where H1(ra) = F{h1(t)}, and ®xx(ra) = 1 for |ra| 3 rao, and ®xx(ra) = 0 for |ra| > rao. If h1(t) = A8(t), (h1 is an ideal amplifier with gain A), then it can be shown:

VAR[h1(x)] = (rao/2n3 a) tan-1(rao/a) + [(1/n) tan-1(rao/a)]2 8.3-18

Note that both variances are increasing functions of rao, the input noise spectrum cutoff frequency. Thus, it appears that it is important not to have the input spectrum exceed the system bandwidth by too much for two important reasons; the first is potential problems with aliasing, and the second is excessive variances in the kernel estimates. See Marmarelis and Naka (1974) for details on the theoretical effects of uncorrelated and measurement noise on the variances of the estimates of the kernels.

A summary of the steps required to calculate the NSUS kernels, h0, h1(t), and h2(x1, t2) in the time domain by the Lee-Schetzen method is shown in Figure 8.3-1. Note that the kernels can be estimated in the frequency domain following the procedure shown in Figure 8.3-2. It is necessary to use the average spectra of x(t) and yo(t) to find estimates of H1(jra) and H2(jra1, jra2) and their inverse transforms, h1(t) and h2(x1, t2). This means that x(t) and yo(t) are broken into N successive sampling epochs, each having 4096 samples, for example. N might be 32.

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