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# JUNIOR PHYSICS LAB II PHYS 332

Rice University

GPA 3.68

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This 6 page Class Notes was uploaded by Ms. Janae Huels on Monday October 19, 2015. The Class Notes belongs to PHYS 332 at Rice University taught by Stanley Dodds in Fall. Since its upload, it has received 20 views. For similar materials see /class/225014/phys-332-rice-university in Physics 2 at Rice University.

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Date Created: 10/19/15

PHYS 332 Junior Physics Laboratory 11 Notes on Fourier Transforms The Fourier transform is a generalization of the Fourier series representation of functions The Fourier series is limited to periodic functions while the Fourier transform can be used for a larger class of functions which are not necessarily periodic Since the transform is essential to the understanding of several exercises we brie y explain some basic Fourier transform concepts A Fourier series Recall the definition of the Fourier series expansion for a periodic function ut If ut T ut then m ut 2601 expiwmt 1 mzim where com 2m T and 1 T2 60 ut ex iwmt dt 2 2 P If ut is a function of time it is natural to call com an angular frequency and to refer to cam as the amplitude of ut at frequency com Together the cam comprise the frequency spectrum of ut Computing the frequency spectrum for a sine or square wave of period T is fairly simple For the sine wave there are only two non zero coefficients mil 3 as one might expect Note though the odd fact that the formalism requires both positive and negative frequencies The spectrum of the square wave is slightly more complicated Cwm i m odd 4 and again contains both positive and negative frequencies The interesting feature here is that the rapid rise and fall of the square wave leads to spectral components at high frequency as demonstrated in Fig 1 This is a general effect and can help one understand the qualitative connection between time and frequency descriptions Sl 52M W Wquot 39 l M Ml N Fig 1 Building up a square wave from Fourier series components Each graph is a plot of Sn the sum of the first 11 terms in the series expansion for increasing values of n B Fourier transforms For many purposes a knowledge of the cam may be more useful than knowing ut itself It is therefore desirable to extend the idea of a frequency spectrum to non periodic functions for example single pulses or finite wave trains This can be done by replacing the sum of Eq 1 by an integral giving 1 0 E fem explwt dw 5 and C00 exp iwt dt 6 where a is now a continuous variable The complex function 0a is called the Fourier transform of ut and is again the amplitude of ut at frequency a Some examples may make the ideas clearer The transform of expia0t is a delta function 6a tag This says that the spectrum of expia0t has a large amplitude only at a mg which seems reasonable since the function is periodic with angular frequency mg The Fourier transform of a pulse centered at t 0 1 ltl 5 12 ut 7 0 Mgtrm is simply sinan 2 7 8 Cw a 2 Note that a short pulse small 7 leads to a significant 0a over a wide frequency range while a long pulse large 7 implies 0a decreases quickly with frequency As with the Fourier series results we might have expected this outcome because a rapidly changing signal must be made up of high frequencies The Fourier transform has many other amusing mathematical properties but only one more is of concern here The Fourier transform of 0a is u t except for a normalization factor Therefore if we transform a function and then transform the result we will almost recover the original function The function 0a contains both amplitude and phase information needed to reconstruct the original signal We are usually interested only in intensity information not phase so we can employ the magnitude squared of the transform Pw lcwl2 9 where Pa is called the power spectral density of the signal with units of voltszHz For a real valued signal the only case of physical interest Pa is an even function of a so all the physical information is contained in the positive frequency domain Integrating Pa over all frequencies we recover the mean square voltage 00 V 2 Pwdw 10 l further reinforcing the connection to intensity More physically the power part of the name comes from the proportionality to voltage squared and it is a spectral density because it is the voltage squared per unit frequency interval Effectively Pa tells us how much power our signal has at a specified frequency So far we have considered functions of one variable which we thought of as a time but there is nothing to prevent a generalization to functions of several variables It turns out to be useful to consider the case of two spatial variables say xy Formally uxy Zijijidw wy expi2nxwx 32 dwxdooy 7 4H 11 Cwxwy uxy exp i27rxwx yay dxdy m m 12 where cox and my are called spatial frequencies with units of length391 Just as ordinary frequency measures how fast a signal changes in time a spatial frequency measures how rapidly a signal varies in space Since the formalism is the same all of our previous observations about the properties of the transforms hold for two variable functions C Sampled data considerations So far we have tacitly assumed that the function ut is known at all values of t but it is more common to sample the input with an analog to digital converter at regular intervals At so that the required computations can be done digitally Also the signal can only be sampled for a finite time period say from T2 to TZ When transforming such sampled data we compute a discrete approximation to the ideal Fourier transform The discrete transform of an N point array u is defined to be N71 nm 6 uex i2ni 13 m E p N quot0 for m 01N 1 For a real input function u 0Nm 2 cm so we obtain N2 distinct points in the frequency spectrum uniformly spaced between f 0 and the Nyquist frequency x 2 12At The spacing between points or the frequency resolution is just lNAt lT where T is the total length of the input record Except as noted below the discrete transform is for our purposes equivalent to the continuous Fourier transform A very efficient algorithm called the Fast Fourier Transform or FFT exists to compute the sum in Eq 13 provided that N is exactly a power of two Fig 2 Sampling of sinusoidal waves Note that there is no way to distinguish between the two sine waves on the basis of the regularly spaced samples The first limitation of the sampling process is the possibility of aliasing as illustrated in Fig 2 Evidently input frequencies from zero to 12At are all distinguishable but if higher frequencies are present they will be confused with some lower frequency An example is shown in Fig 3 where the higher harmonics of the square wave signal are aliased to lower frequencies producing a misleading spectrum The Nyquist frequency x 2 12At is the highest frequency that can be measured with a sample spacing of At Experimentally it is important to avoid aliasing by inserting an analog low pass filter in the input before sampling and then choosing At small enough that the Nyquist frequency is well above the pass limit of the analog filter A more subtle problem arises from the finite length of the sampled record In general the signal will not be exactly zero at the start and end of the record so it appears to change very 105 i i i i 105 E E 0 0 E 10 E 10 IL L 1075 x x x x 75 x x x x 0 100 200 300 400 500 0 100 200 300 400 500 Frequency Hz Frequency Hz Fig 3 Power spectra of square wave signals with a fundamental of 157 Hz Left Without filtering there are numerous spurious peaks due to aliasing Right With proper low pass filtering only the first and third harmonic appear below the Nyquist frequency 5 rapidly when cut off at the ends This is equivalent to introducing spurious high frequencies into the signal which may then be aliased to frequencies below x and appear in the spectrum Depending on the information that is desired it may be possible to improve the measurement by smoothly forcing the data to zero at the ends of the time interval rather than simply truncating This is done by multiplying the input data by a window function before performing the transform For example the Hann window for an N point sample wn 05 05cos27m N 14 is often used for general purpose analysis The effect of a Hann window on the filtered square wave signal is shown in Fig 4 Other window functions can be designed to optimize the resolution for closely spaced frequencies or to obtain accurate amplitude estimates 10S 105 E s 0 0 no 10 E 10 10 5 i i i 10 5 0 100 200 300 400 500 0 100 200 300 400 500 Frequency Hz Frequency Hz Fig 4 Power spectrum of a square wave signal Left without windowing and Right with a Hann window applied

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