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A Fourier sum may be written
|
(6) |
where the complex value
is related to the real frequency
by .
This Fourier sum is a way of building
a continuous function of
from discrete signal values 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,
.
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 ,
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 .
If we choose the base frequency and hence 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
and , to vanish.
Every element (, for example,
or ) 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, should be the -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 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
2013-01-06