What is the Fourier Transform?

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The Fourier Transform is an important mathematical tool in many fields including vibration analysis, audio engineering, and image processing.

 

Why is the Fourier Transform important? The Fourier Transform is used to transform a time domain signal into the frequency domain. This often makes the signal easier to understand.

 

This article will provides a brief history, some background, examples, and applications of the Fourier Transform.

 

History

 

From its beginnings for heat flow analysis, to its wide spread use in many different fields today, the Fourier Transform dates back more than 200 years.  

 

Origins

 

The Fourier Transform takes its name from the French mathematician Jean-Baptiste Joseph Fourier (Figure 1). In 1807 he found that arbitrary functions could be written as a summation of sines and cosines.

 

fourier.pngFigure 1: Jean-Baptiste Joseph Fourier (1768-1830)

Fourier published his findings as part of The Analytical Theory of Heat in 1822.

 

Later it was discovered that it was possible to determine the amplitude of the individual sine and cosine waves making up a Fourier series by using an integral. This became known as the Fourier Transform.

 

The Fourier Transform goes Digital

 

Two types of Fourier Transforms are commonly used today in computer based applications: the Discrete Fourier Transform (DFT) and the Fast Fourier Transform (FFT).

 

1. The Discrete Fourier Transform (DFT) performs a Fourier Transform on a discrete time block. A DFT can be performed on any time signal composed of an arbitrary number of data points.

 

2. The Fast Fourier Transform (FFT) is an algorithm which performs a Discrete Fourier Transform in a computationally efficient manner. It requires a power of two number of samples in the time block being analyzed (e.g. 512, 1024, 2048, and 4096). 

  

For more information about the digital Fourier Transform, see the article Digital Signal Processing: Sampling Rates, Bandwidth, Spectral Lines, and more....

 

Basics of the Fourier Transform

 

Fourier showed that any signal can be represented as a series of sine waves of different amplitude and phase. For example, the signal below (Figure 2) in purple is simply the sum of three sine waves.

 

sine_waves.pngFigure 2: A complex signal (purple, left) is the sum of individual sine waves (red, green, black).The Fourier Transform makes it possible to decompose the original time signal into sinusoids. Each sinusoid has an associated amplitude, phase, and frequency. This is illustrated in the Figure 3 below. 

 

Fourier_3D.pngFigure 3: The Fourier Transform allows a complex time signal (left, right) to be viewed from the frequency domain (right side).

It can be seen in Figure 3 that a complicated looking waveform in time domain can be represented by just three vertical lines in the frequency domain. This simplistic representation in the frequency domain helps in identifying key frequencies.

 

key_fourier1.png

 

No data is lost when moving from the frequency domain to the time domain (or vice versa).

 

Fourier Transform Equation

 

The equation for the Fourier Transform is given in Equation 1:

                                                              Fourier Transform Eqn.pngEquation 1: The Fourier Transform

Where: 

  • S_x(f) is the output of the Fourier Transform in the frequency domain
  • x(t) is the input time domain function
  • 2πf is the frequency in radians per second

 

Complex Numbers

 

The output of a Fourier Transform is a series of complex numbers, each of which corresponds to a frequency, amplitude, and phase on the resulting frequency spectrum.  The complex numbers are of the form a+jb (often referred to as Z). The letter a refers to the real part of the complex number, while b is the imaginary portion. 

 

The complex numbers contain information about the amplitude and phase of the frequency components in the original time signal as shown in Figure 4. The amplitude of the complex number is the hypotenuse. The phase is the angle θ.

 

 

Complex Plane.pngFigure 4: The Complex Plane

Equations 2 and 3 show the relationship between the real and imaginary complex values and the amplitude and phase.

 

amp_and_phase_eqn.pngEquations 2 and 3: Relationship between complex number (a+jb) versus amplitude and phase.

The results of a Fourier Transform can be shown in amplitude and phase (left graph, Figure 5) OR in real and imaginary (right graph, Figure 5).

 

fourier_amp_phase_real_imag.pngFigure 5: The Fourier Transform of a time signal results in a complex spectrum which can be displayed in Amplitude and Phase (left graph) or Real and Imaginary (right graph).

Often, the amplitude and phase representation is used for viewing the spectral information.

 

key_amp_phase.png

  

Because phase is preserved, an inverse Fourier Transform can be performed on the spectral data to restore the original time history.

 

If phase of the spectrum is of no interest, the data can be processed digitally as an Autopower Spectrum.   See the 'Autopower Function...Demystified' knowledge base article for more information.

 

Double Sided Spectrum

 

The Fourier Transform produces a double sided spectrum as shown in Figure 6.

 

double_sided.pngFigure 6: Both the FFT and DFT algorithm produce a double sided spectrum.

A doubled sided spectrum consists of negative and positive frequencies.  Because the limits of the Fourier Transform go from negative to positive infinity in time, so does the frequency range of the resulting spectrum.  The amplitude of the double sided spectrum is half of the peak amplitude in the time domain as shown in Figure 6

 

key_double_side.png

  

By convention, digital data acquisition systems, when performing a FFT or DFT, do not display the negative frequency range.  The amplitude of the positive part of the double sided spectrum is also multiplied by two (except zero Hertz which remains unchanged in amplitude).

 

Example Signals and Their Fourier Transforms

 

Below are some examples of time domain signals and their Fourier Transforms (Figure 7). 

 

Time_Signals_and_Fourier_Transforms.pngFigure 7: Time domain signals (left) and their corresponding Fourier Transforms (left).

Sine Wave

A sine wave is the most fundamental component of a Fourier Transform.  A Fourier Transform of a sine wave produces a single amplitude value with corresponding phase (not pictured) at a single frequency.

 

Damped Transient

If a sine wave decays in amplitude, there is a “smear” around the single frequency. The quicker the decay of the sine wave, the wider the smear. See the article How to Calculate Damping from a FRF for more information.

 

Square Wave

Square wave in frequency domain has odd numbered harmonics that decrease by a fixed amount in amplitude.

 

Impulse

The Fourier Transform of an impulsive signal has a relatively flat amplitude across the frequency spectrum. Impulse signals are often used in modal impact testing because an even distribution of energy throughout the frequency spectrum can excite multiple modes of a structure.

 

Offset

A signal that is a constant value offset from zero amplitude has no frequency content.  In the resulting Fourier Transform, the signal’s amplitude content is at zero Hertz.  A constant value offset signal is often referred to as a DC offset, while signals with dynamic frequency content are considered to have AC content. For more information see the Knowledge Base article 'What is AC and DC coupling'.

 

Random

A random signal has broadband frequency content.  If all frequencies are about the same level, the term white noise is often used to describe the spectrum.

 

Applications of the Fourier Transform

 

The Fourier Transform has a number of different uses from trouble shooting vibration issues to image processing.

 

key_diagnose.png

 

The Fourier Transform can be used to identify the higher frequency components in a signal. These components may pinpoint the cause of unwanted noise or vibration.

 

Some examples follow:

 

Vacuum Cleaner

 

Look at the time series and Fourier Transform of sound pressure data from a problem vacuum cleaner (Figure 8). The vacuum cleaner has a sharp, unpleasant sound when running. The time series does not show any specific issue with this vacuum cleaner.

 vacuum.pngFigure 8: While the time domain signal (top graph) of a vacuum cleaner sound does not indicate an issue, the Fourier Transform (bottom graph) shows a large peak at 5445 Hz.

The FFT of the time signal shows a large peak at 5445 Hz, which is causing the unpleasant sound. This is caused by the fan blade pass. The engineer could address the problem of the noisy fan by adjusting the spacing between the fan blades. Evenly spaced fan blades produce a single pure tone. Unevenly spaced blades will produce multiple tones which are lower in amplitude and distributed over a wider frequency range.

 

Music Analysis

 

When analyzing sound, The amplitude of a frequency spectrum is often displayed in decibels ( a logarithmic quantity).  Decibels is considered a better representation of the human perception of sound than linear amplitude.

 

Consider the Fourier Transform of a rock song versus a pop song. It can be shown that a rock song has more bass (low frequency content) than a pop song (Figure 9).

 

rock_song.pngFigure 9: A Rock Song (red) has higher levels of low frequency than a Pop Song (blue).

The Fourier Transform can detect a "preponderance of bass"! (Hint: 80's movie reference starring Tom Cruise)

 

Bad Bearing

 

The Fourier Transform can be used to identify a potential failure in a bearing. Consider the bearing below (Figure 10). Each component (inner race, outer race, rolling elements) comes in contact with the others at regular intervals during the rotation of the shaft. The condition of the bearing can be monitored with a vibration sensor on its housing.  

 

bearing.pngFigure 10: A bearing can have faults (circled in red) on the outer race, inner race, or rolling elements.The contact between the different parts generates specific frequencies.  In the example of Figure 11, a defect on the outer race of the bearing generates a 119 Hertz frequency. This defect causes the vibration amplitude of the associated frequency to increase versus the baseline condition. 

 

good_versus_bad_bearing.pngFigure 11: Based on the geometry of a bearing, high levels at a specific frequency indicate a failure of a specific component.

Manufacturers of bearings often provide a table of "fault frequencies" associated with each component so that their levels can be monitored while the bearing is in use.

 

Audio Filtering

 

The Fourier Transform can identify the tonal frequency components of a sound. These tones can then be filtered out to improve the listener’s experience. For example, in the 2010 World Cup many fans brought an instrument called a vuvuzela (Figure 12) with them to the stadium.

 

vuvuzela.pngFigure 12: A chorus of vuvuzelas at the 2010 World Cup.

When the vuvuzela is played, it emits a 235 Hz tone which many TV watchers found annoying. The frequency content of the vuvuzela was identified using the Fourier Transform and then filtered out by some broadcasters to improve the viewing experience.

 

Engine Noise and Vibration

 

In noise and vibration troubleshooting, the Fourier Transform is useful because many components on rotating machinery have a unique harmonic signature. Internal combustion engines have many different components which can cause unwanted vibration such as crankshaft imbalance, belt slap, gear rattle, and combustion events (Figure 13).

 

engine.pngFigure 13: An engine has many different potential rotating machinery dynamic issues.

A Fourier Transform on time data from an accelerometer mounted on the engine block can be used to identify the components causing excessive vibration. 

 

Because the speed of an engine changes, the Fourier Transforms are often done at different RPMs and displayed in a colormap (Figure 14). See the Knowledge Base Article Interpreting Colormaps for more information on this process.

 

colormap.pngFigure 14: A series of Fourier Transforms displayed in a colormap can be used to diagnosis issues.In the colormap of Figure 14, the highest levels of sound are due to the combustion events/frequencies produced by the engine. The frequency associated with combustion changes with rpm.  A frequency related to the rpm of a rotating shaft is called an order.

 

For more information on how to calculate a FFT in Simcenter Testlab see the Simcenter Testlab Throughput Processing Tips community article.

 

Thank you for reading!

 

Questions? Email nicholas.divincenzo@siemens.com, contact Siemens GTAC PLM support, or post to this article.

 

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