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A **Fourier sum** may be written

| |
(6) |

where the complex value *Z*
is related to the real frequency
by .This Fourier sum is a way of building
a continuous function of from discrete signal values *b*_{t} in the time domain.
Here we specify both time and frequency domains by a set of points.
Begin with an example of a signal
that is nonzero at four successive instants,
( *b*_{0}, *b*_{1}, *b*_{2}, *b*_{3}).
The transform is
| |
(7) |

The evaluation of this polynomial can be organized as a matrix times a vector,
such as
| |
(8) |

Observe that the top row of the matrix evaluates the polynomial at *Z*=1,
a point where also
.The second row evaluates ,where is some base frequency.
The third row evaluates the Fourier transform for ,and the bottom row for .The matrix could have more than four rows for more frequencies
and more columns for more time points.
I have made the matrix square in order to show you next
how we can find the inverse matrix.
The size of the matrix in (8) is *N*=4.
If we choose the base frequency and hence *W* correctly,
the inverse matrix will be
| |
(9) |

Multiplying the matrix of
(9) with that of
(8),
we first see that the diagonals are +1 as desired.
To have the off diagonals vanish,
we need various sums,
such as 1+*W* +*W*^{2}+*W*^{3}
and 1+*W*^{2}+*W*^{4}+*W*^{6}, to vanish.
Every element (*W*^{6}, for example,
or 1/*W*^{9}) is a unit vector in the complex plane.
In order for the sums of the unit vectors to vanish,
we must ensure that the vectors pull symmetrically away from the origin.
A uniform distribution of directions meets this requirement.
In other words, *W* should be the *N*-th root of unity, i.e.,
| |
(10) |

The lowest frequency is zero, corresponding to the top row of
(8).
The next-to-the-lowest frequency we find by setting *W* in
(10) to .So ; and
for (9) to be inverse to (8),
the frequencies required are

| |
(11) |

** Next:** The Nyquist frequency
** Up:** FOURIER TRANSFORM
** Previous:** FOURIER TRANSFORM
Stanford Exploration Project

12/26/2000