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Perfect Reconstruction (PR)

Module by: Nick Kingsbury. E-mail the author

Summary: This module discusses perfect reconstruction (PR).

We are now able to generalize our analysis for arbitrary filters H0 H0 , H1 H1 , G0 G0 and G1 G1 . Substituting this equation and this equation in our discussion of 2-band filter bank into this earlier equation and then using this equation and this equation from the same discussion, we get:

X ^ ⁢z=12⁢ G0 ⁢z⁢( Y0 ⁢z+ Y0 ⁢−z)+12⁢ G1 ⁢z⁢( Y1 ⁢z+ Y1 ⁢−z)=12⁢ G0 ⁢z⁢ H0 ⁢z⁢X⁢z+12⁢ G0 ⁢z⁢ H0 ⁢−z⁢X⁢−z+12⁢ G1 ⁢z⁢ H1 ⁢z⁢X⁢z+12⁢ G1 ⁢z⁢ H1 ⁢−z⁢X⁢−z=12⁢X⁢z⁢( G0 ⁢z⁢ H0 ⁢z+ G1 ⁢z⁢ H1 ⁢z)+12⁢X⁢−z⁢( G0 ⁢z⁢ H0 ⁢−z+ G1 ⁢z⁢ H1 ⁢−z) X ^ z 1 2 G0 z Y0 z Y0 z 1 2 G1 z Y1 z Y1 z 1 2 G0 z H0 z X z 1 2 G0 z H0 z X z 1 2 G1 z H1 z X z 1 2 G1 z H1 z X z 1 2 X z G0 z H0 z G1 z H1 z 1 2 X z G0 z H0 z G1 z H1 z
(1)
If we require X ^ ⁢z≡X⁢z X ^ z X z - the Perfect Reconstruction (PR) condition - then:
G0 ⁢z⁢ H0 ⁢z+ G1 ⁢z⁢ H1 ⁢z≡2 G0 z H0 z G1 z H1 z 2
(2)
and
G0 ⁢z⁢ H0 ⁢−z+ G1 ⁢z⁢ H1 ⁢−z≡0 G0 z H0 z G1 z H1 z 0
(3)
Identity Equation 3 is known as the anti-aliasing condition because the term in X⁢−z X z in Equation 1 is the unwanted aliasing term caused by down-sampling y0 y0 and y1 y1 by 2.

It is straightforward to show that the expression for H0 H0 , H1 H1 , G0 G0 and G1 G1 , given in this equation, this equation, this equation and this equation for the filters based on the Haar transform, satisfy Equation 2 and Equation 3. They are the simplest set of filters which do.

Before we look at more complicated PR filters, we examine how the filter structures of this figure may be extended to form a binary filter tree (and the discrete wavelet transform).

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