C.M.'s Science Blog: [MT 4] PSD of step-like random processes

In this post we construct a rather general "step-like" random process, consisting of many superposed steps of widely differing lengths tau_i and heights h_i. We are interested in the ensemble averaged power spectral density (PSD) of the process.

First we define a single step (more accurately: box) function of length tau and height 1:

$b_{\tau}(t)=\Theta(t+\frac{\tau}{2})-\Theta(t-\frac{\tau}{2})$ .

The Fourier transform of b(t) is

$b_{\tau}(\omega)=\frac{2 \sin(\omega\tau/2)}{\omega}$

Next we define a Poisson spike train with average spike rate lambda:

$g_{\lambda}(t)=\sum_n \delta(t-t_n)\;\mbox{with}\;\left<t_{n+1}-t_n\right>_n = \frac{1}{\lambda}$

This process has a frequency independent PSD, equal to its rate:

$P_{g_{\lambda}}(\omega)=\lambda$

We next generate a superposition of infinitely many positive, temporaly shifted, equally long and high boxes by convoluting the single box with the spike train:

$f(t)=b_{\tau}(t)\otimes g_{\lambda}(t)$

Note that when the inter-box-intervall 1/lambda is much longer than the single box duration tau, successive boxes will typically not overlap.

The Fourier transform of random process f(t) is

$f(\omega)=b_{\tau}(\omega)\cdot g_{\lambda}(\omega)$

and for the PSD we obtain

$P_f(\omega) = \left| b_{\tau}(\omega)\right|^2 \cdot P_{g_{\lambda}}(\omega)$

which yields in our special case

$P_f(\omega) = \frac{4\lambda}{\omega^2}\sin^2(\omega\tau/2)$

Our present random process contains only positive boxes. For a more symmetric process, we use two independent Poisson spike trains with the same average rate, but with opposite signs:

$f(t)=b_{\tau}(t)\otimes\left( g_{\lambda}^{(1)}(t) - g_{\lambda}^{(2)}(t) \right)$

Since the two spike trains are statistically independent, the PSD of the term in brackets in just the sum of the two individual PSDs:

$P_{g_{\lambda}}(\omega) = P_{g_{\lambda}^{(1)}}(\omega)+P_{g_{\lambda}^{(2)}}(\omega)$

Thus, the PSD of the symmetric process is

$P_f(\omega) = \frac{8\lambda}{\omega^2}\sin^2(\omega\tau/2)$ .

The above process consists only of boxes of the same duration tau. We next consider a superposition of many uncorrelated sub-processes with different tau_i. Again, the total PSD will simply be the sum of all sub-processes. Let the density distribution of the tau_i be Q(tau). We assume further that the range of tau_i is restricted

$\tau_i \in [\tau_1,\tau_2]$

and that Q(tau) is a "smoothly varying function" within the above limits. The total PSD is a weighted integral over all sub-processes:

$P_f(\omega)=\left< P_f(\omega,\tau)\right>_{\tau} = \int_{\tau_1}^{\tau_2}\!d\tau \;Q(\tau) P_f(\omega,\tau)$

One therefore obtains

$P_f(\omega) = \frac{8\lambda}{\omega^2} \int_{\tau_1}^{\tau_2}\!d\tau\;Q(\tau)\sin^2(\omega\tau/2)$ .

For very small frequencies,

$\omega \ll \frac{\pi}{\tau_2}$

the sin^2 function can be replaced by the square of its argument,

$\sin^2(\omega\tau/2)\approx (\omega\tau/2)^2$

yielding

$P_f(\omega\rightarrow 0) = 2\lambda \int_{\tau_1}^{\tau_2}\!Q(\tau)\tau^2d\tau=2\lambda c_1$

where the remaining integral gives just a constant factor c_1.

For very high frequencies,

$\omega \gg \frac{\pi}{\tau_1}$

the sin^2 is a rapidly oscillating function, multiplied by a slowly varying Q(tau)-function. It can then be replaced by its mean, averaged over one oscillation period:

$\sin^2(\omega\tau/2)\approx \frac{1}{\pi} \int_0^{\pi}\! \sin^2(x) dx = \frac{1}{2}$

We thus get

$P_f(\omega\rightarrow \infty) = \frac{8\lambda}{\omega^2} \; \frac{1}{2} \int_{\tau_1}^{\tau_2}\!Q(\tau)d\tau = \frac{4\lambda c_2}{\omega^2}$

where the remaining integral gives another constant factor c_2.

In conclusion, the PSD of step-like random processes with a limited range of step durations is constant (white noise like) for small frequencies (long times) and decays like 1/w^2 for large frequencies (short times).

Accordingly, if the above step-like process would correspond to the force fluctuations within an elastic medium, the bead MSD would have an asymptotic plateau for times much longer than the upper time constant tau_2. Unfortunately, the interesting w^-2 part of the force fluctuations at short times is not easily observable due to the noise floor.

The model could easily be extended to include steps of varying height as well. This would, however, not change the asymptotics of the power spectrum.

The above calculation does, of course, not prove that all processes with a w^-2 power spectrum are step-like, but at least the comment in the Bursac paper appears more reasonable now.

C.M.'s Science Blog

Wednesday, April 16, 2008

[MT 4] PSD of step-like random processes

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