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Finding the optimal dual frame and optimal dual pair for signalreconstruction, which can minimize the reconstruction error when erasure occursduring data transmission, is a deep rooted problem from the perspective offrame theory. In this paper, we consider a new measurement for the erroroperator by taking the average of spectral radius and operator norm withprobabilistic erasure. In this measurement, optimal dual frames are calledProbabilistic Averaged Spectrally Optimal Dual frames, PASOD-frames in shortand optimal dual pair is called PASOD-pair. The properties of the set ofPASOD-frames for a pre-selected frame, has been studied. We prove that the setof all PASOD-frames is convex, closed and compact. We also show that the imageof a PASOD-frame and PASOD-pair under any unitary operator is also aPASOD-frame and PASOD-pair. We provide several equivalent conditions for thecanonical dual to be the unique PASOD-frame for a given frame $F.$ Moreover, weprove non-uniqueness of PASOD-frame under certain condition of the given frame.We also go on to characterize the set of all PASOD-pairs and give severalequivalent conditions for a dual pair to become POD, PSOD and PASOD-pair.

Probability Modelled Averaged Spectrally Optimal Dual Frame and Dual Pair for Erasure