detrend
=======

remove constant, linear or piecewise linear trend from a vector



Calling Sequence
~~~~~~~~~~~~~~~~


::

    y = detrend(x)
    y = detrend(x,flag)
    y = detrend(x,flag,bp)




Arguments
~~~~~~~~~

:x vector or matrix of real or complex numbers (the signal to treat)
: :flag a string equal to "linear" (or "l") for linear or piecewise
  linear treatment or "constant" (or "c") for constant treatment.
: :bp the breakpoints to provide if you want a piecewise linear
  treatment.
: :y output, the signal x with the trend removed from it.
:



Description
~~~~~~~~~~~

This function removes the constant or linear or piecewise linear trend
from a vector `x`. In general this can be useful before a fourier
analysis. If `x` is matrix this function removes the trend of each
column of x.

When `flag` = "constant" or "c" `detrend` removes the constant trend
(simply the mean of the signal) and when `flag` = "linear" or "l" the
function removes the linear trend. By adding a third argument `bp` it
is possible to remove a continuous *piecewise linear* trend. Note that
the "instants" of the signal `x` goes from 0 to m-1 (m = length(x) if
x is a vector and m = size(x,1) in case x is a matrix). So the
breakpoints `bp(i)` must be reals in *[0 m-1]* (breakpoints outside
are simply removed from `bp` vector).

The trend is got by a least square fit of `x` on the appropriate
function space.



Examples
~~~~~~~~


::

    // example #1
    t = `linspace`_(0,16*%pi,1000)';
    x = -20 + t + 0.3*`sin`_(0.5*t) + `sin`_(t) + 2*`sin`_(2*t) + 0.5*`sin`_(3*t); 
    y = detrend(x);
    `clf`_()
    `plot2d`_(t,[x y],style=[2 5])
    `legend`_(["before detrend","after detrend"]);
    `xgrid`_()
    
    // example #2
    t = `linspace`_(0,32*%pi,2000)';
    x = `abs`_(t-16*%pi) + 0.3*`sin`_(0.5*t) + `sin`_(t) + 2*`sin`_(2*t) + 0.5*`sin`_(3*t); 
    y = detrend(x,"linear",1000);
    `clf`_()
    `plot2d`_(t,[x y],style=[2 5])
    `legend`_(["before detrend","after detrend"]);
    `xgrid`_()