All,

To form the "analytic signal" of your original time domain record, X(t), 
you need an FFT algorithm that gives positive and negative frequencies.  
Form the FFT of X, zero the negative frequency components, divide the 
zero frequency value by 2, and transform back to the time domain.  The 
result, Y, is the Hilbert transform of X, the quadrature or 90 degree 
advanced version of X.  The analytic signal is a complex time domain 
signal where for each time sample the real part is X and the imaginary 
part is Y.   The envelop of X is the (real) function formed by taking 
the modulus of the analytic signal.  That is, the envelop at each time 
is the square root of X squared plus Y squared.

The envelop is the instantaneous amplitude of X.  The instantaneous 
phase of X, theta, is arctan(Y/X).  The instantaneous frequency is the 
derivative of theta with respect to time.  These three products are used 
extensively in processing of reflection seismic data, although the 
envelop is by far the most useful.  If X is a pure sinusoid these 
products reduce to the expected values.

A search of the internet for "analytic signal" yields lots of reference 
material.  The classic reference for reflection seismic work is Taner, 
M. T., et al., 1979, Complex Seismic Trace Analysis:  Geophysics, 44, 
1041 - 1063.  It has interesting examples in color.

Chuck Burch
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