Classifying a time series with a Lorentzian power spectral density

Question

I was wondering if there is a general classification for time-varying signal with a Lorentzian shaped power spectral density, specifically with some nonzero center frequency. In terms of equations, what I mean is a power spectral density of the form $S(f) = \frac{a}{1+((f-f_0)/b)^2} + \frac{a}{1+((f+f_0)/b)^2}$ where $a$ is its maximal amplitude, $f_0$ is the center frequency and $b$ is the full width at full maximum. Note that I am talking about a real signal for all that follows.

The reason I am interested in this is because I know that if one has $f_0 = 0$, the timeseries could be seen as an instance of the Ornstein-Uhlenbeck (OU) process. However, I do not know of any process that generates a displaced Lorentzian power spectral density.

I tried thinking about this in terms of the autocorrelation function, but this has not led me very far. For OU this is an exponential decay, and by the Fourier transform this leads to the Lorentzian PSD. But conversely, a displaced Lorentzian again leads to exponentially decaying autocorrelations, multiplied by some phase factor that does not show up in the actual autocorrelations.

However, I am pretty sure there is a physical difference between the two power spectral densities (think about processes where the energy of the Lorentzian is relevant), and I would be interested in looking into the $f_0 \neq 0 $ case a bit further. So I am therefore wondering if there is some OU-type process that generates this, or some classification such as white noise, pink noise, etcetera, that fits this type of spectrum.

Answer 1

While it doesn't exactly give the power spectral density formulated by you, a bimodal (when considering also negative frequencies) power spectral density is given by the time evolution of the position or displacement of a harmonic oscillator driven by Brownian motion. The only modification compared to the Ornstein–Uhlenbeck process is inclusion of inertia, which enables oscillations when the harmonic oscillator is underdamped. An Ornstein–Uhlenbeck process is governed by the differential equation:

$$\dot x(t) = -\theta x(t) + F(t) \tag{1}$$

The dot above the $x(t)$ indicates the derivative of $x(t)$. In an infinitesimal time step $dt$, displacement $x(t)$ relaxes a step $\theta\,dt\,x(t)$ towards zero and is also affected by white noise external forcing $F(t)$. There is no inertia: neither the relaxation term or the external forcing term describe "memory of past velocity". I have poor differential equation handling skills, but my hunch is that if a bit of the past velocity $\dot x(t-dt)$ was mixed into the current velocity, it would give this modified form with inertia:

$$\ddot x(t) = -\gamma_m \dot x(t) - f_m^2 x(t)+F(t) \tag{2}$$

Here $\ddot x(t)$ denotes the second derivative of $x(t)$. Now there are terms both for damping of velocity and for relaxation of the displacement. For the modified process, the power spectral density is of form (Gröblacher 2012, Coffey & Kalmykov 2012):

$$S(f) = \frac{a}{\left(f_m^2-f^2\right)^2 + f^2\gamma_m^2}, \tag{3}$$

where $\gamma_m$ is the width of the resonant peak at half maximum, and $a$ is a normalization constant which has also absorbed the power spectral density of the driving force which is white (constant in frequency) for a Brownian heat bath. The true peak frequency $f_\text{res}$ is (Gröblacher 2012):

$$f_\text{res}=f_m\sqrt{1-\frac{\gamma_m^2}{2f_m^2}}.$$

Figure 1. Power spectral density of the displacement of a massive (as opposed to massless) harmonic oscillator in a heat bath, with $a = 1$, $f_m = 1$ and $\gamma_m=\frac{1}{2}$.

$S(f)$ of Eq. 3 is identical in form to the square of the magnitude frequency response of a two-pole resistor-inductor-capacitor (RLC) filter:

$$|H(if)|^2 = \frac{a}{(f^2 - f_0^2)^2 + f^2\,(f_0/Q)^2}$$

If the filter is fed white noise, the power spectrum of the output will be identical to $S(f)$. It is characterized by a -12 dB/octave (exactly -40 dB/decade) asymptotic slope:

Figure 2. Square of the magnitude frequency response of the two-pole filter, with logarithmic axes. The constants are equivalent to those in Fig. 1.

I'll return to the "just add inertia" argument once more. Consider a discretization of the Ornstein–Uhlenbeck process defined in Eq. 1:

$$\dot x[k] = c_0 x[k] + F[k]$$

We can define velocity and acceleration as:

$$\dot x[k] = x[k] - x[k-1]$$ $$\ddot x[k] = \dot x[k] - \dot x[k-1] = x[k] - 2 x[k-1] + x[k-2]$$

If we add damped inertia to the process as a term $c_1 \dot x[k-1]$:

$$\begin{array}{rrcl} &\dot x[k] &=& c_0 x[k] + c_1 \dot x[k-1] + F[k]\\ \Rightarrow& x[k] - x[k-1] &=& c_0 x[k] + c_1 x[k-1] - c_1 x[k-2] + F[k]\\ \Rightarrow& 0 &=& (c_0 - 1) x[k] + (1 + c_1) x[k-1] - c_1 x[k-2] + F[k]\\ \Rightarrow& c_1 \ddot x[k] &=& (c_1 + c_0 - 1) x[k] + (1 - c_1) x[k-1] + F[k]\\ \Rightarrow& c_1 \ddot x[k] &=& (c_1 + c_0 - 1) x[k] - (c_1 - 1) x[k-1] + F[k]\\ \Rightarrow& c_1 \ddot x[k] &=& (c_1 - 1) \dot x[k] + c_0 x[k] + F[k]\\ \Rightarrow& \ddot x[k] &=& \frac{c_1 - 1}{c_1} \dot x[k] + \frac{c_0}{c_1} x[k] + \frac{1}{c_1}F[k]\\ \end{array}$$

We have arrived at the discrete equivalent of Eq. 2.

References

Gröblacer, Simon (2012) "Quantum Opto-Mechanics with Micromirrors: Combining Nano-Mechanics with Quantum Optics" DOI: 10.1007/978-3-642-34955-3, chapter 2, page 7.

Coffey, William T.; Kalmykov, Yuri P. (2012) "The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering" 3rd ed. ISBN: 978-981-4355-66-8, page 241.