Waves and Fourier sums

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## FT as an invertible matrix

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)

 Waves and Fourier sums

Next: The Nyquist frequency Up: FOURIER TRANSFORM Previous: FOURIER TRANSFORM

2013-01-06