Fourier Analysis

Maths for Materials & Design

Year 2 – Assignment 1

55-5805

James Walker

Contents

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Abstract 1

Introduction. 2

1 – Mathematical Analysis. 3

1 A – Waveform Sketch. 3

1 B – Fourier Coefficients. 3

1 C – Reconstruction. 4

1 D – Gibbs Phenomenon. 6

2 – Tabular Analysis. 8

2 A – The Procedure. 8

2 B – Signal Component Calculation. 10

2 C – Signal Reconstruction. 11

2 D – Known Signal Decomposition. 13

3 – Fast Fourier Transform.. 15

3 A – Frequency Components. 15

3 B – Engineering Applications. 18

4 – Appendices. 20

Appendix 1 – Calculation of M.. 20

Appendix 2 – Calculation of An 21

Appendix 3 – Calculation of Bn 22

Appendix 4 – Tabular Method Results. 24

Appendix 5 – Known Wave Reconstruction Datasets. 26

Appendix 6 – MatLab Code. 26

5 – References 27

Abstract

Signal
processing is useful in a variety of applications, which can include data
compression, image and video compressing and removing noise, interference or
other corruption from a signal. This report details 3 methods of Fourier
Analysis for processing different types of signal.

Introduction

“Fourier
Analysis is the decomposition of a function into a set, possibly infinite, of
simple oscillating functions. Each function will have a different frequency,
phase and amplitude.” Lecture Notes The process is named after the
French Mathematician and Physicist Joseph Fourier who was born in the 18th
century, and developed the principle of representing a function as a sum of
trigonometric functions in order to simplify the study of heat transfer.

Van Veen, B. 2018

Fourier
Analysis has since been developed for use in a wide range of applications and
this report looks at three methods of conducting the process.

Section 1
looks at reconstructing a signal of a known function using integration, which
is restricted in its uses due to being long-winded and requiring the function
to be known.

Section 2
demonstrates the use of a tabular method which has the advantage of being able
to process a series of points rather than a known function, but is inherently
slow and cumbersome due to the number of calculations required, particularly if
there are many frequency components.

Section 3
outlines and demonstrates the use of FFT (Fast Fourier Transform) which
enhances mathematical approaches by making working in the frequency domain as
practical as working with the time and amplitude domain, with the aid of
computer software.

1 – Mathematical Analysis

1 A – Waveform Sketch

T = 3
A = 6

Using the provided dataset, which specifies a time
period (T)
and an amplitude (A),
the graph in Figure 1 was constructed.

Figure 1 – Waveform of
Provided Dataset

1 B – Fourier Coefficients

Full calculations of M, An
& Bn can be found in Appendices 1-3 respectively, their values
are listed below:

Where M is the vertical offset of the
signal, An is the amplitude of the nth Sine function
and Bn the amplitude of the nth Cosine function.

The calculation of An
could have been avoided because the function is symmetrical in the Y-axis therefore
it is an ‘even’ function and An will always be
equal to 0.

The first 8
non-zero Bn coefficients were calculated using the above equation, and are
tabulated in Figure 2.

Figure 2 – Bn Coefficients

1 C – Reconstruction

Following the
calculation of the coefficients, the Fourier Series can be approximated with Equation
1:

Equation
1

Figure 3 – Approximation of the Function

The table shown partly in Figure 3 was created to implement
the formula above in order to create a dataset which runs from t
= -4.5 to 4.5 (as per the original data). The second and third rows contain n
and the corresponding value of Bn as
per Figure 2.

Plotting the results in Figure 3 returns the graph seen in
Figure 4, which very closely resembles the original dataset. The only notable
differences are the slight jagged lines caused by the vast number of points and
the rounded corners caused by the approximation, as can be seen in Figure 5.

Figure 4 – Reconstructed Graph

Figure 5 – Overlay of Original & Approximation

Figure 6 shows the described differences; the rounded peak and the
less smooth lines. The peaks are at 5.85 and -5.85, showing a 2.5% deviation
from the original amplitude.

Figure 6 – Overlay Magnified

1 D – Gibbs Phenomenon

The Gibbs phenomenon occurs through the Fourier analysis of
periodic functions, where the partial sum exceeds the amplitude of the intended
function, as a result of a jump discontinuity. This can be seen in square wave
approximations where the approximated signal will overshoot by typically 9% of
the amplitude after the jump discontinuity (where one x value can have multiple
y values). This can be seen in Figure 7, where the red line shows the
approximation (with a different number of components in each diagram) but the
red line will always surpass the height of the square wave after the change
from negative to positive or vice versa.

Figure 7 – Gibbs Phenomenon MIT

The ripples seen will never disappear, and retain the same
height, however as the number of terms tends to infinity, the width (hence the
area) of the ‘ripples’ tend to 0, resulting in them having a negligible effect.

This ripple effect can have consequences for some square wave AC
welders, such as TIG welders as the current will peak higher than its intended
value, resulting in over penetration of the welded material, however different methods
of producing a ‘square wave’ used for welding non-ferrous metals can be seen
below, with their respective ripple coefficients, the coefficient is defined as
the ratio of maximum current magnitude to its effective value. Inverter welders
and other more modern welders do not use a summation of sine waves; hence the
coefficient is 0.

Figure 8 – Ripple
Coefficients Julian, P. 2003

2 – Tabular Analysis

2 A – The Procedure

The original signal data, sampled at
100Hz, is plotted below in Figure 9.

Figure 9 – Original Signal Data

This signal lasts
for of 0.21 s, calculated by dividing the number of samples (21) by the sample
rate (100 Hz).

Peaks in the signal
represent the component frequencies, therefore the fundamental frequency is
100/21 or 4.76 Hz. Any component frequencies must have a higher frequency than
this.

The full results
table in which the sample data was put through a tabular method of Fourier
analysis is in Appendix 4, which relied on the equations below:

Equation 2

Equation 3

Equation 4

Equation 5

The table starts
with the sample data and the time at which it was sampled, along with the
corresponding values of theta. Theta was calculated by assuming the data
provided shows one full cycle, (2? radians).

The first series of
columns calculates the individual An & Bn
components, these are
summated and multiplied by 2/21 at the bottom of the table to find the overall An
& Bn components,
was per Equations 4 & 5.

Once the components
were found, Equation 3 was used in the second part of the table to find M (0.545) followed by f(t) using Equation 2.

The first 8 terms are
given below:

2 B – Signal Component Calculation

Figure 10 – Frequency Components

The graph in Figure
10 shows the previously listed coefficients graphically, demonstrating that the
even values of n are typically more prominent in this case.

2 C – Signal Reconstruction

The original signal
can be seen in Figure 9. Figure 11 shows the reconstruction when n=3; a very
inaccurate reconstruction.

Figure 11 – n=3

Figure 12 shows the
reconstruction when n=6, by which point the data can be recognised visually as
being similar to the original.

Figure 12 – n=6

Figure 13 shows the
reconstruction when n=10 which is a very accurate reconstruction.

Figure 13 – n=10

The signals were reconstructed using the first and the final
three columns of the table in Appendix 4, and it was found that the function only begins to become
distinguishable on a graph once the sum of the n=1 to n=4 is plotted, prior to
this it appears as a sine wave.

Figure 14 – Overlay

Figure 14 shows
the original, and the points from the other reconstructions for reference, it
can be seen that all the points from n=10 lie on the line of the original, and
create a near perfect replication when plotted (as per Fig. 13).

2 D – Known Signal Decomposition

Figure 15 – Known Waveform

Figure 15 shows the
graph produced from the dataset in Appendix 5.

Again, for comparison the
re-construction is shown with varying values of n in Figure 16.

Figure 16 – Reconstructed Signal

The data points of
n=10 can be found compared with the input values in Appendix 5X. The values are
all 4.7% smaller than their inputs.

When n=6, the
reconstruction becomes distinguishable, but shows some odd features including
overshoots which appear symmetrical in opposite corners, with the closest
representation near the middle of the sample. This may be explained by the
sample being a discreet function (ie. between t = 0 & t = 10), my suspicion is that some of these
features would not appear on a continuous function.

Figure 17 – Overlay

3 – Fast Fourier Transform

3 A – Frequency Components

Fast Fourier Transform (FFT) is used to decompose signals to divide
them into their frequency components (single sinusoidal waves at a particular
frequency) as shown by Figure 18.

Figure 18 – Frequency vs. Time Domain Wikipedia

This is performed by a complex algorithm that initially
performs a discreet Fourier transform (DFT), then FFT uses Fourier analysis to
convert from the time to the frequency domain (as in Figure 18).

The provided
dataset when plotted is shown in Figure 19.

Figure 19 – Original Signal

It is clear to see that not a lot can be interpreted by inspection of
the raw data, so it was put through FFT in MatLab in order to determine the
frequency components. The code is shown in Figure 20.

Fs is the
sampling frequency.

T is the
sampling time interval.

L is the number
of samples.

t is the time
of the whole sample.

JW is the
original dataset.

Figure 20 – MatLab Code

Line 6 performs
the Fourier transform, using the original dataset, outputting a list of complex
numbers, which aren’t very useful (Figure 21). Line 7 converts these to a
double-sided spectrum, then lines 8 & 9 make a single sided spectrum. Line
10 defines the frequency domain, then lines 12-15 plot the results.

Mathworks

Figure 21 – Table y                          Figure 22 – Table P2
Figure 23 – Table P1

Figure 24 – FFT of Provided Sample

The frequency components were found and are listed in Figure
25. This can be seen in graphical form in Figure 24.

Figure 25 – Frequency
Components Table

3 B – Engineering Applications

Fourier analysis is
used in ‘Fourier Transform Infrared Spectroscopy (FTIR), a process used to
determine the composition of a sample material. FTIR is a “non-destructive
microanalytical spectroscopic technique” which uses infrared radiation to
induce vibrations in molecular bonds. This process produces a ‘fingerprint’
which is unique to a particular material, and provides information
(predominantly qualitative) describing the composition of the material sample,
typically the base polymer of the sample. The fingerprint is produced from the
molecules’ transitions between energy levels, which occur at specific frequencies
and can be identified using the absorption spectra displayed by the infrared
light reflected by the sample onto the detector. This spectrum can then be compared
to a library of known spectra in order to identify the material.

FTIR is often used as a first analytical test when
determining a cause of failure, as it determines whether the material is
correct to its drawing specification, and can negate the need for further
testing.

One inadequacy of FTIR is the difficulty in distinguishing
between two similarly structured polymers such as polyethylene terephthalate
and polybutylene terephthalate, in these cases other identification methods
like differential scanning calorimetry can be used in addition.

Another limitation is detecting materials of less than
around 1% concentration in a compound. This detection limit will vary between
spectrometers, depending on their resolution and accuracy, although the process
can be useful for identifying contaminants as the absorption spectra of known
compounds can be subtracted from the results to display absorption spectra not
characteristic of the base resin, which will help to identify any contaminants.
Figure 26 shows an example of 5 known spectra produced from FTIR which could be
used in spectral subtraction.

Jansen

Figure 26 – FTIR
Comparison of Several Polymers Jansen, J

The raw data obtained through FTIR is known as an
interferogram, which appears as a cosine wave which is an electrical signal
provided by the detector. On an interferogram, a range of wavelengths would be
seen resulting in areas of constructive and destructive interference, this
signal is then decomposed using Fourier Analysis to provide a yield spectrum
which identifies the key wavelengths.

Smith, B. 2011

The principle of how the equipment obtains the signal is
shown below in Figure 27.

Figure 27 – Fourier
Transform Infrared Spectrometer Diagram

4 – Appendices

Appendix 1 – Calculation of M

Appendix 2 – Calculation of An

Appendix 3 – Calculation of Bn

Appendix 4 – Tabular Method Results

This method of
Fourier analysis result in a table with many columns, it has had to be split
into two sections for viewing in a paper document.

Appendix 5 – Known Wave Reconstruction Datasets

Time (s)

Input Amplitude

Output Amplitude

0.00

0.00

0.00

0.10

3.00

2.86

0.90

3.00

2.86

1.10

-3.00

-2.86

1.90

-3.00

-2.86

2.10

3.00

2.86

2.90

3.00

2.86

3.10

-3.00

-2.86

3.90

-3.00

-2.86

4.10

3.00

2.86

4.90

3.00

2.86

5.10

-3.00

-2.86

5.90

-3.00

-2.86

6.10

3.00

2.86

6.90

3.00

2.86

7.10

-3.00

-2.86

7.90

-3.00

-2.86

8.10

3.00

2.86

8.90

3.00

2.86

9.10

-3.00

-2.86

9.90

-3.00

-2.86

10.00

0.00

0.00

Appendix 6 – MatLab Code

5 – References

Fast Fourier transform. (2018). Wikipedia.
Retrieved 25 January 2018, from https://en.wikipedia.org/wiki/Fast_Fourier_transform

Fast Fourier transform – MATLAB. (2018). MathWorks.
Retrieved 25 January 2018, from
https://uk.mathworks.com/help/matlab/ref/fft.html

Gibbs’ Phenomenon. (2011). MIT. Retrieved
25 January 2018, from https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-iii-fourier-series-and-laplace-transform/operations-on-fourier-series/MIT18_03SCF11_s22_7text.pdf

Griffiths, P., & Haseth, J. (2007). Fourier
transform infrared spectrometry (2nd ed.). New York, N.Y., etc.:
Wiley.

Hahn, B., & Valentine, D. (2017). Essential
MATLAB for engineers and scientists (6th ed.). Elsevier.

Jansen, J. Characterization of Plastics in Failure
Analysis. Madisongroup.com. Retrieved 27 January 2018, from
https://www.madisongroup.com/publications/Jansen_ASM_Chapte_TMG2011.pdf

Julian, P. (2003). Arc Welding Control.
Cambridge: Woodhead Publishing.

Nave, R. (2018). Fourier Analysis and Synthesis. Hyperphysics.
Retrieved 25 January 2018, from http://hyperphysics.phy-astr.gsu.edu/hbase/Audio/fourier.html

Smith, B. (2011). Fundamentals of Fourier transform
infrared spectroscopy (2nd ed.). Boca Raton, Fla.: CRC Press.

Tolstov, G. (2014). Fourier series. Dover
Pulications.

Van Veen, B. (2018). Fourier Methods in Signal
Processing. All Signal Processing. Retrieved 26 January 2018,
from https://allsignalprocessing.com/fourier-methods-prominent/

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