Fourier Analysis Maths for Materials & Design Year 2 – Assignment 1 55-5805 James Walker Contents Abstract 1Introduction. 21 – Mathematical Analysis. 31 A – Waveform Sketch. 31 B – Fourier Coefficients.
31 C – Reconstruction. 41 D – Gibbs Phenomenon. 62 – Tabular Analysis. 82 A – The Procedure.
82 B – Signal Component Calculation. 102 C – Signal Reconstruction. 112 D – Known Signal Decomposition.
133 – Fast Fourier Transform.. 153 A – Frequency Components.
153 B – Engineering Applications. 184 – Appendices. 20Appendix 1 – Calculation of M.. 20Appendix 2 – Calculation of An 21Appendix 3 – Calculation of Bn 22Appendix 4 – Tabular Method Results.
24Appendix 5 – Known Wave Reconstruction Datasets. 26Appendix 6 – MatLab Code. 265 – References 27 AbstractSignalprocessing is useful in a variety of applications, which can include datacompression, image and video compressing and removing noise, interference orother corruption from a signal. This report details 3 methods of FourierAnalysis for processing different types of signal. Introduction”FourierAnalysis is the decomposition of a function into a set, possibly infinite, ofsimple oscillating functions. Each function will have a different frequency,phase and amplitude.
” Lecture Notes The process is named after theFrench Mathematician and Physicist Joseph Fourier who was born in the 18thcentury, and developed the principle of representing a function as a sum oftrigonometric functions in order to simplify the study of heat transfer.Van Veen, B. 2018 FourierAnalysis has since been developed for use in a wide range of applications andthis report looks at three methods of conducting the process.
Section 1looks at reconstructing a signal of a known function using integration, whichis restricted in its uses due to being long-winded and requiring the functionto be known. Section 2demonstrates the use of a tabular method which has the advantage of being ableto process a series of points rather than a known function, but is inherentlyslow and cumbersome due to the number of calculations required, particularly ifthere are many frequency components. Section 3outlines and demonstrates the use of FFT (Fast Fourier Transform) whichenhances mathematical approaches by making working in the frequency domain aspractical as working with the time and amplitude domain, with the aid ofcomputer software.
1 – Mathematical Analysis1 A – Waveform Sketch T = 3 A = 6 Using the provided dataset, which specifies a timeperiod (T)and an amplitude (A),the graph in Figure 1 was constructed.Figure 1 – Waveform ofProvided Dataset1 B – Fourier CoefficientsFull calculations of M, An& Bn can be found in Appendices 1-3 respectively, their valuesare listed below: Where M is the vertical offset of thesignal, An is the amplitude of the nth Sine functionand Bn the amplitude of the nth Cosine function. The calculation of Ancould have been avoided because the function is symmetrical in the Y-axis thereforeit is an ‘even’ function and An will always beequal to 0. The first 8non-zero Bn coefficients were calculated using the above equation, and aretabulated in Figure 2. Figure 2 – Bn Coefficients1 C – ReconstructionFollowing thecalculation of the coefficients, the Fourier Series can be approximated with Equation1: Equation1 Figure 3 – Approximation of the FunctionThe table shown partly in Figure 3 was created to implementthe 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 nand the corresponding value of Bn asper Figure 2.
Plotting the results in Figure 3 returns the graph seen inFigure 4, which very closely resembles the original dataset. The only notabledifferences are the slight jagged lines caused by the vast number of points andthe rounded corners caused by the approximation, as can be seen in Figure 5. Figure 4 – Reconstructed GraphFigure 5 – Overlay of Original & Approximation Figure 6 shows the described differences; the rounded peak and theless smooth lines. The peaks are at 5.85 and -5.
85, showing a 2.5% deviationfrom the original amplitude. Figure 6 – Overlay Magnified1 D – Gibbs PhenomenonThe Gibbs phenomenon occurs through the Fourier analysis ofperiodic functions, where the partial sum exceeds the amplitude of the intendedfunction, as a result of a jump discontinuity. This can be seen in square waveapproximations where the approximated signal will overshoot by typically 9% ofthe amplitude after the jump discontinuity (where one x value can have multipley values). This can be seen in Figure 7, where the red line shows theapproximation (with a different number of components in each diagram) but thered line will always surpass the height of the square wave after the changefrom negative to positive or vice versa. Figure 7 – Gibbs Phenomenon MITThe ripples seen will never disappear, and retain the sameheight, however as the number of terms tends to infinity, the width (hence thearea) of the ‘ripples’ tend to 0, resulting in them having a negligible effect.
This ripple effect can have consequences for some square wave ACwelders, such as TIG welders as the current will peak higher than its intendedvalue, resulting in over penetration of the welded material, however different methodsof producing a ‘square wave’ used for welding non-ferrous metals can be seenbelow, with their respective ripple coefficients, the coefficient is defined asthe ratio of maximum current magnitude to its effective value. Inverter weldersand other more modern welders do not use a summation of sine waves; hence thecoefficient is 0. Figure 8 – RippleCoefficients Julian, P. 20032 – Tabular Analysis2 A – The Procedure The original signal data, sampled at100Hz, is plotted below in Figure 9.Figure 9 – Original Signal Data This signal lastsfor of 0.
21 s, calculated by dividing the number of samples (21) by the samplerate (100 Hz). Peaks in the signalrepresent the component frequencies, therefore the fundamental frequency is100/21 or 4.76 Hz. Any component frequencies must have a higher frequency thanthis. The full resultstable in which the sample data was put through a tabular method of Fourieranalysis is in Appendix 4, which relied on the equations below: Equation 2 Equation 3 Equation 4 Equation 5 The table startswith the sample data and the time at which it was sampled, along with thecorresponding values of theta.
Theta was calculated by assuming the dataprovided shows one full cycle, (2? radians). The first series ofcolumns calculates the individual An & Bncomponents, these aresummated 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 componentswere 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 aregiven below: 2 B – Signal Component CalculationFigure 10 – Frequency Components The graph in Figure10 shows the previously listed coefficients graphically, demonstrating that theeven values of n are typically more prominent in this case.
2 C – Signal Reconstruction The original signalcan be seen in Figure 9. Figure 11 shows the reconstruction when n=3; a veryinaccurate reconstruction. Figure 11 – n=3 Figure 12 shows thereconstruction when n=6, by which point the data can be recognised visually asbeing similar to the original.
Figure 12 – n=6 Figure 13 shows thereconstruction when n=10 which is a very accurate reconstruction. Figure 13 – n=10The signals were reconstructed using the first and the finalthree columns of the table in Appendix 4, and it was found that the function only begins to becomedistinguishable on a graph once the sum of the n=1 to n=4 is plotted, prior tothis it appears as a sine wave. Figure 14 – OverlayFigure 14 showsthe original, and the points from the other reconstructions for reference, itcan be seen that all the points from n=10 lie on the line of the original, andcreate a near perfect replication when plotted (as per Fig. 13). 2 D – Known Signal DecompositionFigure 15 – Known Waveform Figure 15 shows thegraph produced from the dataset in Appendix 5. Again, for comparison there-construction is shown with varying values of n in Figure 16.Figure 16 – Reconstructed Signal The data points ofn=10 can be found compared with the input values in Appendix 5X.
The values areall 4.7% smaller than their inputs. When n=6, thereconstruction becomes distinguishable, but shows some odd features includingovershoots which appear symmetrical in opposite corners, with the closestrepresentation near the middle of the sample.
This may be explained by thesample being a discreet function (ie. between t = 0 & t = 10), my suspicion is that some of thesefeatures would not appear on a continuous function. Figure 17 – Overlay 3 – Fast Fourier Transform3 A – Frequency Components Fast Fourier Transform (FFT) is used to decompose signals to dividethem into their frequency components (single sinusoidal waves at a particularfrequency) as shown by Figure 18. Figure 18 – Frequency vs. Time Domain WikipediaThis is performed by a complex algorithm that initiallyperforms a discreet Fourier transform (DFT), then FFT uses Fourier analysis toconvert from the time to the frequency domain (as in Figure 18). The provideddataset 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 ofthe raw data, so it was put through FFT in MatLab in order to determine thefrequency components.
The code is shown in Figure 20. Fs is thesampling frequency.T is thesampling time interval.L is the numberof samples.t is the timeof the whole sample.JW is theoriginal dataset.
Figure 20 – MatLab Code Line 6 performsthe Fourier transform, using the original dataset, outputting a list of complexnumbers, which aren’t very useful (Figure 21). Line 7 converts these to adouble-sided spectrum, then lines 8 & 9 make a single sided spectrum. Line10 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 Figure25. This can be seen in graphical form in Figure 24.
Figure 25 – FrequencyComponents Table 3 B – Engineering ApplicationsFourier analysis isused in ‘Fourier Transform Infrared Spectroscopy (FTIR), a process used todetermine the composition of a sample material. FTIR is a “non-destructivemicroanalytical spectroscopic technique” which uses infrared radiation toinduce 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 themolecules’ transitions between energy levels, which occur at specific frequenciesand can be identified using the absorption spectra displayed by the infraredlight reflected by the sample onto the detector. This spectrum can then be comparedto a library of known spectra in order to identify the material. FTIR is often used as a first analytical test whendetermining a cause of failure, as it determines whether the material iscorrect to its drawing specification, and can negate the need for furthertesting. One inadequacy of FTIR is the difficulty in distinguishingbetween two similarly structured polymers such as polyethylene terephthalateand polybutylene terephthalate, in these cases other identification methodslike differential scanning calorimetry can be used in addition. Another limitation is detecting materials of less thanaround 1% concentration in a compound.
This detection limit will vary betweenspectrometers, depending on their resolution and accuracy, although the processcan be useful for identifying contaminants as the absorption spectra of knowncompounds can be subtracted from the results to display absorption spectra notcharacteristic 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 beused in spectral subtraction. JansenFigure 26 – FTIRComparison of Several Polymers Jansen, JThe raw data obtained through FTIR is known as aninterferogram, which appears as a cosine wave which is an electrical signalprovided by the detector. On an interferogram, a range of wavelengths would beseen resulting in areas of constructive and destructive interference, thissignal is then decomposed using Fourier Analysis to provide a yield spectrumwhich identifies the key wavelengths.Smith, B. 2011 The principle of how the equipment obtains the signal isshown below in Figure 27. Figure 27 – FourierTransform Infrared Spectrometer Diagram 4 – AppendicesAppendix 1 – Calculation of M Appendix 2 – Calculation of An Appendix 3 – Calculation of Bn Appendix 4 – Tabular Method ResultsThis method ofFourier analysis result in a table with many columns, it has had to be splitinto 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 – ReferencesFast 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, fromhttps://uk.mathworks.com/help/matlab/ref/fft.html Gibbs’ Phenomenon.
(2011). MIT. Retrieved25 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). Fouriertransform infrared spectrometry (2nd ed.
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(2003). Arc Welding Control.Cambridge: Woodhead Publishing. Nave, R. (2018). Fourier Analysis and Synthesis. Hyperphysics.Retrieved 25 January 2018, from http://hyperphysics.
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