Fourier Analysis

Maths for Materials & Design

Year 2 – Assignment 1

55-5805

James Walker

Contents

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/