Oscillations, Waves, and Interactions - GWDG

48 A. Kohlrausch and S. van de Par
Figure 6. Illustration of the low-noise noise generation. The top panel shows the timedomain
Gaussian noise at the start of the iterative process, the middle panel the lownoise
noise after one iteration, the lower panel, the low-noise noise after 10 iterations. All
waveforms are shown with their respective envelopes.
lope more flat, according to some statistical measure 2 . After a sequence of iterations,
a low-noise noise waveform was obtained with a rather flat temporal envelope and
the initial amplitude spectrum. Thus, summarizing, the method of Pumplin obtained
low-noise noise by modifying the phase spectrum in a special way.
Later on, several alternative manners to generate low-noise noise were proposed
and evaluated by Kohlrausch et al. [31]. We will here describe the method that led
to the lowest degree of fluctuation in the temporal envelope. The method consists of
an iterative process that is initiated by generating a time-discrete Gaussian bandpass
noise. The iterative process then consists of a sequence of straightforward steps.
First the Hilbert envelope of the noise is calculated. Secondly, the noise waveform
is divided by its Hilbert envelope on a sample-to-sample basis in the time domain.
For the rare occasions that the Hilbert envelope is equal to zero, the resulting division
is set to zero. In the third step, a bandpass filtering is applied to remove the new
spectral components outside of the specified bandpass range that were introduced by
the division operation in the previous step. By repeating the iterative steps several
times, a much flatter envelope is obtained.
After the first two steps, calculating the Hilbert envelope and dividing the noise
waveform by its Hilbert envelope, the resulting temporal waveform will have a flat
envelope. The spectrum will also be modified considerably. The division by the Hil-
2 normalized fourth moment of the temporal envelope distribution