The 2-band Filter Bank 2.2 2003/05/01 2003/06/19 Nick Kingsbury ngk10@cam.ac.uk Liqun Wang liqun@rice.edu Nick Kingsbury ngk10@cam.ac.uk 1-D Haar transform 2-D filter bank reconstruction This module derives 2-D filter bank from 1-D Haar transform. Digital filter banks have been actively studied since the 1960s, whereas Wavelet theory is a new subject area that was developed in the 1980s, principally by French and Belgian mathematicians, notably Y. Meyer, I. Daubechies, and S. Mallat. The two topics are now firmly linked and of great importance for signal analysis and compression.

The 2-band Filter Bank Recall the 1-D Haar transform from our previous discussion. y 1 y 2 T x 1 x 2 where T 1 2 1 1 1 -1 We can write this in expanded form as: y 1 1 2 x 1 1 2 x 2 y 2 1 2 x 1 1 2 x 2 More generally if x is a longer sequence and the results are placed in two separate sequences y0 and y1 , we define the process as: y0 n 1 2 x n 1 1 2 x n y1 n 1 2 x n 1 1 2 x n These can be expressed as 2 FIR filters with tap vectors h0 1 2 1 2 and h1 1 2 -1 2 . Hence as z-transforms, and become: Y0 z H0 z X z where H0 z 1 2 z 1 Y1 z H1 z X z where H1 z 1 2 z 1 . (We shall later extend these filters to be more complicated.) In practice, we only calculate y0 n and y1 n at alternate (say even) values of n so that the total number of samples in y0 and y1 is the same as in x. We may thus represent the Haar transform operation by a pair of filters followed by downsampling by 2, as shown in (a). This is known as a 2-band analysis filter bank.

Two-band filter banks for analysis (a) and reconstruction (b).

In this equation in our discussion of the Haar transform, to reconstruct x from y we calculated x T y . For long sequences this may be written: n n even x n 1 1 2 y0 n 1 2 y1 n n n even x n 1 2 y0 n 1 2 y1 n Since y0 n and y1 n are only calculated at even values of n, we may assume that they are zero at odd values of n. We may then combine and into a single expression for x n , valid for all n: x n 1 2 y0 n 1 y0 n 1 2 y1 n 1 y1 n or as z-transforms: X z G0 z Y0 z G1 z Y1 z where G0 z 1 2 z 1 G1 z 1 2 z 1 In the signals Y0 z and Y1 z are not really the same as Y0 z and Y1 z in and because those in and have not had alternate samples set to zero. Also, in X z is the reconstructed output whereas in and it is the input signal. To avoid confusion we shall use X ^ , Y0 ^ and Y1 ^ for the signals in so it becomes: X ^ z G0 z Y0 ^ z G1 z Y1 ^ z We may show this reconstruction operation as upsampling followed by 2 filters, as in (b). If Y0 ^ and Y1 ^ are not the same as Y0 and Y1 , how do they relate to each other? Now n n even y0 ^ n y0 n n n odd y0 ^ n 0 Therefore Y0 ^ z is a polynomial in z, comprising only the terms in even powers of z from Y0 z . This may be written as: Y0 ^ z even n y0 n z n all n 1 2 y0 n z n y0 n z n 1 2 Y0 z Y0 z Similarly Y1 ^ z 1 2 Y1 z Y1 z This is our general model for downsampling by 2, followed by upsampling by 2 as defined in and .