If x and u are uncorrelated, then the messy vector quotient for Hx(jra) ^ ®xy(jra)/®xx(ra), as above.
Rather than generate analog, broadband Gaussian noise to characterize linear systems by the cross-correlation method, use has been made of pseudo-random, binary noise (PRBN) generated by a clocked, digital circuit using sequential and combinational logic (O'Leary and Honrubia, 1975; Graupe, 1976; Chapter 4). Figure 8.2 illustrates a 32-bit PRBN generator designed and built by the author and his graduate students. This system uses a 50-kHz clock with a period of At = 20 |is. The PR digital sequence at the output repeats itself after N = (232 - 1) = 4.29496730 E9 clock cycles, or after NAt = 85.8993459 E3 seconds, or 23.861 hours. The TTL output of the PRBN generator is a TTL square wave with random high and low intervals, clocked at 50 kHz. If HI = 3.6 V is assumed, and LO = 0.2 V, then subtraction of 1.9 Vdc from the output wave will produce a random square wave with zero mean and peak height, a = 1.7 V. In the figure, the zero-mean TTL PRBN wave is passed through a digitally controlled, biquad, analog LPF (Northrop, 1990). The digital control input allows the user to set the break frequency of the biquad while holding the filter gain and damping factor constant.
O'Leary and Honrubia (1975) show that the autocorrelation function for a PRBN random square wave having zero mean and peak amplitude a is given by
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