Fourier Analysis Evan Sheridan, Chris Kervick, Tom Power 11367741 Novemeber 19 01 Abstract Various properties of the Fourier Transform are investigated using the Cassy Lab software, a microphone, electrical oscillator, tuning forks and speaker. Various Fourier transforms are observed using the program. The upper & lower thresholds of hearing are found as well as the limits that the microphone could record. Both rectangular and sinusoidal pulses were investigated and certain properties were confirmed using the Fourier Transform. The diffraction of light was also investigated using the Fourier Transform. 1
Aims To use the Cassy Lab program to construct the Fourier Transforms of various signals. To determine the upper and lower thresholds of frequency for human hearing. Gain an intuitive understanding of how the fourier transform operates. Investigate the relation between the diffraction grating and the Fourier Transform. Backround & Theory The real Fourier Series is: f(x) = a 0 + (a n cos( πnx L )) + (b n sin( πnx L )) n=0 it decomposes a periodic function f(x) = f(x + L) into an infinite sum of sines & cosines. The coefficients are given by : a n = L f(x) cos( πnx L L ) b n = L L L L n=0 f(x) sin( πnx L ) where L is the period of the function. The coefficients are a measure of how much the sine & cosine is present in the fourier expansion of the function. In physical terms, this is the equivalent to the weight of sine & cosine in each harmonic that the wave is composed of. The Fourier Transform is given by: f(k) = while the Inverse Fourier Transform is : f(x) = + + dx π f(x)e ikx dk π f(k)e ikx The Fourier transform maps a function from real space into fourier space. While the Inverse Fourier Transform maps a function from fourier space into real space. As opposed to the fourier series we lift the restriction that the function need be periodic and with this the fourier transform is more powerful than the series because we can now decompose every function into a sum of sines and cosines. It is noteworthy that not all functions transform to functions, some may transform to objects know as distributions which are not functions in the strict sense but useful for modelling certain physical situations. The sampling rate,of a device (in this case: the Cassy Lab program), is how many times the device will measure the value of the wave per unit time, say. Therefore, the larger the sampling rate is the more accurate the reconstruction of the wave becomes. The sampling rate must be greater than f because if it isn t then there will be fewer than two points per cycle and the reconstruction of the signal into a wave will be incorrect. Aliasing occurs when the device assumes that the sampling rate is greater than f but it isn t. It therefore results in the reconstruction of the wrong wave. 1
Experimental Method Experiment 1 This part of the experiment involves using the Cassy Lab program to get the fourier transform of various waves. Connect the oscillator to the Sensor Cassy. Open the Cassy program and set it up, picking the sampling rate such that to avoid aliasing. Switch on the oscillator and set it to an appropriate voltage and frequency (which will be changed throughout). Generate a few waves with the oscillator and observe what their fourier transform looks like, specifically the sine wave and block wave. Experiment Connect the oscillator to the speaker and the microphone to the A inputs of the Sensor Cassy. Generate a block wave in air using the speaker and let the microphone record it. Find the upper and lower thresholds of frequency by lowering the raising the frequency until the sound cannot be heard. Find the maximum and minimum frequencies that the microphone can record. Take two tuning forks near the same frequency and create beats and analyse them. Analyse different voices by comparing male and female voices and then examine the sound of different vowels. Experiment 3 Generate a sine wave using the oscillator. This part of the experiment involves increasing the frequency so choose and appropriate sampling rate to avoid aliasing. Increase the frequency of the signal and observe how the Fourier Transform of the signal changes as the increasing of frequency. Experiment 4 Create a rectangular pulse by setting the oscillator to D.C only. Do this by disconnecting one of the inputs in the Sensor Cassy and set the oscilator on and touch the disconnected input off input terminal for a small length of time. Do this periodically to generate the rectangular pulse. Experiment 5 Generate both a repetitive square pulse and sine pulse using the generator. The sine pulse can be achieved by quickly connecting the inputs into Sensor Cassy. Observe the frequency spectrum for both and measure the positions of the first three minima and maxima on the frequency spectrum.
Results & Analysis Experiment 1 The above plot shows sin(x) and it s Fourier Transform. As expected, we have a single peak at somewhere around the frequency at which we generated. This is because the Fourier transform maps the time domain in the frequency domain and the Fourier Transform, in this case, is the frequency spectrum of the sine wave generated. The Fourier Transform of the sine wave is given by: π (δ(k + 1) δ(k 1)) i The Dirac Delta dependence implies there should be a very sharp, peaked behaviour at a specific frequency which we observe above. In the above plot we have the block wave and it s corresponding fourier transform. It is as we theoretically predict. The peaks in the fourier transform show the amount of the frequencies present in the block wave itself. Experiment We generated a block wave in air using the generator by connecting it to a speaker and we then recorded it using the microphone: 3
As we can see the block wave is a bit shaky but this is because of the interference in the air. The Fourier Transform is what we expect in such circumstances. We then investigated the the upper and lower thresholds of human hearing. This was difficult because of the noise in the room but we successfully got the lower threshold to be : 0-30 Hz. It was difficult to pinpoint exactly when the sound disappeared. The upper threshold was found to be around 0000 Hz, again it was difficult to exactly figure out when the sound couldn t be heard. Given that the range of human hearing is said to be: 0-0000 Hz, we therefore calculated the range of human hearing successfully. The upper and lower limits of which the microphone could record were found to be: X-50000 Hz. In this case the lower limit could not be found because the microphone was sensitive and even though the frequency we output was quite low it still picked up the sounds of other people in the room so we didn t get any discernible lower limit. However, the upper limit was calculated to be 50000 Hz. We tried to get the beat frequency of vibrating two tuning forks against each other. unsuccessful. We got the following results: This was 4
As we can see they aren t beats. We tried many different combinations of the tuning forks but none were successful in creating the beats. We believe that there was too much interference in this room for the microphone to pick up the frequencies. The range of the tuning forks was around 00-500 Hz. Like above the microphone couldn t successfully measure lower frequencies. Although beats weren t recorded they were clearly heard when put against one s ear. We examined the sound of different vowels getting the following plots of their Fourier Transforms: As we can the vowel has a similar behaviour as the block wave in that it initially peaks at some frequency and then peaks at multiples of that frequency. We didn t take a large enough sample above to deduce anything concrete about the acoustics of vowels but at face value it seems that the vowels only have odd frequencies present due to where they peak. Again the graphs are a tad shaky due to the noise of people in the room. Unfortunately we didn t compare a males voice to a females voice. 5
Experiment 3 The above is a plot of the Fourier Transform as a function of frequency. As we can see as we increase the frequency of the signal the plot moves across the axis. In this case the Fourier Transform transforms into the frequency domain and measures how much of a specific frequency is present. As we increase the frequency we expect that presence of a certain frequency to be indicated. The y-axis is essentially measuring how much of a specific frequency is present and the results clearly show that as the frequency is increased the graph is shifted, which is what we expect. Experiment 4 Given that the fourier transform of the rectangular pulse is given by : 6
f(k) = sin(k) k As we can see both transforms above have a such a behaviour,which is what we expect. For the first rectangular pulse (1) we have a shorter period than the second rectangular pulse (). As we can see the longer period we have a more compact Fourier Transform. Therefore, the distance between the successive maxima and minima of the nd transform are less than the first transform. In general, this is true of all Fourier Transforms, i.e a longer period implies a more compact Fourier Transform and a shorter period implies a broader Fourier Transform. For our case, say : f(k) = + L L dx π V e ikx = V kl π (sin( ) ) k Therefore, as we increase L we increase the frequency of the Fourier Transform and thus make the Fourier Transform more compact whilst making the original wave broader. An application of this can be seen in Quantum Mechanics when we take the Fourier Transform of the wave function in position and we transform it to the momentum domain we will get, say, a tightly packed momentum distribution if we have a broad spatial distribution. Essentially, this coincides with the uncertainty principle in that if we know the momentum to a certain accuracy we do not know the position the same accuracy, i.e if the probability distribution(the wave function) in space is broad (large uncertainty in space) then the probability distribution in momentum is small( small uncertainty in momentum). As we can see there is a strong relation between Fourier Transform of the Block wave and the diffraction of light by a square aperture. Essentially, if we define an aperture function A(x) such that it has the same form as the rectangular pulse: { D if L A(x) = < x < L 0 if L < x < L we find that the Fourier Transform of this function is the sinc(x) function. Therefore, the image on the screen is the Fourier Transform of the aperture function. 1 We found the frequencies at which the first 3 maxima and minima appeared : 1 An in depth analysis of this problem can be found: http://www.thefouriertransform.com/applications/diffraction.php 7
Maxima(Hz) Minima (Hz) f 1 0.5 1.1 f 1.61.3 f 3.79 3.4 From above we have that : f(k) = V kl π (sin( ) ) k we wish to find the maxima and minima : where k is just the frequency. Experiment 5 d f(k) dk = 0 We measured the positions of the first 3 minima and of the first 3 maxima for the rectangular wave pulse : Maxima(Hz) Minima (Hz) f 1 0.9 6.6 f 47.8 104.9 f 3 14.8 175.9 Like above we use the same method to get the maxima and minima by taking the derivative of the Fourier Transform and letting it equal zero. A diffraction grating has a width D which is simply the period of the rectangular pulse in out case. As we can see from the Fourier Transform above we have a frequency peak at a regular interval, this again corresponds to the idea of the diffraction grating and by taking the Fourier Transform of the width of the diffraction grating, if we have a square diffraction grating for instance. The above transform illustrates the behaviour of light after passing through a diffraction grating. Again we see the sine pulse transform as we expect it to be a peak as it is described in terms of a dirac-delta function. 8
Discussions & Conclusions On calculating the upper and lower limits that the microphone could record we found it quite difficult to get any accurate results due to the interference because of all the people in the room. This remark also applies to other parts of the experiment that required the use of the microphone. It would be recommended to perform these parts of the experiment in a quite room or when everybody has left in order to get accurate results. Since we used the Cassy Lab Software to generate the Fourier Transforms of the signals we constructed there was little error as long as we avoided aliasing. Therefore, to minimize error it is recommended to initially tweak the settings such that aliasing is always avoided and keep this in mind throughout. It is obvious from each part of the experiment that the Fourier Transform has a huge range of uses. The apparent use may be in signal processing in analysing the reconstructed signal but there are more subtle applications such as in quantum mechanics and in the diffraction of light which exploit the more mathematical aspects of the transform. References http://www.thefouriertransform.com/applications/diffraction.php Optics-Hecht 9