The class of complex random vectors whose covariance matrix is linearlyparameterized by a basis of Hermitian Toeplitz (HT) matrices is considered, andthe maximum compression ratios that preserve all second-order information arederived --- the statistics of the uncompressed vector must be recoverable froma set of linearly compressed observations. This kind of vectors arisesnaturally when sampling wide-sense stationary random processes and features anumber of applications in signal and array processing. Explicit guidelines to design optimal and nearly optimal schemes operatingboth in a periodic and non-periodic fashion are provided by considering two ofthe most common linear compression schemes, which we classify as dense orsparse. It is seen that the maximum compression ratios depend on the structureof the HT subspace containing the covariance matrix of the uncompressedobservations. Compression patterns attaining these maximum ratios are found forthe case without structure as well as for the cases with circulant or bandedstructure. Universal samplers are also proposed to compress unknown HTsubspaces.
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