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A Scrapbook of Digital Signal Processing

Ways of Interpreting the Fourier Transform

There are several ways to view the Fourier Transform. Perhaps the most common are as follows:

CO-ORDINATE TRANSFORMATION (ROTATION) The butterfly used to compute Fourier Transforms can be written as follows:

(a + j b) (cos + j sin) = (a cos - b sin) + j (a sin + b cos)

which is identical to the formula for the rotation of axes, or co-ordinate transformation. The interpretation of this is that each point in a time series is rotated back to a common (base) set of axes in the imaginary plane, prior to being summed with the other rotated points in the series. For each of the Fourier coefficients, the degree of rotation is determined by the frequency corresponding to that coefficient. The summation of these rotated points produces the Fourier coefficient. If the frequency content of the time series corresponds to that of the rotation, the sum will be large. If not, it will be small.

CROSS-CORRELATION Using the following orthogonality relationships (where the notation [] indicates an averaging process):

[ sin(w1t) sin(w2t) ] = [ cos(w1t) cos(w2t) ] = 1/2 for w1=w2, and 0 for w1/=w2
[ sin(w1t) cos(w2t) ] = 0 for all w1 and w2

Each of the Fourier coefficients can be viewed as a cross-correlation between a given time series and a sine or cosine of equal length and of frequency correspoding to that particular Fourier coefficient. Note that the average value of the cross-correlation is zero unless the two frequencies are the same. The averaging process is the Fourier summation. The average will be large or small depending on the degree of correlation.

FILTER COMB The power spectrum output of a Fourier transform can be viewed as a comb of narrow-band filters, the bandwidth of each of which is proportional to the inverse of the time duration of the period being observed (frequency resolution equals inverse of observation time).

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