Completeness of arbitrarily sampled discrete time wavelet transforms

An arbitrarily sampled discrete time wavelet transform is said to be complete if it is uniquely invertible, i.e., if the underlying signal can be uniquely recovered from the available samples of the wavelet transform. We develop easy-to-compute necessary and sufficient conditions and necessary but not sufficient conditions for the completeness of an arbitrarily sampled dyadic discrete time wavelet transform of a periodic signal. In particular, we provide necessary and sufficient conditions for completeness of the sampled wavelet transform when the lowpass filter associated with the dyadic wavelet filter bank has no unit circle zeros other than those at z=1. We present necessary but not sufficient conditions for completeness when the lowpass filter associated with the dyadic wavelet filter bank has arbitrary unit circle zeros. We also provide necessary and sufficient conditions for completeness of a set of samples of both the lowpass approximation to the signal and its wavelet transform. All the conditions we derive use only information about the location of the retained samples and the analyzing wavelet filter bank. They can easily be checked without explicitly computing of the rank of a matrix. Finally, we present a simple signal reconstruction procedure that can be used once we have determined the arbitrarily sampled discrete time wavelet transform is complete.