number is infinite,it is called an Infinite Impulse (IIR) filter. The design problem is the choice of the $h\left(n\right)$ to obtain some desired effect, often to remove noise or separate signals [link] , [link] .

In multirate digital filters, there is an assumed relation between the integer index $n$ in the signal $x\left(n\right)$ and time. Often the sequence of numbers are simply evenly spaced samples of a function of time. Twobasic operations in multirate filters are the down-sampler and the up-sampler. The down-sampler (sometimes simply called a sampler ora decimator) takes a signal $x\left(n\right)$ as an input and produces an output of $y\left(n\right)=x\left(2n\right)$ . This is symbolicallyshown in [link] . In some cases, the down-sampling is by a factor other than two andin some cases, the output is the odd index terms $y\left(n\right)=x(2n+1)$ , but this will be explicitly stated if it is important.