of University Illinois at Urbana-Champaign
Department of Physics
Physics 401 Classical Physics Laboratory
Experiment 10
Fourier Analysis
Table of Contents
Subject
Page
I. Aim-------------------------------------------------------------------------------------II. III. IV. V. Introduction---------------------------------------------------------------------------Theory Fourier series--------------------------------------------------------------Theory discrete Fourier transform-----------------------------------------------The Oscilloscope's FFT analyzer----------------------------------------------------
2 2 2 4 6 8 17 18 19 20 21
VI. Procedures-----------------------------------------------------------------------------VII. Report-----------------------------------------------------------------------------------
References--------------------------------------------------------------------------------------Appendix I - Fourier integral----------------------------------------------------------------Appendix II - LabWindows/CVI simulation program------------------------------------Appendix III Additional exercises---------------------------------------------------------
Revised 9/2006. Copyright 2006 The Board of Trustees of the University of Illinois. All rights reserved.
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I.
Aim
1) To carry out Fourier Analysis, using the FFT of the digital oscilloscope of various common waveforms and pulses. 2) To gain an understanding of the discrete Fourier transform as an extension to Fourier series. 3) To understand the effects of both linear and non-linear operations on Fourier components. II. Introduction
A periodic waveform can be expressed as a sum of sines and cosines whose frequencies are multiples of the fundamental frequency. Fourier analysis is an invaluable tool in experimental as well as theoretical work. III. Theory the Fourier series
In our application of the Fourier series we consider a function of time which has the property F (t + To ) = F (t ) , i.e. it is a periodic function of time with period To . Recall that the frequency f o = 1 To and the angular frequency o = 2 f o . A periodic function can be expressed as the sum of sines and cosines whose frequencies are integer multiples of the fundamental frequency, f o , or more conveniently the angular frequency, o . (Note To = 1 f o ) The expansion is
F (t ) =
where
ao + 2
an cos ( n o t ) +
n =1
1 bn sin ( n o t ) n=
(1)
2 an = To
and
+To / 2
/ 2 F (t ) cos (n o t ) dt T
o
(2)
bn =
2 To
+To / 2
/ 2 F (t ) sin (n o t ) dt . T
o
(3)
The integrals can be done over any interval of length To , e.g. [0, To ] may also be used. Evaluating equation (2) for n = 0 , we see that a o is twice the average of the function F (t ) .
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+To / 2
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1 ao = 2 To
/ 2 F (t ) T
o
dt
(2)
Any function can be expressed as the sum of a symmetric and an anti-symmetric part, G (t ) = G sym (t ) + G an ti sym (t ) , where G sym ( t ) = +G sym (t ) and G an ti sym ( t ) = G an ti sym (t ) . The coefficients, a n , are often called the symmetric coefficients because, if F (t ) is entirely anti-symmetric about the origin, then all the a n are zero. Similarly, the coefficients, bn , are often called the anti-symmetric coefficients because, if F (t ) is entirely symmetric about the origin, then all the bn are zero. Often, in carrying out the integrals of Eq. (2) and Eq. (3) the range of integration can be cleverly reduced, and some integrals set equal to zero, depending on the symmetries of the function F (t ) . An alternate and sometimes more useful way to write the Fourier series is to use complex exponentials instead of cosines and sines. Eq. (1) can be rewritten as
F (t ) =
ao + 2
an
n =1
1 in ot n ot e +e + 2
(
1 ) bn 2i ( e n =1
in o t
e in ot .
)
By collecting terms with the same exponential one obtains
F (t ) =
ao + 2
n =1
1 ( an ibn ) einot + 2
2 (a n =1
1
n
+ ibn ) einot .
If we define complex coefficients, cn , such that
co =
we obtain
ao
2
,
cn =
1 (a ibn ) , 2 n
c n =
1 (a + ibn ) , 2 n
(4a)
F (t ) =
ce n =
n
in o t
.
(4b)
Note that c n = cn* , since an and bn are real. Using Eq. (4), we can calculate a single new Fourier integral for all the cn .
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1 cn = To
+To / 2
F (t ) e T / 2
o
in o t
dt
(5)
Again this integral can be done over any convenient domain of one period. III. Theory the discrete Fourier transform
The waveforms discussed above extend over all time. We make measurements over finite time intervals. The Fourier components are infinite in number. No measurement is possible for an infinite number of components. The Fourier transform is strictly applicable to functions whose domain is < t < + . The discrete Fourier transform may be applied to finite, digitally sampled waveforms. The fundamental basis of the discrete Fourier transform is a mathematical theorem known as the sampling theorem. At the most elementary level the sampling theorem states that if a periodic waveform has no Fourier components beyond a frequency fc (this frequency is called the folding frequency, the critical frequency, or the Nyquist frequency), then the waveform can be completely constructed from a finite number of its samples. Consider a waveform h(t) that is sampled at time interval t as shown in the figure below.
Fig. 1 Waveform sampled at discrete intervals, t The waveform h(t) is determined at N times hk = h (k t ) k = 0, 1, 2, ..., N 1 (6)
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With these N measurements we determine the coefficients of N complex amplitudes, Hn.
hk =
1 N
n=0
H e
n
N 1
2 ik n / N
(7)
The expression above is a discrete version of Eq. 4b. For a strictly periodic waveform with no frequency above the folding frequency, f c = 1 2 N t , the magnitude of these amplitudes are the Fourier coefficients.
Fig. 2 Fourier coefficients FFT The original waveform can be completely constructed from the coefficients with the inverse relation
H n = hk e 2 ik n / N .
k =0
N 1
(8)
The relevant parameters of the discrete Fourier transform are 1) N : total number of discrete samples. 2) t : the time interval between samples. 3) N t = Tmax the total sampling time. Note t = Tmax N 4) 1 t = f s : the sampling frequency. Note f s = 1 t . 5) f c = 1 2 t : the folding frequency. 6) f : the frequency (resolution) increment of the transform. Note f = 1 Tmax .
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V.
The Oscilloscope FFT Analyzer
The digital oscilloscope samples the input waveform. The oscilloscope trace displays 10000 samples equally spaced in time. The sampling rate is 10000 divided by the measurement time, which is selected by the sweep rate or Horizontal Scale. The oscilloscope has a special mathematical function called Fast Fourier Transform (FFT) which can calculate the discrete Fourier Transform of the sampled waveform. The Fourier transform is displayed as Fourier amplitudes versus frequency. Since the algorithm of the FFT requires 2n samples, the oscilloscope uses 1024 time samples which are converted into 512 frequency amplitudes. The phase, i.e. the relative amount of sine and cosine in a particular Fourier component, is ignored. The oscilloscope displays the magnitude of cn . The oscilloscope FFT display is described in exercise 1 below. Note from Eq. (4a)
cn = 1 an 2 + bn 2 . 2
(9)
The figure below shows some of the waveforms that we will analyze. The coefficients of the Fourier series for these waveforms have simple analytical forms. It helps to think about the analytic forms when using the oscilloscope FFT.
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Fig. 3
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VI.
Procedures
In this laboratory exercise you will study the properties of the discrete Fourier transform. The Wavetek Function Generator is the source of the waveforms for most of the laboratory exercises. Connect the output of the Wavetek to the channel 1 input of the digital oscilloscope with a BNC cable. We will use the sine wave, the bipolar square wave, the bipolar triangle wave, and the unipolar square wave with variable duty factor as our signals. The coefficients of the Fourier series for these waveforms have simple closed form analytic expressions. Excel files which show Fourier series for the square wave and triangle wave are on the 401 web site under Tutorials and Lectures, Experiment 10 and in the Common folder of the Physics 401 server, Phyaplportal. These files are from the Physics of Music course, Physics199POM, offered by Professor Errede. See, http://wug.physics.uiuc.edu/courses/phys199pom/ . These files can serve as a template for further investigation. (The files in the 401 Common folder are read only. There are elements in files that are not relevant to this laboratory exercise.) There is also a program written in LabWindows/CVI on the PCs in the Physics 401 laboratory. See Appendix II for a discussion of this program. Note that this program calculates and displays the an and bn coefficients of the Fourier series. The FFT of the digital oscilloscope displays amplitude in dBV or voltage, i.e. it ignores the relative proportion of sine and cosine at a given frequency. Exercise 1. FFT of a sine wave with period commensurate with the sampling time In the first exercise you will use the digital oscilloscope to obtain the discrete Fourier transform of a sine wave under conditions that minimize the problem of leakage. Set up the Wavetek to generate a 1.000 V and 1.000 kHz sine wave. (Wave form #0 in Fig. 3 is a sine wave.) Observe the signal in the time domain, and then, using the FFT function, in the frequency domain. Choose an appropriate sweep rate, e.g. 4 ms/div. This sweep rate corresponds to 250 kSa/s, which is appropriate for this signal. The FFT function requires some setup. Follow the general instructions below for optimum operation. FFT operation:
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1) Make sure your signal peaks do not go off the screen. Off-screen signal peaks will result in FFT waveform errors. 2) Set the Horizontal SCALE control to show 5 or more cycles of the source signal. Showing more cycles provides two benefits: (a) better frequency resolution, and (b) reduce aliasing. 3) Press Vertical MATH button (red color button) to show math menu. FFT is part of the math menu. 4) Push FFT screen button to display FFT side menu. 5) Make sure to select the signal source (for example: Set FFT source to Channel 1 if the signal source is connected to Channel 1). 6) Select the appropriate vertical scale and FFT window. The qualitative properties of the four window filters are listed below. The quantitative action of the window filter is discussed in the HP application note #243, Fundamentals of Signal Analysis, on the Physics 401 web site under Experiment 10 and in many texts. 7) To understand the window, recall that the Fourier Transform integrates over all time. But FFT operates on a finite length time record. In general users may not have control over the shape and phase of a waveform. This has the effect of replicating the finite length time record over all time. Result of it would introduce transients in the waveform. Subsequent frequency domain would have the harmonics spread out, rather than have thin and slender peaks. This effect is called leakage effect. To prevent this, the waveform is forced to end at the end of the time record to assure that no transients will exist when the time record is replicated. Mathematically it is accomplished by multiplying the time record by a WINDOW function. Since a window will modify the time record and will also produce its own undesirable effects in the frequency domain, it is important to choose a proper window. In our scope, there are 4 windows available: Hamming, Hanning, Rectangular and Blackman-Harris. Typically for sine, periodic and narrowband random noise, choose Hamming/Hanning window. Hamming has slightly better frequency resolution than Hanning. Rectangular window is useful for: (1) transients where the signal amplitude before and after the event are nearly equal. (2) measuring the equal amplitude sine waves with very close frequencies. (3) broad-band random noise with slowly varying spectrum. Blackman-Harris window is useful to measure the predominantly single frequency signals to look for higher order harmonics. In this laboratory, determine the best window empirically selecting each window and compare them. 8) You may use the Zoom button (button with the magnifying glass) at Horizontal area of the scope control panel. Zoom range is controlled by the sweep rate or Horizontal SCALE knob. You may use the Cursor to measure the frequency and the amplitude of the signal. You may need a few minutes of experimentations with various controls and functions to understand the measurement technique. A typical waveform would appear as shown in Fig 4.
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Fig. 4 Oscilloscope display of FFT of 1.00 kHz sine wave with Hanning window You will briefly investigate the properties of the windowing functions in the exercises below. 9) Use the Horizontal Scale knob to narrow the zoom range. The zoomed image is shown at the bottom in expanded way. The top image is the original FFT display. Use the Cursor to measure the frequency and the amplitude of the fundamental, since the sine wave has one single frequency component in the frequency domain. Note that the M symbol at the left side of the display shows the zero amplitude level. Changing the Horizontal Scale knob changes the zoomed display. Use it if you need to see the details of each of the spectrum. Change to different windows and observe the corresponding displays. 9) Choose the rectangular window for the FFT, and save the display using the eScope application from your desktop. Rectangular window produces well-defined in this case because the period of
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the waveform (1 ms) is an integral multiple of the sampling period (20 ms). The rectangular window is not useful for most waveforms. The vertical scale is logarithmic (base 10), displayed in dBV (decibels relative to 1 Volt RMS). The measured amplitude is then
20 log Vrms 1V
(10)
where the numerator is the rms value of the signal. Recall that a 2.82 V peak-to-peak sine wave is 1.00 V rms, and thus will read 0.0 dBV on the FFT display. Verify here and in your report that the measured amplitude agrees with the above formula. The noise level is at approximately 50 dBV. The FFT spectrum at this level is erratic. Press the RUN/STOP key to freeze the display. With the cursor, measure the frequency of the peak. Note that peak is broad. The breadth is due to the limited frequency resolution of the FFT. Unfortunately, the details of the FFT are nowhere available in the Tektronix documentation. Examine the FFT spectrum for the presence of harmonic distortion in the Wavetek signal, i. e. evidence of a signal at higher harmonics of the fundamental. The specified harmonic distortion in the Wavetek manual (<1% from 10.00 mHz to 100.0 kHz) is at the limit of the sensitivity of this oscilloscope. Is there evidence for harmonic distortion? A high quality (and complicated) spectrum analyzer is available in the laboratory for your instructor to demonstrate the harmonic distortion of the Wavetek. The display screen is similar to the digital oscilloscope. Observe the amplitudes of the harmonics of a 1.000 kHz, 1.000 V sine wave. They are very small but measurable with a good quality spectrum analyzer. Exercise 2. Demonstrate aliasing with a sine wave whose is period incommensurate with the sampling time In the third exercise you will obtain the discrete Fourier transform of the 12.63 kHz sine wave under conditions that demonstrate the problem of aliasing. Begin with a sampling rate of 250 kSa/s (4 ms/div). Find the frequency displayed the by FFT with the cursors. (The time domain display at this sampling rate shows about 40 cycles.) Change the sampling rate in steps from
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a) b) c) d)
250 kSa/s (4 ms/div) to 100 kSa/s (10 ms/div), 100 kSa/s (10 ms/div) to 50 kSa/s (20 ms/div), 50 kSa/s (20 ms/div) to 25 kSa/s (40 ms/div), 25 kSa/s (50 ms/div) to 10 kSa/s (100 ms/div).
In each step find the frequency displayed by the FFT with the cursors. Determine at which sampling rate aliasing has occurred. Explain here and in your report why aliasing occurs for this waveform at this sampling rate. Aliasing can also be convincingly demonstrated by changing the frequency of the signal for a fixed sampling rate. Choose a sampling rate, and then find a range of frequencies which demonstrate aliasing. Record the sampling rate and the frequencies. Exercise 3. Find FFT of a square wave (Caution: The square wave has significant high frequency Fourier components. Thus the conditions of the sampling theorem, namely, that the waveform have no Fourier strength beyond the folding frequency, are not satisfied. The amplitudes of the lower frequency components are still reliably shown. The baseline of the FFT is erratic due to the fold over from the high frequency components.) Set the Wavetek to a bipolar square wave at 1.000 kHz and amplitude 1.000 V. Leakage is artificially minimized with this choice of frequency. (Wave form #1 in the Fig. 3 is a square wave.) Use the time domain to verify that you have the correct waveform. The FFT spectrum should show many peaks. For a complicated waveform the FFT displays
20 log
an 2 + bn 2 2 1V
where the an and bn are the coefficients of the discrete Fourier transform. Measure the amplitudes of the harmonics up to n = 11. Recall that the coefficients of the Fourier series are 4 n for odd n and zero for even n . (See the Fourier simulation program or the eScope data file.) For your lab report answer the questions below.
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(a) Verify that the frequency components in the oscilloscope FFT agree with the harmonics from the analytic expression or in the computer simulation. (b) Note which harmonics are zero. (In your report state what property of the waveform makes these harmonics zero. The answer is not obvious.) (c) Use the expression above to calculate the amplitude of each peak in the FFT (non-zero harmonics) in volts from the measured amplitude in dBV. Do the calculations in Excel. Then use Excel to make a plot of the amplitude of the non-zero harmonics versus harmonic number, n, to verify that your data are sensible (d) For your report plot the amplitude of the non-zero harmonics versus harmonic number, n, on with linear axes and with log-log axes. On the same graph also plot the amplitude of the harmonics versus harmonic number of the Fourier series. Verify that there is agreement between the measured FFT and the Fourier series. (e) For your report determine the slope of your log-log plot. (f) For your report justify the value of the slope using the closed form solution for the Fourier coefficients.
Exercise 4. Find FFT of a triangle wave Set the Wavetek to a triangle wave at the same frequency (1.000 kHz) and amplitude (1.000 V).
(Waveform #2 in Fig. 3 is a triangle wave.) Use the time domain to verify that you have the correct waveform. Carry out the same procedures on the triangle wave as for the square wave above.
Exercise 5. Find the response of linear network When a periodic waveform is modified by linear elements (e.g. resistors, capacitors and inductors) the phase (i.e. relative magnitudes of the an and bn ) and the amplitude of the Fourier
components are changed. However, no new frequencies are generated. To illustrate this principle, use the circuit shown in Fig. 5. 1) Set the Wavetek for a 1.000 kHz bipolar square wave, and R to 1 kilohm. First set C = 0 F and measure the harmonics in the signal up to n = 7. Next, leaving R unchanged, set C = 0.1 F . Note the change in the waveform and the FFT. Again measure the amplitudes of the harmonics in the signal up to n = 7. Save the display for later reference.
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R Wavetek output Oscillocope input 11.11 K max
C = 0.1 F
Fig. 5 The circuit is a voltage divider for which the ratio of the complex impedances 1 1 iC = . 1 1 + i RC R+ iC
(11)
This ratio can be written in polar form:
r= 1 1 + ( RC )
2
= tan 1 ( RC ) .
(12)
Our FFT measures only amplitudes not phases so we can ignore the phase. The signal then at each angular frequency = n o is attenuated by the factor
r= 1 1 + (no RC ) 2
.
(13)
Let Vn (C = 0.1 F ) and Vn (C = 0 F ) be the measured amplitudes with and without the capacitor. Then the ratio of Vn (C = 0.1 F ) to Vn (C = 0 F ) for each harmonic is equal to the attenuation factor.
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For your lab report calculate the amplitudes of the components in volts from the measurements in dBV. Calculate the attenuation of each component. Compare the calculated attenuation to the measured attenuation, i.e. the ratio of the amplitudes measured with C = 0.1 F to C = 0 F .
Exercise 6. Find the response of non-Linear network Set the Wavetek to a 1.000 kHz, 1.000 V sine wave. Replace the capacitor with a diode, and leave
R = 1 kilohm. Observe the signal across the diode in the time domain. Note the effect of the diode. The input signal has only one frequency. The diode is a non-linear element, and it introduces new frequencies. Measure the amplitudes of the harmonics up to n = 7. Save the display for future reference. For your report give an explanation for the origin of the signals at the new frequencies.
Exercise 7. Find the frequencies in an unknown signal A signal is distributed to each bench. The black terminal is ground and the red terminal is the
signal. Connect the terminals to the oscilloscope with appropriate leads. Observe the signal in the time domain and in the frequency domain. Measure the amplitudes and frequencies of all components above the noise level in the FFT spectrum. Save the display for future reference. Calculate the amplitude of the components in volts from the measurements in dBV. VII. Report
For your laboratory report complete the analyses described in the exercises above. Answer the questions asked in the exercises. Describe the results of each exercise. Note that each graph should have a title, labeled axes with appropriate units.
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References http://www.me.psu.edu/me82/Learning/FFT/FFT.html These web pages are an excellent introduction to the discrete Fourier transform. The discussion is at an appropriate level and appropriate length. Fundamentals of Signal Analysis, Agilent Technologies Application Note 243. Link to the application note is on the Physics 401 web site under Experiment 10. Numerical Recipes, William H. Press et al., Cambridge University Press. Chapter 12 of Numerical Recipes is on the FFT. The book is on the web at http://www.library.cornell.edu/nr/bookcpdf.html
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Appendix I Fourier Integral Recall the expression for the Fourier series using complex exponential from above.
F (t ) =
c ein t n =
o
n
cn =
1 T
+T / 2
/ 2 F (t ) e T
in0t
dt
In exercise 5 we let the period T get larger and larger. Equivalently, the fundamental frequency, o , becomes smaller and smaller. In the limit T the Fourier coefficients cn c( ) , i.e. they become a continuum. The Fourier series becomes a Fourier integral.
F (t ) = c( ) eit d
+
1 c( ) = 2
+
F (t ) e
it
dt
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Appendix II LabWindows/CVI Fourier Simulation Program There is a LabWindows/CVI program that calculates the coefficients of the Fourier series on the Physics 401 PCs. The program is foursim2 and can be run from a shortcut on the desktop. If there is no desktop link, simply search for the program. Alternatively, click on the Shortcut to CVI or Start -> Program -> LabWindows_CVI -> LabWindows_CVI to launch the CVI software. At the menu panel, click on File -> Open -> Project[*.prj] to load the foursim2.prj program. [ Or, it may have already been loaded if you see its name in the project window. ] Press Shift-F5 or click at the menu panel Run -> Run Project to run the program. Explore the front panel of the program. There are a variety different waveforms with several different amplitudes available to analyze. The fundamental frequency is set to 1 Hz. The program computes 25 Fourier coefficients and displays them on the screen. It can also display the harmonics and coefficients in graphical plots. The regeneration of the original wave from the harmonics is another feature of this program. Watch how the wave is regenerated by adding harmonics. You are encouraged to run the program for all waveforms, as time allows. Each waveform takes a minute or less to run. The program also lets you print the results in paper format so that you could analyze them for your report. A word of caution in the interpretation of the results is in order. Following equations (2) and (3) above, the program does a numerical integration to obtain the coefficients of the Fourier series. The coefficients are displayed in scientific notation. Due to accuracy of the integration technique coefficients which should be identically zero are not zero. In almost all cases it is easy to see from the large negative exponent that the coefficients is actually zero.
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