Fourier Analysis

Chris Kervick (11355511)With Evan Sheridan and Tom Power

November 2012

Abstract

Some properties of the Fourier Transform were investigate usingthe CassyLab computer program and input device, an oscillator, aspeaker and a microphone. These properties included; the frequencycomposition of various sound waves; the Fourier transform of a blockwave and its relation to diffraction of light. All results were found tobe consistent with theoretical prediction.

Aims

• To investigate the properties of the Fourier Transform

Introduction and Theory

It was discovered by Joseph Fourier circa 1820 that any periodically repeatingfunction can be represented as a sum of sine and cosine waves known as theFourier Series. This technique, however, can also apply to functions withno periodicity (or an “infinite” period), where the sum becomes an integral.This is known as a Fourier Transform. Often, it transforms a function oftime, f(t) into a function of frequency.

The CassyLab program uses an algorithm called the Fast Fourier Trans-form to display a function in either its time domain (where the x-axis is time)or the frequency domain.

Using this program, one can input a wave (via an oscilloscope, microphoneetc.) into CassyLab, which will then use the Fast Fourier Transform todisplay the frequencies present in the wave. From this, we can see whichharmonics are present in the wave. To ensure, however, that the programrecreates the wave correctly, we must obey Nyquist’s thoerem which states

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that for the reconstruction of a wave of frequency f , the sampling rate mustbe greater than 2f . Otherwise, there will be fewer than two points per cycle,and the program will incorrectly attempt to recreate the wave under the aliasof another signal.

Experimental Method

Experiment 1

Using CassyLab’s automatic Fast Fourier Transform setting, we viwed boththe time spectrum and frequency spectrum of various waves via various inputmethods. Each time, a good choice of settings must be made on the program.These settings are:

• Measuring interval - The number of measurements of the wave takenper second. This must be at least more than double the frequency toensure an accurate reconstruction of the wave.

• xNumber - The amount of measurements the program takes in total.This must be chosen such that, for a given measurement interval, thetotal measurement time is reasonable.

• Voltage range - This must be chosen to be greater than that of theamplitude of the wave, so that the entire wave can be displayed on-screen.

Upon each successful measurement, we saved a bitmap file of the graphsin both the time and frequency domain.

Experiment 2

The oscillator was connected to the speaker, and the microphone to the Sen-sorCassy. We generated and recorded various sound waves via the speaker,both sinusoidal and square.

By varying the frequency of the sound wave emitted by the speaker, weattempted to determine the upper and lower thresholds of human hearing,and the maximum and minimum frequencies detectable by the microphone.However, there was far too much background noise in the room to obtainparticularly accurate data.

We also attempted to measure the beat frequency between two similar-frequency tuning forks. Again, however, there was too much backgroundnoise in the room to be able to detect or measure the beat frequency.

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Finally, we analysed the frequency spectrum of the human voice, byspeaking different vowels into the microphone and recording the relevantdata.

Experiment 3

A sine wave was generated by the oscillator, and the frequency spectrumdisplayed on-screen. The frequency of the sine wave was increased in smallsteps, and the change in the frequency spectrum displayed was noted.

Experiment 4

A rectangular pulse was generated as follows:

• The oscillator was set to D.C. only

• One of the sensor inputs was disconnected

• Recording was started and the disconnected input was touched off theinput terminal for less than a second

We then measured the frequencies at which the first three maxima andminima occur.

Experiment 5

We generated both a block pulse and a sine pulse by setting the oscillator tothe relevant settings and following the procedure below:

• One of the sensor inputs was disconnected

• Recording was started

• The disconnected inout was touched off the input terminal for a shortperiod of time

We then measured the frequencies at which the first thee maxima and minimaoccurred in both cases.

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Results and Analysis

Experiment 1

Below is an example of a sine wave (left) and its Fourier transform (right).As it is a pure sine wave, only one frequency is present, as is evident fromthe single peak in the transform.

Experiment 2

Below is an example of a block wave in air, produced by a speaker, and itsFourier transform. The reason it (and by extension, its transform) differsfrom the block wave produced directly by the oscillator is that the speakeris incapable of perfectly reproducing a block wave, nor cannot it produce a“negative” amplitude sound.

Below are examples of the frequency spectra of various vowels. As we cansee, different vowels contain different overtones. This is what contributes tothe “harshness” or “narrowness” attributed to certain vowels.

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The vowel “ah” as in “lab”

The vowel “ee”

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The vowel “oh” as in “snow”

Experiment 3

Here we can see a successive frequency spectra for an oscillator producing asignal of increasing frequency. Obviously, because frequency is plotted on thex-axis, as the frequency increases, the peak in the Fourier transform movesacross the screen.

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Experiment 4

Below is an image of a singular block wave and its transform. Its transformis a sinc function, as can be derived theoretically.

The first three minima and maxima are given in the table below.

Theoretically, the first maximum occurs at an indeterminate value, which

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Number Minimum (Hz) Maximum (Hz)1 1.12 0.052 2.3 1.613 3.42 2.79

obviously our program cannot display.It can be shown that the Fourier transform of an Aperture Function gives

the Fraunhofer (or far-field) diffraction pattern. As the aperture function ofa square aperture is a singular block wave, the Fourier transform of this func-tion, displayed above, gives the familar diffraction pattern of light through asquare aperture.

Experiment 5

Below is a graph of a repeated block wave and its transform:

As the aperture function of a diffraction grating is a repetititve blockpulse, the Fourier transform of this function, displayed above, gives the fami-lar diffraction pattern of light through a diffraction grating.

Below is a graph of a sine pulse and its Fourier transform:

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Again, the Fourier transform of this function gives only a single peak asonly one frequency is present.

Discussion and Conclusions

• We successfully showed the frequency composition and overtones presentin various vowels

• Owing to the large amount of background noise, we were unsuccessful inour attempts to detect the maximum or minimum thresholds of hearingfor either ourselves or the microphone. We were also unable to detector measure beats using tuning forks.

• We showed how the Fourier transform varies with the frequency of theinput wave.

• We found the Fourier transform of a singular block pulse, which cor-rectly corresponded to the diffraction pattern of light through a squareaperture.

• We found the Fourier transform of a repetitive block wave, which cor-rectly corresponded to the diffraction of light through a diffraction grat-ing.

References

Theory and Problems of Optics by Eugene Hecht, Schaum, 1975

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