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Understanding Fourier Transforms: A Comprehensive Guide

Learn about the various equations and approaches to Fourier Transforms, analyzing signals across a continuum, resolving spectral components, and understanding the inverse Fourier Transform. Explore characteristics of the Continuous and Discrete Fourier Transforms, including Nyquist frequency and Spectrum Analyzer approach.

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Understanding Fourier Transforms: A Comprehensive Guide

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  1. Unit 5 The Fourier Transform

  2. Courtesy of Professor Alan M. Nathan,University of Illinois at Urbana-Champaign

  3. Introduction Many, many different equations for Fourier Transforms . . . . . . . . 1. Mathematician Approach 2. Engineering Approach Even within each approach there are many equations.

  4. Continuum of Signals • Stationary vibration signals can be placed along a continuum in terms of the their qualitative characteristics • A pure sine oscillation is at one end of the continuum • A form of broadband random vibration called white noise is at the other end • Reasonable examples of each extreme occur in the physical world • Most signals, however, are somewhere in the middle of the continuum

  5. Frequency Example

  6. Resolving Spectral Components • The time history appears to be the sum of several sine functions • What are the frequencies and amplitudes of the components? • Resolving this question is the goal of this Unit

  7. Equation of Previous Signal At the risk of short-circuiting the process, the equation of the signal in Figure 1 is The signal thus consists of three components with frequencies of 10, 16, and 22 Hz, respectively. The respective amplitudes are 1.0, 1.5, and 1.2 G. In addition, each component could have had a phase angle. In this example, the phase angle was zero for each component. Thus, we seek some sort of "spectral function" to display the frequency and amplitude data. Ideally, the spectral function would have the form shown in the next figure.

  8. Textbook Fourier Transform The Fourier transform X(f) for a continuous time series x(t) is defined as where - < f <  The Fourier transform is continuous over an infinite frequency range. Note that X(f) has dimensions of [amplitude-time].

  9. Textbook Inverse Fourier Transform The inverse transform is Thus, a time history can be calculated from a Fourier transform and vice versa.

  10. Magnitude and Phase Note that X(f) is a complex function. It may be represented in terms of real and imaginary components, or in terms of magnitude and phase. The conversion to magnitude and phase is made as follows for a complex variable V.

  11. Real Time History • Practical time histories are real • Note that the inverse Fourier transform calculates the original time history in a complex form • The inverse Fourier transform will be entirely real if the original time history was real, however.

  12. Fourier Transform of a Sine Function The transform of a sine function is purely imaginary. The real component, which is zero, is not plotted. f=0

  13. Characteristics of the Continuous Fourier Transform • The real Fourier transform is symmetric about the f = 0 line • The imaginary Fourier transform is anti-symmetric about the f = 0 line

  14. Characteristics of the ContinuousFourier Transform • This Unit so far has presented Fourier transforms from a mathematics point of view • A more practical approach is needed for engineers . . .

  15. Discrete Fourier Transform • An accelerometer returns an analog signal • The analog signal could be displayed in a continuous form on a traditional oscilloscope • Current practice, however, is to digitize the signal, which allows for post-processing on a digital computer • Thus, the Fourier transform equation must be modified to accommodate digital data • This is essentially the dividing line between mathematicians and engineers in regard Fourier transformation methodology

  16. Discrete Fourier Transform Equations The Fourier transform is The corresponding inverse transform is

  17. Discrete Fourier Transform Relations Note that the frequency increment f is equal to the time domain period T as follows The frequency is obtained from the index parameter k as follows where k is the frequency domain index

  18. Nyquist Frequency • The Nyquist frequency is equal to one-half of the sampling rate • Shannon’s sampling theorem states that a sampled time signal must not contain components at frequencies above the Nyquist frequency • Otherwise an aliasing error occurs

  19. Half AmplitudeDiscrete This is the imaginary component of the transform. The real component is zero.

  20. Previous Fourier Transform Plot • Note that the sine wave has a frequency of 1 Hz • The total number of cycles is 512, with a resulting period of 512 seconds • Again, the Fourier transform of a sine wave is imaginary and anti-symmetric • The real component, which is zero, is not plotted

  21. Spectrum Analyzer Approach • Spectrum analyzer devices typically represent the Fourier transform in terms of magnitude and phase rather than real and imaginary components • Furthermore, spectrum analyzers typically only show one-half the total frequency band due to the symmetry relationship • The spectrum analyzer amplitude may either represent the half-amplitude or the full-amplitude of the spectral components • Care must be taken to understand the particular convention of the spectrum analyzer • Note that the half-amplitude convention has been represented in the equations thus far

  22. Full Amplitude Fourier Transform The full-amplitude Fourier transform magnitude would be calculated as

  23. Full Amplitude Fourier Transform • Note that k = 0 is a special case • The Fourier transform at this frequency is already at full-amplitude • For example, a sine wave with an amplitude of 1 G and a frequency of 1 Hz would simply have a full-amplitude Fourier magnitude of 1 G at 1 Hz, as shown in the next figure

  24. Full Amplitude Fourier TransformTime History Example

  25. Full Amplitude One-Sided

  26. Unit 5 Exercise 1 A time history has a duration of 20 seconds. What is the frequency resolution of the Fourier transform?

  27. Unit 5 Exercise 2 Identify the natural frequencies in the following files from previous exercises by taking a Fourier transform: tuning_fork.txt, channel.txt, drop.txt Use: vibrationdata.m > Fourier transform > External ASCII text file (each of the three above txt files, one at a time) Mean Removal=yes Window=rectangular Max Frequency (Hz): tuning_fork.txt & channel.txt 1000 Hz, drop.txt 100 Hz Y-axis Label: tuning_fork.txt - unscaled sound pressure, channel.txt & drop.txt - Accel (G)

  28. Unit 5 Exercise 3 Use vibrationdata.m to generate a sine function: Amplitude = 2 phase = 0 deg Duration = 50 sec sample rate = 200 Frequency = 5 Hz Save signal to Matlab workspace – pick you own output name Then take Fourier transform using: vibrationdata.m Use max frequency = 50 Hz Verify spectral line (5 Hz, 2 G)

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