0.3 Filter banks and the discrete wavelet transform  (Page 2/6)

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

We talk about digital filtering and down-sampling because that is exactly what [link] and [link] do. These equations show that the scaling and wavelet coefficients atdifferent levels of scale can be obtained by convolving the expansion coefficients at scale $j$ by the time-reversed recursion coefficients $h(-n)$ and $h_1(-n)$ then down-sampling or decimating (taking every other term, the even terms) to give the expansion coefficients at the next levelof $j-1$ . In other words, the scale- $j$ coefficients are “filtered" by two FIR digital filters with coefficients $h(-n)$ and $h_1(-n)$ after which down-sampling gives the next coarser scaling and waveletcoefficients. These structures implement Mallat's algorithm [link] , [link] and have been developed in the engineering literature on filter banks, quadrature mirror filters (QMF), conjugatefilters, and perfect reconstruction filter banks [link] , [link] , [link] , [link] , [link] , [link] , [link] and are expanded somewhat in Chapter: Filter Banks and Transmultiplexers of this book. Mallat, Daubechies, and others showed the relation of wavelet coefficient calculation and filter banks.The implementation of [link] and [link] is illustrated in [link] where the down-pointing arrows denote a decimation or down-sampling by two and the other boxes denote FIR filtering or aconvolution by $h(-n)$ or $h_1(-n)$ . To ease notation, we use both $h\left(n\right)$ and $h_0\left(n\right)$ to denote the scaling function coefficients for the dilation equation [link] .

As we will see in Chapter: The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients , the FIR filter implemented by $h(-n)$ is a lowpass filter, and the one implemented by $h_1(-n)$ is a highpass filter. Note the average number of data points out of this system is thesame as the number in. The number is doubled by having two filters; then it is halved by the decimation back to the original number. This means thereis the possibility that no information has been lost and it will be possible to completely recover the original signal. As we shall see, thatis indeed the case. The aliasing occurring in the upper bank can be “undone" or cancelled by using the signal from the lower bank. This isthe idea behind perfect reconstruction in filter bank theory [link] , [link] .

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

The frequency response of a digital filter is the discrete-time Fourier transform of its impulse response (coefficients) $h\left(n\right)$ . That is given by