# Superposition of Lorentzians with fixed height pulses

In the last post we have seen that 1/f noise can be obtained from superposition of Lorentzian power spectral densities. Though we have imposed an odd requirement of scaling pulse height for different characteristic rates \( \lambda \). In this post let us explore how to achieve the same thing with fixed height pulses.

## Fixing the height of the pulses

Let us now assume that \( a = 1 \) for all possible \( \lambda \). Under this assumption, because \( \bar{\nu} = \lambda / 2 \), uniform distribution of \( \lambda \) will not result in 1/f noise (instead we will have \( S(f) \sim \ln(f) \)). What other distribution should we pick?

An intuitive guess would be (bounded) Pareto distribution with \( \alpha = 0 \). The exact probability density function then is given by (for \( \lambda_{\text{min}} \leq \lambda \leq \lambda_{\text{max}} \)):

\begin{equation} p(\lambda) = \frac{1}{\ln\left(\frac{\lambda_{\text{max}}}{\lambda_{\text{min}}}\right)} \cdot \frac{1}{\lambda} . \end{equation}

Where the term with the logarithm serves as a normalization constant.

Then, if each individual signal has a Lorentzian spectral density with characteristic rate sampled from the bounded Pareto distribution, the power spectral density of superposition:

\begin{equation} S\left(f\right)=\frac{1}{\ln\left(\frac{\lambda_{\text{max}}}{\lambda_{\text{min}}}\right)} \int_{\lambda_{\text{min}}}^{\lambda_{\text{max}}}\frac{1}{\lambda}\cdot\frac{\bar{\nu}}{\lambda^{2}+\pi^{2}f^{2}} d \lambda = \frac{1}{\ln\left(\frac{\lambda_{\text{max}}}{\lambda_{\text{min}}}\right)}\cdot\frac{\operatorname{arccot}\left(\frac{\pi f}{\lambda_{\text{max}}}\right)-\operatorname{arccot}\left(\frac{\pi f}{\lambda_{\text{min}}}\right)}{2 \pi f}. \end{equation}

In the above we have used the fact that for every individual Lorentzian \( \bar{\nu} = \lambda / 2 \). The obtained result is almost identical to the one from the previous post with a difference being in the normalization constant and the leftover \( \frac{1}{2} \) from the \( \bar{\nu} \).

## Interactive app

As previously, to effectively show the key point we do not generate the signals themselves. Here we just add the Lorentzian shapes, which should emerge as a result of long duration detailed simulations. As previously, we encourage you to explore how adjusting the bounds of the \( \lambda \) distribution changes the spectral density of the superposition.