The Fast Fourier Transform (FFT)
5. Fast Fourier Transform (FFT) Definition (Piecewise Continuous). The function interval , if there exists values continuous in each of the open intervals hand limits at each of the values , for Definition (Fourier Series). If , then the Fourier Series
is piecewise continuous on the closed with such that f is , for and has left-hand and right.
is periodic with period for is
and is piecewise continuous on
, are given by the so-called Euler's formulae:
where the coefficients
, and . Theorem (Fourier Expansion). Assume that are piecewise continuous on .
is the Fourier Series for . If , then is convergent for all
The relation of discontinuity of
where
holds for all , then
is continuous. If
is a point
, where denote the left-hand and right-hand limits, respectively. With this understanding, we have the Fourier Series expansion: . Definition (Fourier Polynomial). If on , then the Fourier Polynomial
is periodic with period and is piecewise continuous for of degree m is ,
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The Fast Fourier Transform (FFT)
where the coefficients
are given by the so-called Euler's formulae: ,
and . is periodic with period Example 1. Assume that by for . Find the Fourier polynomial of degree n = 5. Solution 1.
, i.e.
, and is defined
The Fast Fourier Transform for data. The FFT is used to find the trigonometric polynomial when only data points are given. We will demonstrate three ways to calculate the FFT. The first method involves computing sums, similar to "numerical integration," the second method involves "curve fitting," the third method involves "complex numbers." Computing the FFT with sums. Given data points
where
and
over [0,2L] where
. Also given that , to that the data is periodic with period construct the FFT polynomial over [0,2L] of degree m.
for . We shall
, based on
The abscissa's form n subintervals of equal width are for and for The construction is possible provided that
. .
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. The coefficients
The Fast Fourier Transform (FFT)
Remark. Notice that the sums
involve only
ordinates
.
Example 2. Given the 12 equally spaced data points
which can be extended periodically over , if we define . Find the Fourier polynomial of degree n = 5 for the 12 equally spaced points over interval Use numerical sums to find the coefficients. Solution 2.
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.
The Fast Fourier Transform (FFT)
Example 1. Assume that is periodic with period by for . Find the Fourier polynomial of degree n = 5. Solution 1.
, i.e.
We need to define this function individually in each sub-interval.
Now plot the function and the Fourier polynomial.
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, and is defined
The Fast Fourier Transform (FFT)
Remark. Observe that the Fourier polynomial has period
.
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The Fast Fourier Transform (FFT)
Example 2. Given the 12 equally spaced data points which can be extended periodically over , if we define . Find the Fourier polynomial of degree n = 5 for the 12 equally spaced points over interval Use numerical sums to find the coefficients. Solution 2. Note. The data are computed using the function
The ordinates
for
We can adjust the subscript so that the math
.
, which is the function that was used in examples 1. Construct the data points to be used.
are used in computing the sums for constructing the coefficients
for
,
are used in computing the sums for constructing the coefficients
Remark. Notice that precisely 12 data points are used in computing the coefficients, and a point corresponding to Construct the Fourier polynomial using the coefficients
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and the Fourier polynomial.
and
.
is not used.
,
.
The Fast Fourier Transform (FFT)
Caveat. Since the data has period , if data points were used at both end of the interval then the "numerical sums" would weight the endpoints twice, whereas all other function values would have weight 1. If 13 data points were used which included the right end point, then a wrong answer will result, and spurious terms appear in the trigonometric polynomial. Look at the wrong answer. Wrong Answer. Observe. The upper limit of summation has been changed from
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to the wrong value
.
