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This work is motivated by the search for discrete wavelet transform (DWT) with near shift-invariance. After examining the elements of the property, we introduce a new notion of shift-invariance, which is particularly informative for multirate systems that are not shift-invariant in the strict sense. Briefly speaking, a discrete-time system is mu-shift-invariant if a shift in input results in the output being shifted as well. However, the amount of the shift in output is not necessarily identical to that in input. A fractional shift is also acceptable and can be properly specified in the Fourier domain. The mu-shift-invariance can be interpreted as invariance of magnitude spectrum with linear phase offset of output with respect to shift in input. It is stronger than the shiftability in position, which is equivalent to insensitivity of energy to shift in input. Under this generalized notion, the expander is always mu-shift-invariant. The M-fold decimator is mu-shift-invariant for input with width of frequency support not more than 2pi/M ; equivalently, the output contains no aliasing term in some frequency band with length of 2pi . We generalize the transfer function description of linear shift-invariant systems for mu -shift-invariant systems. We then perform mu-shift-invariance analysis of 2-band orthogonal DWT and of the 2-band dual-tree complex wavelet transform (DT-CWT). The analysis in each case provides clarifications to early understanding of near shift-invariance. We show that the DWT is mu-shift-invariant if and only if the conjugate quadrature filter (CQF) is analytic or antianalytic. For the DT-CWT, the CQFs must have supports included within [-2pi/3, 2pi/3], in addition to the well-know half-sample delay condition at higher levels and the one-sample delay condition at the first level.
Date of Publication: July 2009