6. Fourier Series and Fourier Transform
(1) Fourier series Fourier coefficients
(2) Fourier series with period T Fourier coefficients
Replacing x in (1) above with x = 2πft = ωt and a period of 2π = T;
(3) Complex Fourier series and complex Fourier coefficients
Euler's formula
Considering this, (2) is:
<Supplement>
On the other hand, f(t) is a real function
Therefore
The complex Fourier coefficients of a real function are Cm. Once you calculate m = 1 to M, all you need to do is take the complex conjugate Cm*.
Furthermore, the real and imaginary parts of Cm are separated into positive and negative frequency components by complex Fourier series expansion, and their values are 1/2 of the Fourier coefficients am and bm.
(4) Fourier transform, inverse Fourier transform
By expanding T to infinity, this method allows even non-periodic waveforms (aperiodic functions) to be expanded into Fourier series.
The formula for calculating the Fourier coefficients Cm is also considered a function, and this is called the Fourier transform. f(t) is considered a function that performs the inverse calculation of the Fourier transform, and is called the inverse Fourier transform. The Fourier transform and inverse Fourier transform are considered as a pair of equations.
Fourier transform
Inverse Fourier Transform
The real part Re and the imaginary part Im are
(5) Discrete Fourier Transform Inverse Fourier Transform
For practical purposes, T will be finite. Also, in an FFT analyzer, a finite number N of data samples are taken from the AD-to-D conversion, and the Fourier transform is performed. Since sampling results in discrete values (non-continuous, discrete values), if we use N instead of T (where h is the sampling interval time), we consider the k-th term of the discrete Fourier transform and the inverse Fourier transform (since the frequency is kω).
The real part Re and the imaginary part Im are
(6) Discrete Fourier series, Fourier coefficients
To find the Fourier coefficients from sampled discrete values, expand (5) above and calculate using the following formula. The relationship between the Fourier series, Fourier coefficients, and their real and imaginary parts is shown for comparison.
<Fourier series, Fourier coefficients>
The Fourier coefficient of the k-th term is:
If we consider the Fourier transform of the k-th term as 2/N such that it matches the Fourier coefficients an and bn, then:
The formula for the k-th term's Fourier coefficient, or the formulas Re [F (kω)] and Im [F (kω)], can be used to calculate it in a spreadsheet program.
In ω = 2πfo, fo is the reciprocal of the period T. T = Nh, where N is the number of samples and h is the sample time interval. Therefore, if h is constant, N is related to the frequency resolution fo.
(7) Example of calculating Fourier coefficients
In the previous step, we combined the cosine functions of 10Hz and 20Hz;
I've created some sample data.
Let's create time-series data from the equation f(t) and calculate the Fourier coefficients of f(t) at 10Hz by substituting the following values.
- Sample frequency 1000Hz,
- Sample time h = 1/1000 (s),
- Sample size N = 2048
- t = nh
- n = 0、1、2、・・・2047
as
The following are calculated in order in columns A through G: n, nh, f (nh), cos (20πnh), sin (20πnh), f (nh) cos (20πnh), and f (nh) sin (20πnh).
and;
The spectra a and b at 10Hz were obtained for N = 500 (n = 0 to 499) and N = 2048 (n = 0 to 2047).
To find the case:
If N = 500 (Nh = 0.5s), then a = 0.866 and b = -0.500.
If N = 2048 (Nh = 2.048s), then a = 0.853 and b = -0.508.
Calculation process in Microsoft Excel
From the equation f(t), the spectrum at 10 Hz is:
a = cos 30 deg = 0.866 b = sin 30 deg = 0.5
Since we know this, the results match when N = 500, but there is a slight difference when N = 2048. This is due to the "periodicity" assumption of the Fourier series.
When N = 500 (0.5 s between n = 0 and 499), it corresponds to exactly 5 periods of 10 Hz, but when N = 2048, it is not an integer multiple of 10 Hz.
As shown in Figure A, if N is chosen so that it equals exactly one period (or an integer multiple thereof), the same waveform can be drawn continuously. However, for N that does not equal one period (or an integer multiple thereof), the waveform becomes discontinuous (Figure B).
If there are discontinuities, the waveform becomes continuous by taking the midpoint between them, and the approximate calculation is performed using a waveform that is smoothly connected at the midpoint.
Now, the Fourier coefficients for 10 Hz;
a = 0.866 b = -0.5
twist,
This is the result. Furthermore, this equation also has the following from Re = 0.866 and Im = -b = 0.5:
It can be seen that this is the case.
Similarly, when we find the other frequency components, the observed waveform f(t) is:
This is the result.
This is the concept of an FFT analyzer.
(Excerpt from the email newsletter issued on May 24, 2007)
