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NATIONAL COLLEGE OF SCIENCE & TECHNOLOGY Amafel Bldg. Aguinaldo Highway Dasmariñas City, Cavite EXPERIMENT 5 Fourier Theory – Frequency Domain and Time DomainPagara, Sheila Marie P. September 06, 2011Signal Spectra and Signal Processing/BSECE 41A1 Score: Engr. Grace Ramones Instructor
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Objectives: 1. Learn how a square wave can be produced from a series of sine waves at different frequencies and amplitudes. 2. Learn how a triangular can be produced from a series of cosine waves at different frequencies and amplitudes. 3. Learn about the difference between curve plots in the time domain and the frequency domain. 4. Examine periodic pulses with different duty cycles in the time domain and in the frequency domain. 5. Examine what happens to periodic pulses with different duty cycles when passed through low- pass filter when the filter cutoff frequency is varied.
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Sample ComputationFrequency (f) (Step 4 & 11)Duty cycle (D) (Step 19 & 27)Difference (Question in Step 19 )First zero crossing point (f) (Step 31 )Bandwidth (BW)( Step 31 )
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Data Sheet:Materials:One function generatorOne oscilloscopeOne spectrum analyzerOne LM 741 op-ampTwo 5 nF variable capacitorsResistors: 5.86 kΩ, 10 kΩ, and 30 kΩTheory:Communications systems are normally studies using sinusoidal voltage waveforms to simplify theanalysis. In the real world, electrical information signal are normally nonsinusoidal voltage waveforms,such as audio signals, video signals, or computer data. Fourier theory provides a powerful means ofanalyzing communications systems by representing a nonsinusoidal signal as series of sinusoidal voltagesadded together. Fourier theory states that a complex voltage waveform is essentially a composite ofharmonically related sine or cosine waves at different frequencies and amplitudes determined by theparticular signal waveshape. Any, nonsinusoidal periodic waveform can be broken down into sine orcosine wave equal to the frequency of the periodic waveform, called the fundamental frequency, and aseries of sine or cosine waves that are integer multiples of the fundamental frequency, called theharmonics. This series of sine or cosine wave is called a Fourier series.Most of the signals analyzed in a communications system are expressed in the time domain, meaning thatthe voltage, current, or power is plotted as a function of time. The voltage, current, or power isrepresented on the vertical axis and time is represented on the horizontal axis. Fourier theory provides anew way of expressing signals in the frequency domain, meaning that the voltage, current, or power isplotted as a function of frequency. Complex signals containing many sine or cosine wave components areexpressed as sine or cosine wave amplitudes at different frequencies, with amplitude represented on thevertical axis and frequency represented on the horizontal axis. The length of each of a series of verticalstraight lines represents the sine or cosine wave amplitudes, and the location of each line along thehorizontal axis represents the sine or cosine wave frequencies. This is called a frequency spectrum. Inmany cases the frequency domain is more useful than the time domain because it reveals the bandwidthrequirements of the communications system in order to pass the signal with minimal distortion. Testinstruments displaying signals in both the time domain and the frequency domain are available. Theoscilloscope is used to display signals in the time domain and the spectrum analyzer is used to display thefrequency spectrum of signals in the frequency domain.In the frequency domain, normally the harmonics decrease in amplitude as their frequency gets higheruntil the amplitude becomes negligible. The more harmonics added to make up the composite waveshape,the more the composite waveshape will look like the original waveshape. Because it is impossible todesign a communications system that will pass an infinite number of frequencies (infinite bandwidth), aperfect reproduction of an original signal is impossible. In most cases, eliminate of the harmonics does not
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significantly alter the original waveform. The more information contained in a signal voltage waveform(after changing voltages), the larger the number of high-frequency harmonics required to reproduce theoriginal waveform. Therefore, the more complex the signal waveform (the faster the voltage changes), thewider the bandwidth required to pass it with minimal distortion. A formal relationship betweenbandwidth and the amount of information communicated is called Hartley’s law, which states that theamount of information communicated is proportional to the bandwidth of the communications systemand the transmission time.Because much of the information communicated today is digital, the accurate transmission of binarypulses through a communications system is important. Fourier analysis of binary pulses is especiallyuseful in communications because it provides a way to determine the bandwidth required for theaccurate transmission of digital data. Although theoretically, the communications system must pass all theharmonics of a pulse waveshape, in reality, relatively few of the harmonics are need to preserve thewaveshape.The duty cycle of a series of periodic pulses is equal to the ratio of the pulse up time (tO) to the timeperiod of one cycle (T) expressed as a percentage. Therefore,In the special case where a series of periodic pulses has a 50% duty cycle, called a square wave, the plot inthe frequency domain will consist of a fundamental and all odd harmonics, with the even harmonicsmissing. The fundamental frequency will be equal to the frequency of the square wave. The amplitude ofeach odd harmonic will decrease in direct proportion to the odd harmonic frequency. Therefore,The circuit in Figure 5–1 will generate a square wave voltage by adding a series of sine wave voltages asspecified above. As the number of harmonics is decreased, the square wave that is produced will havemore ripples. An infinite number of harmonics would be required to produce a perfectly flat square wave. Figure 5 – 1 Square Wave Fourier Series
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XSC1 Ext T rig V6 + R1 J1 _ A B 10.0kΩ + _ + _ 10 V Key = A V1 R2 J2 10 Vpk 10.0kΩ 1kHz Key = B 0° V2 R3 J3 4 155 0 8 160 14 13 12 R7 109 02 3 100Ω 3.33 Vpk 10.0kΩ 3kHz Key = C 0° V3 R4 J4 2 Vpk 10.0kΩ 5kHz Key = D 0° V4 R5 J5 1.43 Vpk 10.0kΩ 7kHz 0° Key = E V5 J6 R6 1.11 Vpk 10.0kΩ 9kHz Key = F 0° .The circuit in Figure 5-2 will generate a triangular voltage by adding a series of cosine wave voltages. Inorder to generate a triangular wave, each harmonic frequency must be an odd multiple of thefundamental with no even harmonics. The fundamental frequency will be equal to the frequency of thetriangular wave, the amplitude of each harmonic will decrease in direct proportion to the square of theodd harmonic frequency. Therefore,Whenever a dc voltage is added to a periodic time varying voltage, the waveshape will be shifted up bythe amount of the dc voltage. Figure 5 – 2 Triangular Wave Fourier Series
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XSC1 Ext T rig V6 + R1 J1 _ A B 10.0kΩ + _ + _ 15 V Key = A V1 R2 J2 10 Vpk 10.0kΩ 1kHz 90° V2 Key = B R3 J3 13 12 1 2 3 4 5 8 9 11 0 R7 6 0 1.11 Vpk 100Ω 10.0kΩ 3kHz 90° V3 Key = C R4 J4 0.4 Vpk 10.0kΩ 5kHz 90° V4 Key = D R5 J5 0.2 Vpk 10.0kΩ 7kHz 90° Key = E For a series of periodic pulses with other than a 50% duty cycle, the plot in the frequency domain will consist of a fundamental and even and odd harmonics. The fundamental frequency will be equal to the frequency of the periodic pulse train. The amplitude (A) of each harmonic will depend on the value of the duty cycle. A general frequency domain plot of a periodic pulse train with a duty cycle other than 50% is shown in the figure on page 57. The outline of peaks if the individual frequency components is called envelope of the frequency spectrum. The first zero-amplitude frequency crossing point is labelled fo = 1/to, there to is the up time of the pulse train. The first zero-amplitude frequency crossing point fo) determines the minimum bandwidth (BW) required for passing the pulse train with minimal distortion. Therefore,A f=1/to f 2/to Frequency Spectrum of a Pulse Train
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Notice than the lower the value of to the wider the bandwidth required to pass the pulse train withminimal distortion. Also note that the separation of the lines in the frequency spectrum is equal to theinverse of the time period (1/T) of the pulse train. Therefore a higher frequency pulse train requires awider bandwidth (BW) because f = 1/TThe circuit in Figure 5-3 will demonstrate the difference between the time domain and the frequencydomain. It will also determine how filtering out some of the harmonics effects the output waveshapecompared to the original3 input waveshape. The frequency generator (XFG1) will generate a periodicpulse waveform applied to the input of the filter (5). At the output of the filter (70, the oscilloscope willdisplay the periodic pulse waveform in the time domain, and the spectrum analyzer will display thefrequency spectrum of the periodic pulse waveform in the frequency domain. The Bode plotter willdisplay the Bode plot of the filter so that the filter bandwidth can be measured. The filter is a 2-pole low-pass Butterworth active filter using a 741 op-amp. Figure 5-3 Time Domain and Frequency Domain XFG1 XSC1 C1 XSA1 Ext T rig + 2.5nF 50% _ Key=A A _ B _ IN T + + R1 R2 741 30kΩ 30kΩ 42 OPAMP_3T_VIRTUAL 0 6 0 31 R3 C2 R4 5.56kΩ 10kΩ XBP1 2.5nF 50% Key=A R5 IN OUT 10kΩProcedure:Step 1 Open circuit file FIG 5-1. Make sure that the following oscilloscope settings are selected: Time base (Scale = 200 µs/Div, Xpos = 0, Y/t), Ch A (Scale = 5V/Div, Ypos = 0, DC), Ch B (Scale = 50 mV/Div, Ypos = 0, DC), Trigger (Pos edge, Level = 0, Auto). You will generate a square wave curve plot on the oscilloscope screen from a series of sine waves called a Fourier series.Step 2 Run the simulation. Notice that you have generated a square wave curve plot on the oscilloscope screen (blue curve) from a series of sine waves. Notice that you have also plotted the fundamental sine wave (red). Draw the square wave (blue) curve on the plot and the fundamental sine wave (red) curve plot in the space provided.
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Step 3 Use the cursors to measure the time periods for one cycle (T) of the square wave (blue) and the fundamental sine wave (red) and show the value of T on the curve plot. = T1 = 1.00 ms T2 = 1.00 msStep 4 Calculate the frequency (f) of the square wave and the fundamental sine wave from the time period. = f = 1 kHzQuestions: What is the relationship between the fundamental sine wave and the square wave frequency(f)? = They are the sameWhat is the relationship between the sine wave harmonic frequencies (frequencies of sine wavegenerators f3, f5, f7, and f9 in figure 5-1) and the sine wave fundamental frequency (f1)? = Both are all odd functions.What is the relationship between the amplitude of the harmonic sine wave generators and the amplitudeof the fundamental sine wave generator? = The amplitude of the odd harmonics decrease in direct proportion to odd harmonic frequency.Step 5 Press the A key to close switch A to add a dc voltage level to the square wave curve plot. (If the switch does not close, click the mouse arrow in the circuit window before pressing the A key). Run the simulation again. Change the oscilloscope settings as needed. Draw the new square wave (blue) curve plot on the space provided.
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Question: What happened to the square wave curve plot? Explain why. = The square wave curve plot moved upwards because of the additional dc voltage.Step 6 Press the F and E keys to open the switches F and E to eliminate the ninth and seventh harmonic sine waves. Run the simulation again. Draw the new curve plot (blue) in the space provided. Note any change on the graph.Step 7 Press the D key to open the switch D to eliminate the fifth harmonics sine wave. Run the simulation again. Draw the new curve plot (blue) in the space provided. Note any change on the graph.Step 8 Press the C key to open switch C and eliminate the third harmonic sine wave. Run the simulation again.Question: What happened to the square wave curve plot? Explain. = The square wave curve plot became sinusoidal. Because the sine wave harmonics are missing, the sine wave fundamental was the only wave left.
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Step 9 Open circuit file FIG 5-2. Make sure that the following oscilloscope settings are selected: Time base (Scale = 200 µs/Div, Xpos = 0, Y/t), Ch A (Scale = 5V/Div, Ypos = 0, DC), Ch B (Scale = 100 mV/Div, Ypos = 0, DC), Trigger (Pos edge, Level = 0, Auto). You will generate a triangular wave curve plot on the oscilloscope screen from a series of sine waves called a Fourier series.Step 10 Run the simulation. Notice that you have generated a triangular wave curve plot on the oscilloscope screen (blue curve) from the series of cosine waves. Notice that you have also plotted the fundamental cosine wave (red). Draw the triangular wave (blue) curve plot and the fundamental cosine wave (red) curve plot in the space provided.Step 11 Use the cursors to measure the time period for one cycle (T) of the triangular wave (blue) and the fundamental (red), and show the value of T on the curve plot. = T1 = 1.00 ms = T2 = 1.00 msStep 12 Calculate the frequency (f) of the triangular wave from the time period (T). = f = 1 kHzQuestions: What is the relationship between the fundamental frequency and the triangular wavefrequency? = The frequencies are the same.What is the relationship between the harmonic frequencies (frequencies of generators f 3, f5, and f7 infigure 5-2) and the fundamental frequency (f1)? = They are all odd functions.What is the relationship between the amplitude of the harmonic generators and the amplitude of thefundamental generator? = The amplitude of the odd harmonics decrease in direct proportion to odd harmonic frequency.
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Step 13 Press the A key to close switch A to add a dc voltage level to the triangular wave curve plot. Run the simulation again. Draw the new triangular wave (blue) curve plot on the space provided.Question: What happened to the triangular wave curve plot? Explain. = The triangular wave curve plot moved upwards because of the additional dc voltage.Step 14 Press the E and D keys to open switches E and D to eliminate the seventh and fifth harmonic sine waves. Run the simulation again. Draw the new curve plot (blue) in the space provided. Note any change on the graph.Step 15 Press the C key to open the switch C to eliminate the third harmonics sine wave. Run the simulation again.Question: What happened to the triangular wave curve plot? Explain. = The triangular wave curve plot became sinusoidal. The sine wave harmonics are missing, that is why the sine wave fundamental was only wave left.Step 16 Open circuit FIG 5-3. Make sure that following function generator settings are selected: Square wave, Freq = 1 kHz, Duty cycle = 50%, Ampl – 2.5 V, Offset = 2.5 V. Make sure that the following oscilloscope settings are selected: Time base (Scale = 500 µs/Div, Xpos = 0, Y/T), Ch A (Scale = 5 V/Div, Ypos = 0, DC), Ch B (Scale = 5 V/Div, Ypos = 0, DC), Trigger (pos edge, Level = 0, Auto). You will plot a square wave in the time domain at the input and output of a two-pole low-pass Butterworth filter.Step 17 Bring down the oscilloscope enlargement and run the simulation to one full screen display, then pause the simulation. Notice that you are displaying square wave curve plot
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in the time domain (voltage as a function of time). The red curve plot is the filter input (5) and the blue curve plot is the filter output (7)Question: Are the filter input (red) and the output (blue) plots the same shape disregarding anyamplitude differences? = Yes.Step 18 Use the cursor to measure the time period (T) and the time (fo) of the input curve plot (red) and record the values. = T= 1 ms to = 500.477µsStep 19 Calculate the pulse duty cycle (D) from the to and T = D = 50.07%.Question: How did your calculated duty cycle compare with the duty cycle setting on the functiongenerator? = The difference is 0.07%.Step 20 Bring down the Bode plotter enlargement to display the Bode plot of the filter. Make sure that the following Bode plotter settings are selected; Magnitude, Vertical (Log, F = 10 dB, I = -40 dB), Horizontal (Log, F = 200 kHz, I = 100 Hz). Run the simulation to completion. Use the cursor to measure the cutoff frequency (fC) of the low-pass filter and record the value. = fC = 21.197Step 21 Bring down the analyzer enlargement. Make sure that the following spectrum analyzer settings are selected: Freq (Start = 0 kHz, Center = 5 kHz, End = 10 kHz), Ampl (Lin, Range = 1 V/Div), Res = 50 Hz. Run the simulation until the Resolution frequencies match, then pause the simulation. Notice that you have displayed the filter output square wave frequency spectrum in the frequency domain, use the cursor to measure the amplitude of the fundamental and each harmonic to the ninth and record your answers in table 5-1. Table 5-1 Frequency (kHz) Amplitude f1 1 5.048 V f2 2 11.717 µV f3 3 1.683 V f4 4 15.533 µV f5 5 1.008 V f6 6 20.326 µV f7 7 713.390 mV f8 8 25.452 µV f9 9 552.582 mV
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Questions: What conclusion can you draw about the difference between the even and odd harmonics for asquare wave with the duty cycle (D) calculated in Step 19? = The frequency domain consists of a fundamental and odd harmonics while the even harmonics are almost zero.What conclusions can you draw about the amplitude of each odd harmonic compared to the fundamentalfor a square wave with the duty cycle (D) calculated in Step 19? = The amplitude of each odd harmonic decreases as the fundamental frequency for a square wave.Was this frequency spectrum what you expected for a square wave with the duty cycle (D) calculated inStep 19? = Yes.Based on the filter cutoff frequency (fC) measured in Step 20, how many of the square wave harmonicswould you expect to be passed by this filter? Based on this answer, would you expect much distortion ofthe input square wave at the filter? Did your answer in Step 17 verify this conclusion? = There are square waves. Yes, because the more number of harmonics square wave the more distortion in the input square wave.Step 22 Adjust both filter capacitors (C) to 50% (2.5 nF) each. (If the capacitors won’t change, click the mouse arrow in the circuit window). Bring down the oscilloscope enlargement and run the simulation to one full screen display, then pause the simulation. The red curve plot is the filter input and the blue curve plot is the filter output.Question: Are the filter input (red) and output (blue) curve plots the same shape, disregarding anyamplitude differences? = No.Step 23 Bring down the Bode plotter enlargement to display the Bode plot of the filter. Use the cursor to measure the cutoff frequency (Fc of the low-pass filter and record the value. = fc = 2.12 kHzStep 24 Bring down the spectrum analyzer enlargement to display the filter output frequency spectrum in the frequency domain, Run the simulation until the Resolution Frequencies match, then pause the simulation. Use cursor to measure the amplitude of the fundamental and each harmonic to the ninth and record your answers in Table 5-2. Table 5-2 Frequency (kHz) Amplitude f1 1 4.4928 V f2 2 4.44397µV f3 3 792.585 mV f4 4 323.075 µV
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f5 5 178.663mV f6 6 224.681 µV f7 7 65.766 mV f8 8 172.430 µV f9 9 30.959 mVQuestions: How did the amplitude of each harmonic in Table 5-2 compare with the values in Table 5-1? = Compare with the previous table, the value of the amplitude of the harmonics is lower.Based on the filter cutoff frequency (fc), how many of the square wave harmonics should be passed by thisfilter? Based on this answer, would you expect much distortion of the input square wave at the filteroutput? Did your answer in Step 22 verify this conclusion? = Based on the fc, there should be less than 5 square wave harmonics.. Yes, there have much distortion in the input square wave at the filter output.Step 25 Change the both capacitor (C) back to 5% (0.25 nF). Change the duty cycle to 20% on the function generator. Bring down the oscilloscope enlargement and run the simulation to one full screen display, then pause the simulation. Notice that you have displayed a pulse curve plot on the oscilloscope in the time domain (voltage as a function of time). The red curve plot is the filter input and the blue curve plot is the filter output.Question: Are the filter input (red) and the output (blue) curve plots the same shape, disregarding anyamplitude differences? = Yes.Step 26 Use the cursors to measure the time period (T) and the up time (to) of the input curve plot (red) and record the values. = T= 1 ms to =Step 27 Calculate the pulse duty cycle (D) from the to and T. = D = 19.82%Question: How did your calculated duty cycle compare with the duty cycle setting on the functiongenerator? = The calculated duty cycle compare with the duty cycle setting on the function generator are almost the same. Their difference is only 0.18%Step 28 Bring down the Bode plotter enlargement to display the Bode plot of the filter. Use the cursor to measure the cutoff frequency (fC) of the low-pass filter and record the value. = fC = 21.197 kHzStep 29 Bring down the spectrum analyzer enlargement to display the filter output frequency spectrum in the frequency domain. Run the simulation until the Resolution Frequencies
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match, then pause the simulation. Draw the frequency plot in the space provided. Also draw the envelope of the frequency spectrum. 5.041 kHzQuestion: Is this the frequency spectrum you expected for a square wave with duty cycle less than 50%? = Yes.Step 30 Use the cursor to measure the frequency of the first zero crossing point (fo) of the spectrum envelope and record your answer on the graph. = fo = 5.041 kHzStep 31 Based on the value of the to measured in Step 26, calculate the expected first zero crossing point (fo) of the spectrum envelope. = fo = 5.045 kHzQuestion: How did your calculated value of fo compare the measured value on the curve plot? = They have a difference of 0.004 HzStep 32 Based on the value of fo, calculate the minimum bandwidth (BW) required for the filter to pass the input pulse waveshape with minimal distortion. = BW = 5.045 kHzQuestion: Based on this answer and the cutoff frequency (fc) of the low-pass filter measure in Step 28,would you expect much distortion of the input square wave at the filter output? Did your answer in Step25 verify this conclusion? = I would expect less distortion of the input square wave at the filter. The higher the BW, the lesser the distortion formed.Step 33 Adjust the filter capacitors (C) to 50% (2.5 nF) each. Bring down the oscilloscope enlargement and run the simulation to one full screen display, then pause the simulation. The red curve plot is the filter input and the blue curve plot is the filter output.Question: Are the filter input (red) and the output (blue) curve plots the same shape, disregarding anyamplitude differences? = No.
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Step 34 Bring down the Bode plotter enlargement to display the Bode plot of the filter. Use the cursor to measure the cutoff frequency (fc) of the low-pass filter and record the value. = fc = 4.239 kHzQuestions: Was the cutoff frequency (fc) less than or greater than the minimum bandwidth (BW) requiredto pass the input waveshape with minimal distortion as determined in Step 32? = It is greater than the minimum bandwidth (BW).Based on this answer, would you expect much distortion of the input pulse waveshape at the filteroutput? Did your answer in Step 33 verify this conclusion? = No, there will have much distortion.Step 35 Bring down the spectrum analyzer enlargement to display the filter output frequency spectrum inthe frequency domain. Run the simulation until the Resolution Frequencies match, then pause thesimulation.Question: What is the difference between this frequency plot and the frequency plot in Step 29? = The difference between this frequency plot and the frequency plot in Step 29 is that the amplitude in this frequency plot is much lower than before.
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Conclusion: Fourier Series is obtain through the summation of the sine and cosine function. Therefore, anynonsinusoidal waves can be transformed into sine and cosine function. A Fourier series is composed offundamental sine wave and harmonics, so as the number of harmonics is decreased the wave will havemore ripples and if the harmonics are completely gone, the wave will become the fundamental sine wave.Meanwhile, adding more harmonics to the sine wave the more the waveshape becomes a square wave ortriangular wave. Add to that, when a dc supply is added, the graph will move upward. For duty cycle otherthan 50%, there will be series of even and odd harmonics produced. The bandwidth of the graph isinversely proportional to the distortion. The wider the bandwidth the less the distortion will be. Lastly,the higher the increment of the capacitor, the lower the amplitude of each frequency.
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