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

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

which, after changing variables

If we denote

then

is expressible at a scale of

where the

which, by using Equation 3 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

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 convolving
the 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

There is a large literature on digital filters and how to design them
[19], [18].
If the number of filter coefficients

In multirate digital filters, there is an assumed relation between the
integer index

In down-sampling, there is clearly the possibility of losing information since half of the data is discarded. The effect in the frequency domain (Fourier transform) is called aliasing which states that the result of this loss of information is a mixing up of frequency components [19], [18]. Only if the original signal is band-limited (half of the Fourier coefficients are zero) is there no loss of information caused by down-sampling.

We talk about digital filtering and down-sampling because
that is exactly what Equation 9 and Equation 10 do.
These equations show that the scaling and wavelet coefficients at
different levels of scale can be obtained by convolving the expansion
coefficients at scale

As we will see in Chapter: The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients , the FIR filter implemented by

This splitting, filtering, and decimation can be repeated on the scaling
coefficients to give the two-scale structure in Figure 3.
Repeating this on the scaling coefficients is called *iterating the
filter bank*. Iterating the filter bank again gives us the three-scale
structure in Figure 4.

The frequency response of a digital filter is the discrete-time Fourier
transform of its impulse response (coefficients)

The magnitude
of this complex-valued function gives the ratio of the output to the input
of the filter for a sampled sinusoid at a frequency of

The first stage of two banks divides the spectrum of

For any practical signal that is bandlimited, there will be an upper scale

down to as low a resolution,

which is a finite scale version of (Reference).
We will discuss the choice of