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# The 2-band Filter Bank

Module by: Nick Kingsbury. E-mail the author

Summary: 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.

( y1 y2 )=T( x1 x2 ) y 1 y 2 T x 1 x 2
(1)
where T=12( 11 1-1 ) T 1 2 1 1 1 -1

We can write this in expanded form as:

y1=12x1+12x2 y 1 1 2 x 1 1 2 x 2
(2)
y2=12x112x2 y 2 1 2 x 1 1 2 x 2
(3)
More generally if xx is a longer sequence and the results are placed in two separate sequences y0 y0 and y1 y1 , we define the process as:
y0 n=12xn1+12xn y0 n 1 2 x n 1 1 2 x n
(4)
y1 n=12xn112xn y1 n 1 2 x n 1 1 2 x n
(5)
These can be expressed as 2 FIR filters with tap vectors h0 =( 1212 ) h0 1 2 1 2 and h1 =( 12-12 ) h1 1 2 -1 2 . Hence as z-transforms, Equation 4 and Equation 5 become:
Y0 z= H0 zXz Y0 z H0 z X z
(6)
where H0 z=12(z-1+1) H0 z 1 2 z 1
Y1 z= H1 zXz Y1 z H1 z X z
(7)
where H1 z=12(z-11) H1 z 1 2 z 1 . (We shall later extend these filters to be more complicated.)

In practice, we only calculate y0 n y0 n and y1 n y1 n at alternate (say even) values of nn so that the total number of samples in y0 y0 and y1 y1 is the same as in xx.

We may thus represent the Haar transform operation by a pair of filters followed by downsampling by 2, as shown in Figure 1(a). This is known as a 2-band analysis filter bank.

In this equation in our discussion of the Haar transform, to reconstruct xx from yy we calculated x=TTy x T y . For long sequences this may be written:

n,n=even:xn1=12 y0 n+12 y1 n n n even x n 1 1 2 y0 n 1 2 y1 n
(8)
n,n=even:xn=12 y0 n12 y1 n n n even x n 1 2 y0 n 1 2 y1 n
(9)
Since y0 n y0 n and y1 n y1 n are only calculated at even values of nn, we may assume that they are zero at odd values of nn. We may then combine Equation 8 and Equation 9 into a single expression for xn x n , valid for all nn:
xn=12( y0 n+1+ y0 n)+12( y1 n+1 y1 n) x n 1 2 y0 n 1 y0 n 1 2 y1 n 1 y1 n
(10)
or as z-transforms:
Xz= G0 z Y0 z+ G1 z Y1 z X z G0 z Y0 z G1 z Y1 z
(11)
where
( G0 z=12(z+1))( G1 z=12(z1)) G0 z 1 2 z 1 G1 z 1 2 z 1
(12)
In Equation 11 the signals Y0 z Y0 z and Y1 z Y1 z are not really the same as Y0 z Y0 z and Y1 z Y1 z in Equation 6 and Equation 7 because those in Equation 6 and Equation 7 have not had alternate samples set to zero. Also, in Equation 11 Xz X z is the reconstructed output whereas in Equation 6 and Equation 7 it is the input signal.

To avoid confusion we shall use X ^ X ^ , Y0 ^ Y0 ^ and Y1 ^ Y1 ^ for the signals in Equation 11 so it becomes:

X ^ z= G0 z Y0 ^ z+ G1 z Y1 ^ z X ^ z G0 z Y0 ^ z G1 z Y1 ^ z
(13)
We may show this reconstruction operation as upsampling followed by 2 filters, as in Figure 1(b).

If Y0 ^ Y0 ^ and Y1 ^ Y1 ^ are not the same as Y0 Y0 and Y1 Y1 , how do they relate to each other?

Now

n,n=even: y0 ^ n= y0 n n n even y0 ^ n y0 n
(14)
n,n=odd: y0 ^ n=0 n n odd y0 ^ n 0
(15)
Therefore Y0 ^ z Y0 ^ z is a polynomial in zz, comprising only the terms in even powers of zz from Y0 z Y0 z . This may be written as:
Y0 ^ z=even n y0 nzn=all n12( y0 nzn+ y0 nzn)=12( Y0 z+ Y0 z) 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
(16)
Similarly
Y1 ^ z=12( Y1 z+ Y1 z) Y1 ^ z 1 2 Y1 z Y1 z
(17)
This is our general model for downsampling by 2, followed by upsampling by 2 as defined in Equation 14 and Equation 15.

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A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

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