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Up:Translation
Invariance and the Previous:Introduction
For simplicity, we consider the case of multiplier 2 and a signal,
, of length, or period,
.
The general higher multiplier case is described in Appendix A.
The discrete wavelet transformation is defined by the 2p low
pass coefficients
and
2p high pass coefficients
,
where
and
The wavelet transform represents the signal of length n by a
low pass,
, and a high
pass,
, component, each
of length n/2.
with summation on
.
The direct sum
is a unitary transformation,
,
on vectors in
.
The adjoint operations
verify the identity
The unit shift operation,
,
on period n vectors can be represented by the circulant matrix [4]
and
for
, is also a
unitary transformation on.
The inverse ofis its
transpose
.
It is generally true that
Therefore, even shifts of order 2k of the data produce order k shifts in the low and high pass components.
From the above,
Therefore, even shifts are simply related to even shifts, and odd shifts are simply related to odd shifts. Even shifts are not simply related to odd shifts. Therefore, the modes in the wavelet transform are not translation invariant. The energy in the wavelet modes can change rapidly with regard to amplitude and order for different even/odd translations [7], [8], [10].