User:Orcioni/sandbox

Power and variance
This section want to answer to this question:

Relation of variance and power of signal

Let's start with some definitions.

If we are speaking of the variance of $$X$$, $$X$$ should be a random variable (r.v.), but since it is also indexed by a real number t, we should speak of a stochastic process (s.p.) (i.e. a r.v. indexed by a real variable).

For better understanding some of the following concepts we need to make explicit the random nature of $$X$$, by introducing the probability space $$(\Omega,

\mathcal{F}, P)$$

where

$$\Omega$$ is the sample space,

$$\mathcal{F} $$ is the sigma-algebra of events

$$P$$ is a probability function

and defining $$X$$ as a function $$X: \Omega\rightarrow \R $$.

So by writing $$X = X(\omega, t)$$; we can expressly indicate $$X(\omega_1, t)$$ as a trajectory, $$X(\omega_2, t)$$ as another trajectory, and so on. We can also omit the explicit dependence from the space of samples by writing $$X(t)$$.

Given the stochastic process $$X(t)$$, we can define its mean value as

$$M_X(t) = E[X(t)]$$,

the variance as

$$V_X(t) = E[ (X(t) - M_X(t))^2 ]$$,

and the autocorrelation function as

$$R_{XX}(t, s) = E[ X(t)X(s) ]$$

where $$E$$ is the expected value, i.e. the mean value performed in the space of event (we will omit by now how to compute it).

We can note that the mean value and the variance may depends on time. To avoid this we need a stationary r.v.

A stochastic process is said second order stationary or weak-sense stationary if

$$E[X(t)] = M_X =$$ constant;

$$E[ X(t)X(s) ] = R_{XX}(t - s) = R_{XX}(\tau)$$

in this case

$$ R_{XX}(0) = V_X $$ if  $$ M_X = 0 $$.

Now we have constant mean value and variance.

We can introduce the mean and autocorrelation in time, instead of in probability space, as

$$M_x(\omega) = \lim_{T\rightarrow\infty} \int_{-T/2}^{T/2} X(\omega,t)\text{d} t $$

$$ R_{xx}(\omega, \tau) = \lim_{T\rightarrow\infty} \int_{-T/2}^{T/2} X(\omega,t)X(\omega,t+\tau) \text{d} t$$.

as made explicit by $$\omega$$ the temporal mean and autocorrelation are r.v.

In the case of a weak-sense stationary s.p. we will have

$$ E[ M_x(\omega)] = M_X $$

$$ E[ R_{xx}(\omega, \tau) ] = R_{XX}(\tau) $$.

Only the expected values of temporal mean and autocorrelation are equal to their counterparts computed in the sample space.

For a time series $$s(t)$$ the power can be defined as

$$P_s = \lim_{T\rightarrow\infty} \int_{-T/2}^{T/2} s(t)^2\text{d} t $$

so in the case of the s.p. $$X(\omega, t)$$ we will have

$$P_x(\omega) = \lim_{T\rightarrow\infty} \int_{-T/2}^{T/2} X(\omega,t)^2\text{d} t $$

and so

$$P_x(\omega) = R_{xx}(\omega, 0) $$

Now we just have to get rid of the fact that also the power is a r.v. This is why ergodicity helps us.

A s.p. $$X(t)$$ is ergodic if

$$ M_x = M_X $$

$$ R_{xx}(\tau) = R_{XX}(\tau) $$

except for a set of null measure. We can note that, with respect to the weak-stationarity, these equality are valid without the expected values.

So for an ergodic s.p. we can conclude that

$$ R_{XX}(0) = P_x $$.

This means that making the temporal mean and autocorrelation for a non-ergodic signal, these are dependent on the realization used.

Instead a realization of an ergodic s.p. (and also the output of an ergodic system) takes all possible values, as well as the other realizations, allowing to determine ensemble characteristic with the observation of only one realization during time.

As a conclusion we can say that the variance is equal to the power of a s.p. if the mean value is zero and the s.p. is ergodic.

Neural decoding
How individual neurons or networks encode information is the subject of numerous studies and research. In central nervous system it mainly happens by altering the spike firing rate (frequency encoding) or relative spike timing (time encoding). Time encoding consists of altering the random inter-spikes intervals (ISI) of the stochastic impulse train in output from a neuron. Homomorphic filtering was used in this latter case to obtain ISI variations from the power spectrum of the spike train in output from a neuron with or without the use of neuronal spontaneous activity. The ISI variations were caused by an input sinusoidal signal of unknown frequency and small amplitude, i.e. not sufficient, in absence of noise to excite the firing state. The frequency of the sinosoidal signal was recovered by using homomorphic filtering based procedures.