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The Unevenly Spaced Lomb-Fourier Transform

Syntax	[a, b] = lombft(t, z, f)
See Also	dft , fft

Description
Computes Lomb-Fourier transform of the data sequence
     { (t , z ): j = 1, ... , N }
         j   j
where t and z are column vectors with length N. The vector t is integer, real, or double-precision and z is integer, real, double-precision or complex. The column vector f is integer, real, or double-precision and all of its elements are greater than zero. The Lomb-Fourier transform solves the following problem:
                      N                                          2
                     ---  |                    2 pi i f(j) t(j) |
zhat(f ) =  argmin   >    | z   - (a + i b) * e                 |
      j   (a + i b)  ---  |  j                                  |
                     j=1
where i denotes the square root of minus one. The return values a and b are double-precision column vectors with the same length as f and such that
     zhat(f ) = a(f ) + i b(f )
           j       j         j

Example
The following program plots the square of the absolute value of the Lomb-Fourier transform of a sine wave on unevenly spaced data:

#
clear
#
ranseed
N      = 100
Tmax   = sqrt(double(N))
f0     = sqrt(double(N))
t      = Tmax * rand(N, 1)
z      = sin(2 * PI * f0 * t)
f      = seq(N / 2) 
#
[a, b] = lombft(t, z, f)
gplot(f, a^2 + b^2)
gxaxis("log")
gtitle("Lomb-Fourier Transform Squared")