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Contents

  1. 1/f noise in Power spectrum Leads to Long-memory effects
  2. Superposition of Random(Poisson) processes Lead to

1/f noise in Power spectrum Leads to Long-memory effects

Whilst it is relatively simple to yield power-law (scale-free) relationships in the correlation function (see below), it is of particular interest if the relationship is of the form, 

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.. so called '1/f noise', 'flicker noise', or 'pink noise' (see w: link on parent page). The reasoning is as follows:

  1. We have the temporal correlation function: 

    latex error! exitcode was 2 (signal 0), transscript follows:

    and the cosine transformed power spectrum 

    latex error! exitcode was 2 (signal 0), transscript follows:
    ,

  2. Now, suppose 

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    and 

    latex error! exitcode was 2 (signal 0), transscript follows:
    ,

  3. Notice that given 

    latex error! exitcode was 2 (signal 0), transscript follows:

    on the LHS we have 

    latex error! exitcode was 2 (signal 0), transscript follows:
    for the power spectrum, whilst on the right hand side we have 
    latex error! exitcode was 2 (signal 0), transscript follows:
    which is a linear function of 
    latex error! exitcode was 2 (signal 0), transscript follows:
    . Now 
    latex error! exitcode was 2 (signal 0), transscript follows:
    is the interval of correlation, or in other words the period of correlation. Hence, since we assume 
    latex error! exitcode was 2 (signal 0), transscript follows:
    , we are actually assuming, 
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    ;

  4. Combine this with the cosine transform of f in the power spectrum and we have that (LHS ~ RHS):

    latex error! exitcode was 2 (signal 0), transscript follows:

That is, if we find that 

latex error! exitcode was 2 (signal 0), transscript follows:
(as in flicker noise), then this implies that 
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, which causes the assumed relationship for 
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to break down and instead be replaced by slow logarithmic decay. Very low-frequency signals have significant impact on inter-temporal events. Hence, the interest in 1/f distributions in the power function.

Ref: p.9 of Jensen

Superposition of Random(Poisson) processes Lead to

Leviathan: SciNotes/SelfOrganizedCriticality/FurtherNotes (last edited 2010-08-13 23:44:17 by sangus)