Exercise
1-3
Band-Limiting
EXERCISE OBJECTIVE
When you have completed this exercise, you will be familiar with band-limiting.
You will be able to verify the effects of band-limiting on pulse signals as seen in
both the frequency domain and the time domain. You will establish the
relationship between pulse rise time and bandwidth.
DISCUSSION OUTLINE
The Discussion of this exercise covers the following points:
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DISCUSSION
Spectral analysis
All-pass filter
Ideal low-pass filter
Practical low-pass filter
Cutoff frequency
Filtering effects
Spectral analysis
Spectral analysis shows that a pulse waveform can be considered to consist of a
large number of frequency components, each having its own frequency and
power level. In order to transmit a pulse signal through a system without
distortion, all of the frequency components must be transmitted equally well.
They may all be amplified or attenuated uniformly, but the relative magnitude of
each frequency component must not be altered. If all significant frequencies are
not transmitted, the pulse waveform will suffer appreciable distortion.
Figure 1-21 shows a signal being transmitted through a system. By comparing
the signal at the input and the output of the system, one can determine how
effectively the system transmits different frequencies, and obtain the frequency
response characteristics of the system. In general, the frequency response of a
system includes an amplitude response (amplitude versus frequency) as well as
a phase response (phase versus frequency). We will limit ourselves to the
amplitude response.
Figure 1-21. Determining the frequency response characteristics of a system.
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Ex. 1-3 – Band-Limiting  Discussion
All-pass filter
Figure 1-22 shows the frequency response curve of an all-pass system. This
figure shows that the gain of an all-pass system is the same for all frequencies.
This means that all frequencies are transmitted equally well.
Figure 1-22. All-pass system frequency response.
Completely unlimited all-pass transmission is not possible to obtain. In practical
systems, there is always some limit to the frequencies that can be transmitted
without attenuation. The limiting of the range of frequencies that are transmitted
through a system is called bandwidth limitation or band-limiting.
In order to understand the operation of band-limited systems, we will consider a
system which transmits equally well, all frequencies within a certain range, but
completely blocks frequencies outside of this range. This type of system is called
an ideal, or brick-wall filter.
Ideal low-pass filter
Figure 1-23 shows the frequency response of an ideal low-pass filter. All
frequencies from 0 Hz to the maximum frequency transmitted, called the cutoff
frequency, are transmitted equally well, but frequencies greater than the cutoff
frequency are not transmitted at all. The range of frequencies that are transmitted
is called the bandwidth B of the system. For a low-pass filter, the bandwidth is
equal to the cutoff frequency. Such an ideal low-pass filter will eliminate all
frequency components greater than the cutoff frequency. A signal which contains
no frequency components beyond a certain range is said to be strictly
bandlimited.
Figure 1-23.Ideal low-pass filter frequency response.
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© Festo Didactic 39862-00
Ex. 1-3 – Band-Limiting  Discussion
Practical low-pass filter
Ideal filters are impossible to construct. Practical filters have a fairly constant
frequency response within a certain range, and gradually attenuate frequencies
beyond this range. Figure 1-24 shows the frequency response of a practical lowpass filter. Frequencies well below the cutoff frequency are transmitted uniformly,
whereas higher frequencies are attenuated. The amount of attenuation of the
high frequencies varies with the frequency: the higher the frequency, the more
the attenuation. This gradual increase in attenuation over a range of frequencies
is called rolloff. The rate of attenuation is related to the order of the filter. In
general, the order of a filter is a function of the number of reactive components
(inductors and capacitors) in the filter circuit. The higher the filter order, the
sharper the rolloff.
Figure 1-24.Practical low-pass filter frequency response.
Cutoff frequency
The cutoff frequency of a practical filter is the frequency where the power of the
transmitted signal is reduced by one-half, or 3 dB. This corresponds to an
attenuation in amplitude to 1/ 2 times the original amplitude. The bandwidth (or
bandpass) of a practical system is the range of frequencies over which the gain
remains within 3 dB. This is often called the "half-power bandwidth" or " – 3 dB
bandwidth" (see Figure 1-24).
Depending on the design and order of the filter, the frequency response may
increase slightly just before it begins to drop, or there may be slight ripples in the
frequency response curve.
Filtering effects
Figure 1-25 shows the effect of band-limiting a narrow-pulse signal using a lowpass filter. Before band-limiting, the spectrum is fairly flat over a certain range, as
shown by the dotted line. After band-limiting, the attenuation of each harmonic
depends on its frequency – the higher the frequency, the more the attenuation.
The degree of attenuation at the cutoff frequency of the filter is equal to 3 dB.
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Ex. 1-3 – Band-Limiting  Procedure Outline
Figure 1-25. Band-limiting using a low-pass filter.
These frequency domain characteristics have their counterpart in the time
domain. In pulses which have been distorted by band-limiting, the transition of
the amplitude from zero to the final value is not instantaneous. Instead, this
transition takes a certain time. The rise time is usually defined as the time
required for the voltage to rise from 10% to 90% of its final value.
The rise time of a pulse depends on the bandwidth of the system through which it
has been transmitted. For this reason, bandwidth is a measure of how rapidly a
signal in the system can change. As the bandwidth B is decreased, the voltage
rises more slowly. The rise time is inversely proportional to the bandwidth. For
practical low-pass filters of order greater than second order, the rise time is
approximately equal to 1/(2B). Therefore, band-limiting limits the minimum rise
time of transmitted pulses.
PROCEDURE OUTLINE
The Procedure is divided into the following sections:
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PROCEDURE
Set-up and connections
Observing a pulse signal
Observing a pulse frequency spectrum
Filtering in the frequency domain
Rise time
Set-up and connections
1. Turn on the RTM Power Supply and the RTM and make sure the RTM power
LED is lit.
File f Restore Default Settings
returns all settings to their
default values, but does not
deactivate activated faults.
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.
Double-click to select SWapp
If the software is already running, choose Exit in the File menu and restart
LVCT to begin a new session with all faults deactivated.
b
3. Make the external connections for Pulses 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.
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Click the PULSES button to show the required external connections.
© Festo Didactic 39862-00
Ex. 1-3 – Band-Limiting  Procedure
Observing a pulse signal
4. Make the following Generator Settings on Function Generator A:
Function ......................................................... Pulse
Frequency (Hz) .............................................. 1 000
Duty Cycle (%)............................................... 1
5. Click the PAM Generator tab in order to display the PAM Generator diagram.
Show the Probes bar (click
in the toolbar or choose View f Probes Bar).
Connect the probes as follows:
Oscilloscope Probe
Connect to
Signal
1
TP1
AUDIO INPUT
E
TP6
Buffered CLOCK INPUT
6. Show the Oscilloscope (click
in the toolbar or choose Instruments f
Oscilloscope). Figure 1-26 shows an example of settings and what you
should observe.
Oscilloscope Settings:
Channel 1........................... 500 mV/div
Persistence ............................ 4 Traces
Time Base ........................... 0.5 ms/div
Trigger Slope ............................. Rising
Trigger Level ............................... 0.5 V
Trigger Source ................................Ext
Figure 1-26. Pulse signal with 1% duty cycle.
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The pulses appear to fade in and out on the oscilloscope due to their short
duration. Turning on the persistence allows the pulses to be observed more
easily.
Observing a pulse frequency spectrum
7. Show the Probes bar (click
in the toolbar or choose View f Probes Bar).
Connect the Spectrum Analyzer Probe to TP1.
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Ex. 1-3 – Band-Limiting  Procedure
Show the Spectrum Analyzer (click
in the toolbar or choose
Instruments f Spectrum Analyzer). Figure 1-27 shows an example of
settings and what you should observe.
Spectrum Analyzer Settings:
Maximum Input ....................... -30 dBV
Scale Type ........................ Logarithmic
Scale .................................... 5 dBV/div
Averaging ........................................... 4
Time Window ....................... Hamming
Frequency Span ................... 2 kHz/div
Reference Frequency .................0 kHz
Figure 1-27. Pulse signal spectrum with 1% duty cycle.
8. Is the spectrum fairly flat? Explain.
Filtering in the frequency domain
9. Store the current spectrum in Memory location M1. Move the Spectrum
Analyzer probe to TP2 and set the Low-pass filter to the Max value of 8 kHz.
Compare the filtered and unfiltered signal spectrum.
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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.
Have all of the harmonics been attenuated to the same degree? Explain.
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© Festo Didactic 39862-00
Ex. 1-3 – Band-Limiting  Procedure
10. The degree of attenuation at the cutoff frequency of a filter is 3 dB. Use the
horizontal cursors to determine which harmonic has been attenuated by
approximately 3 dB. Is the result consistent with the frequency setting of the
Low-pass Filter?
11. Connect the probes as follows in order to observe the filtered signal in both
the time and frequency domains:
Probe
Connect to
Signal
Oscilloscope 1
TP1
AUDIO INPUT
Oscilloscope 2
TP2
Low-pass Filter output
Spectrum Analyzer
TP2
Low-pass Filter output
12. Make the following Generator Settings on Function Generator A:
Function ......................................................... Pulse
Frequency (Hz) .............................................. 1 000
Duty Cycle (%)............................................... 50
Set the Low-pass Filter order to 4th and capture the oscilloscope and
spectrum analyzer output for each of the following Cutoff Frequency values:
1. 8 kHz
2. 6 kHz
3. 4 kHz
4. 2 kHz
5. 1 kHz
6. 300 Hz
How does reducing the bandwidth affect the spectrum?
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Ex. 1-3 – Band-Limiting  Procedure
Rise time
13. Return the settings on Function Generator A to the following if they have
been modified:
Function ......................................................... Pulse
Frequency (Hz) .............................................. 1 000
Duty Cycle (%) ............................................... 50
Set the Low-pass Filter frequency to the maximum value of 8 kHz. Turn off
channel 1 on the oscilloscope so that only the filtered signal is displayed.
Adjust the settings on the oscilloscope so that the display is similar to
Figure 1-28. Use the single refresh option on the oscilloscope to freeze the
display for measuring purposes.
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The horizontal trigger
at the bottom of the oscilloscope screen should be
dragged to the center of the screen.
To refresh and freeze the display, click the
button in the instrument toolbar.
This refreshes the display once and freezes it. You can also press F5 or
, press F6 or choose View f
choose View f Single Refresh. Click
Continuous Refresh to resume normal operation.
Oscilloscope Settings:
Channel 2........................... 200 mV/div
Time Base ............................. 50 Ps/div
Trigger Slope ............................. Rising
Trigger Level ............................... 0.5 V
Trigger Source ............................. Ch 2
Figure 1-28. Oscilloscope display for measuring rise time.
14. For the bandwidth values (Low-pass filter frequencies) shown in Table 1-2,
capture the output and use the oscilloscope cursors to determine the
rise time tr (from 10% to 90%) of the pulse voltage as shown in Figure 1-29.
Record the values of tr in Table 1-2. Calculate the values of 1/B and record
them in the table as well.
a
Use the horizontal cursors to mark the 90% and 10% levels and then use the
vertical cursors to determine the time between the two levels. The position of
the horizontal cursors should not require adjusting as the filter bandwidth is
modified.
The oscilloscope window can be maximized for easier viewing.
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© Festo Didactic 39862-00
Ex. 1-3 – Band-Limiting  Procedure
90% Crossing
100%
90%
10% Crossing
Rise Time
10%
0%
Figure 1-29. Measuring rise time.
Table 1-2. Rise time measurements.
Bandwidth B (kHz)
1/B (Ps)
Rise Time tr (Ps)
8
6
4
2
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Ex. 1-3 – Band-Limiting  Conclusion
15. Using the values in Table 1-2, plot the rise time tr versus 1/B in Figure 1-30.
Figure 1-30. Relationship between rise time and bandwidth.
Describe the relationship between the rise time tr and the bandwidth B.
16. When you have finished using the system, exit the LVCT software and turn
off the equipment.
CONCLUSION
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In this exercise, you observed that a practical band-limited system transmits
frequencies within its bandwidth uniformly (within 3 dB) but attenuates
frequencies outside this range. You observed that attenuating the high frequency
components of a pulse signal increases the distortion of the pulses, and that the
pulse rise time is inversely proportional to the bandwidth.
© Festo Didactic 39862-00
Ex. 1-3 – Band-Limiting  Review Questions
REVIEW QUESTIONS
1. What is the half-power bandwidth of a practical low-pass filter?
2. How does band-limiting a pulse signal with a low-pass filter affect the
spectrum of the signal?
3. Does a practical low-pass filter completely eliminate frequency components
outside its bandwidth? Explain.
4. How does band-limiting a pulse signal with a low-pass filter affect the
waveform (shape) of the pulses?
5. What is the relation between rise time and bandwidth?
© Festo Didactic 39862-00
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