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

Random signal transformations

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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

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White noise fed into a linear, time-invariant filter to simulate the 1st and 2nd moments of an arbitrary random process.

White noise can simulate any wide-sense stationary, continuous-time random process   with constant mean   and covariance function

 

and power spectral density

 

We can simulate this signal using frequency domain techniques.[clarification needed]

Because   is Hermitian symmetric and positive semi-definite, it follows that   is real and can be factored as

 

if and only if   satisfies the Paley-Wiener criterion.

 

If   is a rational function, we can then factor it into pole-zero form as

 

Choosing a minimum phase   so that its poles and zeros lie inside the left half s-plane, we can then simulate   with   as the transfer function of the filter.

We can simulate   by constructing the following linear, time-invariant filter

 

where   is a continuous-time, white-noise signal with the following 1st and 2nd moment properties:

 
 

Thus, the resultant signal   has the same 2nd moment properties as the desired signal  .

Whitening a continuous-time random signal

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An arbitrary random process x(t) fed into a linear, time-invariant filter that whitens x(t) to create white noise at the output.

Suppose we have a wide-sense stationary, continuous-time random process   defined with the same mean  , covariance function  , and power spectral density   as above.

We can whiten this signal using frequency domain techniques. We factor the power spectral density   as described above.

Choosing the minimum phase   so that its poles and zeros lie inside the left half s-plane, we can then whiten   with the following inverse filter

 

We choose the minimum phase filter so that the resulting inverse filter is stable. Additionally, we must be sure that   is strictly positive for all   so that   does not have any singularities.

The final form of the whitening procedure is as follows:

 

so that   is a white noise random process with zero mean and constant, unit power spectral density

 

Note that this power spectral density corresponds to a delta function for the covariance function of  .