Activity: Fourier Analysis

In today's activity, you will use both electric signals and computer simulations to study Fourier series and Fourier transforms.

Equipment needed: Each group needs a function generator with power cord, an Universal Lab Interface, a cord connecting them, and a computer running LoggerPro and CUPS.

Fourier Series - Exploration

Connect the OUTPUT connection of the function generator to Din 1 of the ULI. Turn on the function generator, and make sure the ULI is plugged in and turned on. Depress the 50 (Hz) button on the function generator, and make sure that the DC OFFSET and -20dB are depressed while the DUTY CYCLE and CMOS LEVEL are up (off). Using the COURSE and FINE FREQUENCY controls, set the frequency of the function generator to somewhere around 30 Hz. Turn the DC OFFSET knob so it is slightly to the right (clockwise) of pointing straight up. Turn the OUTPUT LEVEL knob nearly all the way to the left (counterclockwise). Start with the sine wave button (not square or triangular waves) depressed.

Open the LoggerPro file fourier.mbl. Click Collect. When the data has been collected, you should see several cycles of a sine wave in the left-hand graph. The DC OFFSET knob may be used to keep the voltage from dropping below zero (turning clockwise will raise entire wave), and the OUTPUT LEVEL knob will adjust the amplitude of the wave. Adjust these knobs, collect data, and repeat until you have a wave extending from around 0.1 V to 1.0 V.

1. Collect data for your 30 Hz sine wave. The right-hand graph is the Fourier spectrum for you sine wave. Sketch it, including axes and scale markings on your graph.

2. The Fourier spectrum should have a peak, with the peak's frequency given in a box at the top of the graph. What is this peak frequency?

3. Change the frequency of the frequency generator to about 70 Hz and recollect the data. How does the Fourier spectrum differ?

4. Change the frequency to 50 Hz and recollect the data. What happens to the Fourier spectrum?

5. Based on your observations, what does a Fourier spectrum represent?

STOP. The class will discuss these results before continuing.

Before You Start:
Answer the following questions to the best of your ability before doing the experiment.

The analytical Fourier series for a square wave is given by

For a square wave of 50 Hz, what will be the lowest frequency f₀ with a peak in the Fourier spectrum?
Will the Fourier spectrum of that wave have a peak at 100 Hz? How do you know?
List 2 or 3 (other) frequencies at which the Fourier spectrum will have peaks. Justify your answer.

After you have thought about your answers, compare notes with your group members. Does everyone have the same predictions, or are there differing opinions?

Fourier Series - Square Waves

6. Depress the square wave button on the function generator, and set the frequency to 50.00 Hz. Collect data in LoggerPro, and look at the Fourier spectrum displayed in the FFT window. Sketch this graph, including axes and scale markings.

7. Click on the x = ? button (or select Examine from the Analyze menu). Move the cursor through the FFT window and identify the frequencies at which the spectrum peaks. Is there a peak at 100 Hz? How do the peak frequencies match your predictions?

8. Look at the first four (lowest frequency) peaks, and write down the maximum F-V value for each peak, along with the peak's frequency.

9. Compare the relative heights with the analytic expression for a square wave Fourier Series given above. For example, is the second peak 1/3 as high as the first? If you have discrepencies, what might be causing them?

You may disconnect the equipment and shut down LoggerPro once you are satisfied with your results.

STOP. The class will discuss these results before continuing.

Fourier Transforms - Pulses
Open the folder CUPS on the desktop. Start Cupswo.exe from the Cupswo folder. When the menu comes up, choose the first option: "Fourier Analysis and Fourier Transforms".
CUPS will not let you minimize it with the mouse. Be sure to read the following instructions (or bring them up on another computer) before starting the cupswo.exe program. Alternatively, you can press Alt-Tab to bring back your Windows desktop.

Under the 1-D DFT menu, choose the first option: "Single Pulse". The graph at the top of the screen represents the voltage pulse. The graph at the bottom of the screen is the Fourier transform of this pulse. Click the button "Real Transform" in the Fourier graph. This should leave a green line that is symmetric about the vertical axis.

10. Using the sliding controls on either side of the top graph, set the "Posn" to 0, and the "Width" to 40. Sketch the resulting Fourier spectrum (bottom graph), making sure to label both your graph's axes and the scale markings.

11. Decrease the Width of the pulse. What happens to the Fourier spectrum as the width pulse decreases?

12. Sketch the Fourier spectrum when the Width is set to 6, making sure to label both your graph's axes and the scale markings.

13. Which pulse width (bigger or smaller) could be used to represent a bit at a faster bit rate?

14. Would the faster-bitrate bits then be composed of a smaller or larger range of frequencies as indicated by the Fourier spectrum of the bit?

15. The bandwidth of a medium refers to the range of frequencies that can be reliably propagated through that medium without distortion. Given your answers to the questions above, explain why a large bandwidth is preferable in a signal carrier.

16. In your own words, describe the differences and similarities between Fourier Series and Fourier Transforms.

17. If you reach this point in the activity before your instructor stops the class for discussion, play around with the different features of the CUPS simulation. Below are some questions to consider:

Physical signals are not exactly square. Under the 1-D DFT menu, choose "Gaussian". As before, change the Posn to 0 and select "Real transform". How does the Fourier spectrum of the Gaussian pulse compare with the Fourier spectrum of the square pulse?

Under the Fourier menu, choose Square Wave. Click on Run/Stop to produce the Fourier spectrum in the graphs on the bottom. Do you see any differences between the Fourier spectrum of this square wave and the one you analyzed with LoggerPro? What might be the sources of those differences?

Under the Fourier menu, choose Propagate. Leave the settings at their default values and click OK. Watch what happens to your green signal when you click on Run/Stop to simulate the wave traveling over a distance. Does the wave retain its shape?

Compare the behavior of the Square Wave signal with that of a sine wave signal by choosing Input Function from the Fourier menu. Type sin(x) in the input box, then click OK. Run/Stop to produce the Fourier Spectrum, then Propagate. How does the distortion of the sine wave compare to the distortion of the square wave?

1.	Collect data for your 30 Hz sine wave. The right-hand graph is the Fourier spectrum for you sine wave. Sketch it, including axes and scale markings on your graph.
2.	The Fourier spectrum should have a peak, with the peak's frequency given in a box at the top of the graph. What is this peak frequency?
3.	Change the frequency of the frequency generator to about 70 Hz and recollect the data. How does the Fourier spectrum differ?
4.	Change the frequency to 50 Hz and recollect the data. What happens to the Fourier spectrum?
5.	Based on your observations, what does a Fourier spectrum represent?

6.	Depress the square wave button on the function generator, and set the frequency to 50.00 Hz. Collect data in LoggerPro, and look at the Fourier spectrum displayed in the FFT window. Sketch this graph, including axes and scale markings.
7.	Click on the x = ? button (or select Examine from the Analyze menu). Move the cursor through the FFT window and identify the frequencies at which the spectrum peaks. Is there a peak at 100 Hz? How do the peak frequencies match your predictions?
8.	Look at the first four (lowest frequency) peaks, and write down the maximum F-V value for each peak, along with the peak's frequency.
9.	Compare the relative heights with the analytic expression for a square wave Fourier Series given above. For example, is the second peak 1/3 as high as the first? If you have discrepencies, what might be causing them?

10.	Using the sliding controls on either side of the top graph, set the "Posn" to 0, and the "Width" to 40. Sketch the resulting Fourier spectrum (bottom graph), making sure to label both your graph's axes and the scale markings.
11.	Decrease the Width of the pulse. What happens to the Fourier spectrum as the width pulse decreases?
12.	Sketch the Fourier spectrum when the Width is set to 6, making sure to label both your graph's axes and the scale markings.
13.	Which pulse width (bigger or smaller) could be used to represent a bit at a faster bit rate?
14.	Would the faster-bitrate bits then be composed of a smaller or larger range of frequencies as indicated by the Fourier spectrum of the bit?
15.	The bandwidth of a medium refers to the range of frequencies that can be reliably propagated through that medium without distortion. Given your answers to the questions above, explain why a large bandwidth is preferable in a signal carrier.
16.	In your own words, describe the differences and similarities between Fourier Series and Fourier Transforms.
17.	If you reach this point in the activity before your instructor stops the class for discussion, play around with the different features of the CUPS simulation. Below are some questions to consider:
	Physical signals are not exactly square. Under the 1-D DFT menu, choose "Gaussian". As before, change the Posn to 0 and select "Real transform". How does the Fourier spectrum of the Gaussian pulse compare with the Fourier spectrum of the square pulse? Under the Fourier menu, choose Square Wave. Click on Run/Stop to produce the Fourier spectrum in the graphs on the bottom. Do you see any differences between the Fourier spectrum of this square wave and the one you analyzed with LoggerPro? What might be the sources of those differences? Under the Fourier menu, choose Propagate. Leave the settings at their default values and click OK. Watch what happens to your green signal when you click on Run/Stop to simulate the wave traveling over a distance. Does the wave retain its shape? Compare the behavior of the Square Wave signal with that of a sine wave signal by choosing Input Function from the Fourier menu. Type sin(x) in the input box, then click OK. Run/Stop to produce the Fourier Spectrum, then Propagate. How does the distortion of the sine wave compare to the distortion of the square wave?