Question 10.14: It is desired to use a DFT program to measure the PSD of ran......
It is desired to use a DFT program to measure the PSD of random pressure fluctuations generated by a jet engine, in the frequency band from 25 to 800 Hz. It is known from experience that in this case the spectrum will be fairly flat, so that it will be sufficient to plot it at 25 Hz intervals. A data sample 1 second long, sampled at 2048 samples per second, is available. An anti-aliasing filter with a substantially flat response up to 800 Hz, and negligible response at the Nyquist frequency, 1024 Hz, is used before sampling. Show how the scatter can be reduced by smoothing.
Using a DFT with N = 2048 points, sampled at 2048 samples/s, then T = 1 and B = 1 / T = 1. From Eq. (10.87):
\frac{\sigma }{\mu } \approx \frac{1}{\sqrt{BT} } = 1 (A)However, since it is acceptable, in this case, to present the power spectrum at intervals of 25 Hz, it may be smoothed by averaging in groups of twenty five. Now, in effect, B = 25, and T = 1, as before, so,
\frac{\sigma }{\mu } \approx \frac{1}{\sqrt{BT} } = \frac{1}{\sqrt{25} } = 0.2 (B)Thus the scatter that would be apparent if the PSD were plotted at 1 Hz intervals is considerably reduced.
It is important to note that the PSD values must be formed before smoothing takes place. The Fourier transformof randomdata is itself random, and would average to zero.