Necessary and Sufficient Conditions for Perfect Reconstruction
It can be shown [287] that the most general conditions for perfect reconstruction are that
![$\displaystyle \zbox {\bold{R}(z)\bold{E}(z) \eqsp c z^{-K} \left[\begin{array}{cc} \bold{0}_{(N-L)\times L} & z^{-1}\bold{I}_{N-L} \\ [2pt] \bold{I}_L & \bold{0}_{L \times (N-L)} \end{array}\right]}$](http://www.dsprelated.com/josimages_new/sasp2/img2132.png) |
(12.63) |
for some constant
and some integer 
, where 
is any
integer between 0
and 
.
Note that the more general form of
above can be regarded as a (non-unique) square root of a vector unit delay, since
![$\displaystyle \left[\begin{array}{cc} \bold{0}_{(N-L)\times L} & z^{-1}\bold{I}_{N-L} \\ [2pt] \bold{I}_L & \bold{0}_{L \times (N-L)} \end{array}\right]^2 \eqsp z^{-1}\bold{I}_N.$](http://www.dsprelated.com/josimages_new/sasp2/img2136.png) |
(12.64) |
Thus, the general case is the same thing as
 |
(12.65) |
except for some channel swapping and an extra sample of delay in some channels.
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Sufficient Condition for Perfect Reconstruction
