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Necessary and Sufficient Conditions for Perfect Reconstruction
It can be shown [287] that the most general conditions
for perfect reconstruction are that

(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 (nonunique) square root of a vector unit delay, since

(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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