### 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 (non-unique) *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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Sufficient Condition for Perfect Reconstruction