# The distribution of peak and rms of response in time domain

I am trying to model the pdf of the peak and rms of the time domain response $$y(t)$$ in a reverberation chamber. The system is stimulated by pulsed linear chirp $$x(t)$$ for 200 ns that goes linearly from 4-5 GHz and its Fourier transform is denoted by $$X(f)$$. The frequency response of the system, $$H(f)$$ is a random variable and is measured 50 times for 50 different tuner steps over a frequency range of 1 to 6 GHz. We know the distribution of $$H(f)$$ over a narrow frequency band (~500 MHz) is given by Lomax distribution. Now we know that we can use the following equation to calculate the time-domain response $$y(t)$$. $$y(t) = \mathscr{F}^{-1}\big\{X(f)*H(f) \big\}$$

Where $$\mathscr{F}^{-1}$$ is the inverse Fourier transform. On the other hand, we have measured $$y(t)$$ in the time domain as well. As you might imagine, $$y(t)$$ is a random variable. We are interested in the distribution of the peak and rms of this variable. Here is a plot of 200 measurements in the time-domain.

Now you might be wondering why we did the convolution at first, the answer is that since we know the distribution of $$H(f)$$, we would like to link that to the distribution of the peak and rms of y(t) so we understand how the spectrum of the pulsed chirp and characteristics of $$H(f)$$ (modal density, frequency spacing, etc.) influence the distribution of the peak and rms of $$y(t)$$

• I'm still trying to decode this. Are you tracking the peak with a sliding maximum and the rms with a sliding average on the square of the signal? And then you want a "distribution" of values of both? Like a histogram of peak values and another histogram of rms values? Apr 27, 2023 at 5:04
• so, $H(f)$ is the Fourier transform of $y(t)$, right? Apr 27, 2023 at 8:47
• I added plenty of details. In hindsight, my original post wasn't descriptive enough. Apr 28, 2023 at 8:17
• robert bristow-johnson I know how to get the histogram. That's trivial. I would like to gain insight into the best approach to model peaks and rms of y(t). A random variable is fully specified by its CDF, so ultimately we would like to find an analytical expression for that. Apr 28, 2023 at 8:19
• If $$y(t) = \mathscr{F}^{-1}\big\{X(f) \circledast H(f) \big\}$$ then $$y(t) = x(t) \cdot h(t)$$ Apr 28, 2023 at 14:01