# 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 $$\lambda_{\text{min}} \leq \lambda \leq \lambda_{\text{max}}$$):

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

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:

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

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.