nj = tj m/ j x n+ L-1 / k=0 L-1 / k=0 v m/ k x tk m/ k - v m/ j x x 3) The "pow-R" Method in [30] Real world signals are not white and this method aims at exploiting this prior. Non-whiteness depends on how average energy is distributed in the signal space. Such an information is contained in the correlation matrix K x = E 6xx <@ that is the starting point of this method. Consider K x and note that if x were white, then all the eigenvalues of K x would be equal. Hence, one can measure non-whiteness with localization [31], that quantifies the deviation of the actual eigenvalues of K x from their equidistributed version Lx = / c trm^KK xh - n1 m n-1 j=0 xj 2 = tr ^ K 2x h -1 tr 2 ^ K x h n (7) Localization goes from 0 (white signals) to 1 - 1/n (signals whose energy concentrates along a single direction). The method does not yield deterministic matrices A. This slightly complicates the design flow but provides some advantages that can be exploited, for example, for effective implementations. It assumes that A is a random matrix made of independent and identically distributed non-white rows. If we indicate the generic row as a < and the corresponding measurement with y = a < x, then the method aims at identifying rows with an high rakeness, i.e., with the ability of collecting the largest possible amount of energy from the signal and transfer it to the measurement3. Since the power of the measurement is E 6 y 2@ = E 6x < aa < x@ = tr ^ K x K a h the natural design parameter is the correlation matrix K a = E 6aa <@ that is fixed by solving Z ] Ka * 0 ] K a = KIEEE Circuits and Systems Magazine - Q1 2020
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