Transforms using random matrices have been found to have many applications. We are concerned with the projection of a signal onto Gaussian-distributed random orthogonal bases. We also would like to easily invert the process through transposes in order to facilitate iterative reconstruction. We derive an efficient method to implement random unitary matrices of larger sizes through a set of Givens rotations. Random angles are hierarchically generated on-the-fly and the inverse merely requires traversing the angles in reverse order. Hierarchical randomization of angles also enables reduced storage. Using the random unitary matrices as building blocks we introduce random paraunitary systems (filter banks). We also highlight an efficient implementation of the paraunitary system and of its inverse. We also derive an adaptive under-decimated system, wherein one can control and adapt the amount of projections the signal undergoes, in effect, varying the sampling compression ratio as we go along the signal, without segmenting it. It may locally range from very compressive sampling matrices to (para) unitary random ones. One idea is to adapt to local sparseness characteristics of non-stationary signals.
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