Letting n go to infinity does not take us to the expectation .The problem is, as we increase n, we increase the frequency resolution but not the statistical resolution (i.e., the fluctuation around ). To increase the statistical resolution, we need to simulate ensemble averaging. There are two ways to do this:
The second method is illustrated in Figure 10. This figure shows a noise burst of 240 points. Since the signal is even, the burst is effectively 480 points wide, so the autocorrelation is 480 points from center to end: the number of samples will be the same for all cases. The spectrum is very rough. Multiplying the autocorrelation by a triangle function effectively smooths the spectrum by a sincsquared function, thus reducing the spectral resolution (). Notice that is equal here to the width of the sincsquared function, which is inversely proportional to the length of the triangle ().
However, the first taper takes the autocorrelation width from 480 lags to 120 lags. Thus the spectral fluctuations should drop by a factor of 2, since the count of terms s_{k} in is reduced to 120 lags. The width of the next weighted autocorrelation width is dropped from 480 to 30 lags. Spectral roughness should consequently drop by another factor of 2. In all cases, the average spectrum is unchanged, since the first lag of the autocorrelations is unchanged. This implies a reduction in the relative spectral fluctuation proportional to the square root of the length of the triangle ().
Our conclusion follows:
The tradeoff among resolutions of time, frequency, and spectral amplitude is
