Exercise 2-2. Spectral Characteristics of PAM Signals EXERCISE OBJECTIVE DISCUSSION OUTLINE DISCUSSION. Sampling

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1 Exercise 2-2 Spectral Characteristics of PAM Signals EXERCISE OBJECTIVE When you have completed this exercise, you will be familiar with the spectral characteristics of PAM signals. You will be able to verify how the sampling rate and the duty cycle of the sampling signal affect the spectra of PAM signals. DISCUSSION OUTLINE The Discussion of this exercise covers the following points: Sampling Aperture distortion DISCUSSION Sampling To understand the frequency spectrum of a PAM signal we must first examine the frequency spectra of both message and sampling signals. Figure 2-12 shows the frequency spectrum of a message signal. In Figure 2-12, the message signal contains frequency components ranging from 0 Hz to the maximum frequency fmax. This signal is strictly band-limited to fmax, because it contains no frequency components beyond this frequency. Figure Message signal spectrum. Figure 2-13 shows the spectrum of a sampling signal consisting of ideal (infinitesimally narrow) pulses. The pulse repetition rate is fs. This spectrum consists of a fundamental frequency component (or first harmonic) equal to fs, and an infinite number of harmonics at multiples of fs. Because the pulses in the sampling signal are infinitesimally narrow, the frequency components in the spectrum are all at the same power level (the envelope of the spectrum is flat). Figure Sampling signal spectrum (ideal pulses). Festo Didactic

2 Ex. 2-2 Spectral Characteristics of PAM Signals Discussion When the message signal in Figure 2-12 is sampled using the sampling signal in Figure 2-13, the resulting PAM signal has a spectrum which is shown in Figure The first element in this spectrum is the message signal itself. In addition to this, the PAM spectrum contains a number of replicas of the message signal spectrum. These replicas are exact copies of the message signal spectrum, except that they are displaced in frequency. Each replica is centered on a harmonic of the sampling signal. The envelope of the PAM spectrum is flat. Figure PAM signal spectrum ideal sampling. Because the replicas are centered on the harmonics of the sampling signal, their position in the spectrum depends on the sampling rate. If the sampling rate is increased, the replicas in the PAM signal spread apart. If it is decreased, they move together. The first replica contains frequency components ranging from fs - fmax. to fs + fmax. The second replica contains frequency components ranging from 2fs - fmax to 2fs + fmax, and so on. Changing fmax causes the frequency range of each replica to change. If the pulse width of the sampling signal is finite, rather than infinitesimally small, the envelope of the sampling signal spectrum is curved, as in Figure 2-15 Figure Sampling signal spectrum (pulse width = ). When this sampling signal is used with natural sampling, the PAM spectrum resembles Figure 2-16 With natural sampling the attenuation of each replica varies with the envelope of the sampling signal. In each replica, the attenuation is uniform. Figure PAM signal spectrum, natural sampling. 56 Festo Didactic

3 Ex. 2-2 Spectral Characteristics of PAM Signals Discussion When flat-top sampling is used, the PAM spectrum resembles Figure 2-17 The attenuation in each replica is not uniform but follows the envelope of the sampling signal. There is also some attenuation of the higher frequencies of the message signal as shown in Figure Figure PAM signal spectrum, flat-top sampling. Aperture distortion The attenuation of the higher frequencies of the message signal when flat-top sampling is used is called aperture distortion. This is indicated by the gray area in Figure Aperture distortion is significant only when the duty cycle of the sampling signal is large, and the sampling rate is relatively low. Because the envelope of the spectrum resembles the shape of a (sin x)/x function, aperture distortion is also called (sin x)/x distortion. Figure Aperture distortion indicated by the gray area. Each frequency component in the message signal produces two frequency components in each replica. This can best be observed using a sine wave message signal. Figure 2-19 shows the spectrum of a PAM signal generated by sampling a sine wave message signal having a frequency fm at a sampling rate fs. Festo Didactic

4 Outline Figure PAM signal spectrum. In the first replica, the two frequency components are fs - fm and fs + fm. In the second, they are 2fs - fm and 2fs + fm, and so on. The frequency components of the n th replica are nfs - fm and nfs + fm. PROCEDURE OUTLINE The Procedure is divided into the following sections: Set-up and connections Input signal spectrum PAM signal spectrum Harmonics and replicas PROCEDURE Set-up and connections 1. Turn on the RTM Power Supply and the RTM and make sure the RTM power LED is lit. File Restore Default Settings returns all settings to their default values, but does not deactivate activated faults. Double-click to select SWapp 2. Start the LVCT software. In the Application Selection box, choose PAM and click OK. This begins a new session with all settings set to their default values and with all faults deactivated. b If the software is already running, choose Exit in the File menu and restart LVCT to begin a new session with all faults deactivated. 3. Make the Default external connections shown on the System Diagram tab of the software. For details of connections to the Reconfigurable Training Module, refer to the RTM Connections tab of the software. b Click the Default button to show the required external connections. 4. As an option, connect a conventional oscilloscope or spectrum analyzer to the PAM Generator OUTPUT using BNC T-connector. 58 Festo Didactic

5 Input signal spectrum 5. Make the following Generator Settings: Function Generator A: Function... Sine Frequency (Hz) Function Generator B: Function... Pulse Frequency (Hz) Duty Cycle (%)... 5 Apply Duty Cycle to Clock... On 6. Click the PAM Generator tab in order to display the PAM Generator diagram. On the PAM Generator, ensure that Nat. is selected for the Mode for natural sampling. Show the Probes bar (click in the toolbar or choose View Probes Bar). Connect the probes as follows: Probe Connect to Signal Spectrum Analyzer TP1 AUDIO INPUT Spectrum Analyzer Settings: Maximum Input dbv Scale Type... Logarithmic Scale... 5 dbv/div Averaging... 4 Time Window... Hamming Frequency Span... 2 khz/div Reference Frequency... 0 khz 7. Show the Spectrum Analyzer (click in the toolbar or choose Instruments Spectrum Analyzer). Figure 2-20 shows an example of settings and what you should observe. Figure Sine wave message signal spectrum. Festo Didactic

6 8. Why is there only one line in the spectrum? 9. Vary the frequency of the message signal (Function Generator A Frequency). What happens to the spectrum of the message signal as the frequency is increased or decreased? Spectrum Analyzer Settings: Maximum Input dbv Scale Type... Logarithmic Scale... 5 dbv/div Averaging... 4 Time Window... Hamming Frequency Span... 5 khz/div Reference Frequency... 0 khz PAM signal spectrum 10. Set the frequency of the message signal back to 1 khz. Move the Spectrum Analyzer probe to TP8 in order to observe the spectrum of the PAM Generator OUTPUT. The display should be similar to Figure Figure PAM signal spectrum. 60 Festo Didactic

7 What do the spectra of the message signal and the PAM signal have in common? What is different about them? (Ignore differences in the height of the lines). 11. Vary the frequency of the message signal. Describe how this affects the spectrum of the PAM signal. Spectrum Analyzer Settings: Maximum Input dbv Scale Type... Logarithmic Scale... 5 dbv/div Averaging... 4 Time Window... Hamming Frequency Span khz/div Reference Frequency... 0 khz 12. Readjust the message signal frequency to 1 khz and increase the Frequency Span on the Spectrum Analyzer so that the display resembles Figure Figure PAM signal spectrum. Festo Didactic

8 Vary the sampling rate (Function Generator B Frequency). Describe what happens to the PAM spectrum as the sampling rate is increased or decreased. 13. Readjust the message signal frequency to 1 khz and the sampling rate to 10 khz, so that the display again resembles Figure Adjust the pulse duty cycle in increments from a low value of 5% to a high value of 95%. Describe the effect on the spectrum of the PAM signal. With natural sampling, does the shape of each replica change as the duty cycle of the sampling signal is varied? Explain. 14. On the PAM Generator, set the Mode to Flat. Adjust the pulse duty cycle in increments from a low value of 5% to a high value of 95%. With flat-top sampling, does the shape of each replica change as the duty cycle of the sampling signal is varied? Explain. 62 Festo Didactic

9 15. Figure 2-23 and Figure 2-24 show a PAM signal spectrum. Draw the envelopes of the spectra using dotted lines. Figure PAM signal spectrum.. Figure PAM signal spectrum. Identify which sampling mode is used for each figure. Figure 2-23 sampling mode: Figure 2-24 sampling mode: Harmonics and replicas 16. Ensure that the Mode of the PAM Generator is set to Nat. Adjust the duty cycle of the sampling signal so that the fourth replica is missing from the PAM signal spectrum. Festo Didactic

10 Store the Spectrum Analyzer display in Memory location M1. b Click M1 or M2 in the instrument toolbar to store the current display in Memory 1 or Memory 2. Use the Memories setting to show the contents of Memory 1, Memory 2, or both. Move the Spectrum Analyzer probe to TP6 in order to observe the sampling signal spectrum. Spectrum Analyzer Settings: Maximum Input dbv Scale Type... Logarithmic Scale... 5 dbv/div Averaging... 4 Time Window... Hamming Frequency Span khz/div Reference Frequency... 0 khz Turn on Memory location M1 to observe the PAM signal and sampling signal spectra simultaneously. The display should resemble Figure b a It may help to toggle Memory location M1 off and on to observe the relative positions of the spectra. Small spikes in the signal floor should be ignored. Figure PAM and sampling signal spectra. For each harmonic in the spectrum of the sampling signal, how many replicas are there in the PAM signal spectrum? Describe the position of each replica in the PAM signal spectrum with respect to the harmonics in the sampling signal spectrum. 64 Festo Didactic

11 Compare the height of the fourth harmonic in the sampling signal spectrum with that of the fourth replica in the PAM signal spectrum. 17. Calculate the frequencies of the spectral lines using the formulae in the table below and enter them in Table 2-1 Table 2-1. Replica frequencies. Lines Formulae (fs = 100 khz) Calculated Frequency (khz) Observed Frequency (khz) Message Signal fm 1 st replica fs - fm fs + fm 2 nd replica 2fs - fm 2fs + fm 3 rd replica 3fs - fm 3fs + fm Do the observed frequencies correspond to the calculated frequencies within an experimental error of 10%? What would be the frequencies of the lines in the fifth replica? 18. Make the following Generator Settings: Function Generator A: Function... Triangular Frequency (Hz) Function Generator B: Function... Pulse Frequency (Hz) Duty Cycle (%) Apply Duty Cycle to Clock... On Move the Spectrum Analyzer probe to TP1. Capture the spectrum of the message signal. It should resemble Figure Festo Didactic

12 Spectrum Analyzer Settings: Maximum Input... 0 dbv Scale Type... Logarithmic Scale... 5 dbv/div Averaging... 4 Time Window... Square Frequency Span... 5 khz/div Reference Frequency... 0 khz Spectrum Analyzer Settings: Maximum Input... 0 dbv Scale Type... Logarithmic Scale... 5 dbv/div Averaging... 4 Time Window... Square Frequency Span... 5 khz/div Reference Frequency... 0 khz Figure khz triangular message signal. Move the Spectrum Analyzer probe to TP6. Capture the spectrum of the sampling signal. It should resemble Figure Figure Sampling signal spectrum. Move the Spectrum Analyzer probe to TP8. Capture the spectrum of the PAM Generator OUTPUT. It should resemble Figure Festo Didactic

13 Ex. 2-2 Spectral Characteristics of PAM Signals Conclusion Spectrum Analyzer Settings: Maximum Input... 0 dbv Scale Type... Logarithmic Scale... 5 dbv/div Averaging... 4 Time Window... Square Frequency Span... 5 khz/div Reference Frequency... 0 khz Figure PAM Generator OUTPUT. On the Spectrum analyzer capture of the PAM Generator OUTPUT, identify the message signal spectrum and a replica of the message signal spectrum. Where is the replica positioned with respect to the sampling signal spectrum? 19. When you have finished using the system, exit the LVCT software and turn off the equipment. CONCLUSION In this exercise, you compared the spectra of a sine wave and a triangle wave message signal, a pulse sampling signal and the resulting PAM signals. You observed that there are replicas of the message signal in the PAM signal spectrum, and that they are centered on the harmonics in the sampling signal spectrum. You verified how varying the sampling rate, the pulse duty cycle and the message frequency affect PAM signals. REVIEW QUESTIONS 1. Describe the spectrum of a PAM signal by comparing it to the spectra of the message and sampling signals. 2. What is the effect on a PAM spectrum of decreasing the sampling rate? Festo Didactic

14 Ex. 2-2 Spectral Characteristics of PAM Signals Review Questions 3. What would be the effect on the PAM signal spectrum of increasing the maximum frequency of the message signal? 4. Describe the difference between a PAM signal using natural sampling and one using flat-top sampling. 5. If a sine wave at 3 Hz is sampled at a frequency of 8 khz, what will be the frequencies of the components in the first replica? 68 Festo Didactic