The purpose of this work is to investigate perfect reconstruction
underlying range space of operators in finite dimensional Hilbert spaces
by a new matrix method. To this end, first we obtain more structures of
the canonical $K$-dual. % and survey optimal $K$-dual problem under
probabilistic erasures. Then, we survey the problem of recovering and
robustness of signals when the erasure set satisfies the minimal
redundancy condition or the $K$-frame is maximal robust. Furthermore,
we show that the error rate is reduced under erasures if the $K$-frame
is of uniform excess. Toward the protection of encoding frame (K-dual)
against erasures, we introduce a new concept so called $(r,k)$-matrix
to recover lost data and solve the perfect recovery problem via matrix
equations. Moreover, we discuss the existence of such matrices by using
minimal redundancy condition on decoding frames for operators. We
exhibit several examples that illustrate the advantage of using the new
matrix method with respect to the previous approaches in existence
construction. And finally, we provide the numerical results to confirm
the main results in the case noise-free and test sensitivity of the
method with respect to noise.
Matrix methods for perfect signal recovery underlying range space of operators
