User:Jorge Stolfi/Temp/White noise simulation and whitening

Random signal transformations
We cannot extend the same two concepts of simulating and whitening to the case of continuous time random signals or processes. For simulating, we create a filter into which we feed a white noise signal. We choose the filter so that the output signal simulates the 1st and 2nd moments of any arbitrary random process. For whitening, we feed any arbitrary random signal into a specially chosen filter so that the output of the filter is a white noise signal.

Simulating a continuous-time random signal


White noise can simulate any wide-sense stationary, continuous-time random process $$x(t) : t \in \mathbb{R}\,\!$$ with constant mean $$\mu$$ and covariance function


 * $$K_x(\tau) = \mathbb{E} \left\{ (x(t_1) - \mu) (x(t_2) - \mu)^{*} \right\} \mbox{ where } \tau = t_1 - t_2$$

and power spectral density
 * $$S_x(\omega) = \int_{-\infty}^{\infty} K_x(\tau) \, e^{-j \omega \tau} \, d\tau.$$

We can simulate this signal using frequency domain techniques.

Because $$K_x(\tau)$$ is Hermitian symmetric and positive semi-definite, it follows that $$S_x(\omega) $$ is real and can be factored as
 * $$S_x(\omega) = | H(\omega) |^2 = H(\omega) \, H^{*} (\omega) $$

if and only if $$S_x(\omega)$$ satisfies the Paley-Wiener criterion.
 * $$ \int_{-\infty}^{\infty} \frac{\log (S_x(\omega))}{1 + \omega^2} \, d \omega < \infty $$

If $$S_x(\omega)$$ is a rational function, we can then factor it into pole-zero form as
 * $$S_x(\omega) = \frac{\Pi_{k=1}^{N} (c_k - j \omega)(c^{*}_k + j \omega)}{\Pi_{k=1}^{D} (d_k - j \omega)(d^{*}_k + j \omega)}.$$

Choosing a minimum phase $$H(\omega)$$ so that its poles and zeros lie inside the left half s-plane, we can then simulate $$x(t)$$ with $$H(\omega)$$ as the transfer function of the filter.

We can simulate $$x(t)$$ by constructing the following linear, time-invariant filter
 * $$\hat{x}(t) = \mathcal{F}^{-1} \left\{ H(\omega) \right\} * w(t) + \mu $$

where $$w(t)$$ is a continuous-time, white-noise signal with the following 1st and 2nd moment properties:
 * $$ \mathbb{E}\{w(t)\} = 0$$
 * $$ \mathbb{E}\{w(t_1)w^{*}(t_2)\} = K_w(t_1, t_2) = \delta(t_1 - t_2).$$

Thus, the resultant signal $$\hat{x}(t)$$ has the same 2nd moment properties as the desired signal $$x(t)$$.

Whitening a continuous-time random signal


Suppose we have a wide-sense stationary, continuous-time random process $$x(t) : t \in \mathbb{R}\,\!$$ defined with the same mean $$\mu$$, covariance function $$K_x(\tau)$$, and power spectral density $$S_x(\omega)$$ as above.

We can whiten this signal using frequency domain techniques. We factor the power spectral density $$S_x(\omega)$$ as described above.

Choosing the minimum phase $$H(\omega)$$ so that its poles and zeros lie inside the left half s-plane, we can then whiten $$x(t)$$ with the following inverse filter
 * $$H_{inv}(\omega) = \frac{1}{H(\omega)}.$$

We choose the minimum phase filter so that the resulting inverse filter is stable. Additionally, we must be sure that $$H(\omega)$$ is strictly positive for all $$\omega \in \mathbb{R}$$ so that $$H_{inv}(\omega)$$ does not have any singularities.

The final form of the whitening procedure is as follows:
 * $$w (t) = \mathcal{F}^{-1} \left\{ H_{inv}(\omega) \right\} * (x(t) - \mu)$$

so that $$w(t)$$ is a white noise random process with zero mean and constant, unit power spectral density


 * $$S_{w}(\omega) = \mathcal{F} \left\{ \mathbb{E} \{ w(t_1) w(t_2) \} \right\} = H_{inv}(\omega) S_x(\omega) H^{*}_{inv}(\omega) = \frac{S_x(\omega)}{S_x(\omega)} = 1.$$

Note that this power spectral density corresponds to a delta function for the covariance function of $$w(t)$$.
 * $$K_w(\tau) = \,\!\delta (\tau)$$