ECE 2111 Signals and Systems
Spring 2009, UMD
Experiment 9: Sampling
Objective:
In this experiment the properties and limitations of the sampling theorem are investigated. A
specific sampling circuit will be constructed and tested using a variety of input signals and sampling
signals.
Equipment and Material:
- Rohde & Schwarz FSH3 Spectrum analyzer
- 2-Function Generators
- Oscilloscope
- LF 398 sample-and-hold chip
- MF4CN-100 fourth order Butterworth low-pass filter
- 4- 10 μF capacitors (  50 V)
BACKGROUND
1. - Sampled Signals
Analog signals, which are the most familiar type of signal, are continuous functions of time in the
sense that their amplitudes are defined explicitly for every instant of time. However, there is another
important class of signals, usually referred to as sampled signals, for which the amplitude is defined
(non-zero) only for a certain discrete instant of time. Fig. 4.1 displays an example of both the analog
and a sampled signal. Sampled signals are used in pulse-modulation communication systems, in
sampled data control systems, and when digital computers are used as part of an analog system.
Fig. 4.1 Examples of: (a) an analog signal; (b) a sampled signal
The process of generating sampled signals, sometimes called pulse-amplitude modulation is
illustrated in Fig. 4.2. The analog input signal, xi(t), is multiplied by the uniform pulse train. xs(t),
and the resulting output signal, xo(t), is non-zero only when xi(t) and xs(t) are both non-zero. The
analog signal, xi(t), is said to have been sampled by the sampling signal, xs(t).
An equivalent method of describing the sampling process is the single-pole, single-throw switch
shown in Fig. 4.3.
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Fig.4.2. Generation of Sampled Signals
Fig.4.3 Representation of the Sampling Process
2.-Sampling Theorem
Sampled signals such as xo(t) in Fig. 4.2(d) are useful only if they contain the same information as
the original signal, xi(t), as shown in Fig.4.2(b). That is to say, xi(t) must be recoverable from
xo(t). The conditions under which such a recovery of the original signal constitute a statement of the
sampling theorem. Briefly these conditions are:
1. - The original signal xi(t) must be a band-width limited function (i.e., have no frequency
components outside the frequency interval [ -fb to + fb ], and
2. - The frequency of the sampling signal, xs(t), must be greater than 2fb.
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Fig.4.4. Illustration of the sampling theorem
Consider the signal xi(t) shown in Fig. 4.4 (a), which has a band-width limited spectrum, also
shown in Fig. 4.4 (a). This signal is sampled by the uniform impulse train, xδs(t), shown in Fig. 4.4
(b). The spectrum of xδs(t), is itself a uniform impulse train, Xδs(f), in the frequency domain, as is
shown in Fig. 4.4 (b) (from Fourier analysis).
The output signal, xδo(t), is easily found in the time domain, as is illustrated in Fig. 4.4(c). The
output spectrum, Xδs(f), is found by the convolution of Xi(f) and Xδs(f). This is due to the fact (from
Fourier analysis) that multiplication in the time domain is equivalent to convolution in the
frequency domain. Thus
When the convolution indicated by eq. (1) is carried out, the output, Xδo(f), shown in Fig. 4.4(c)
results.
Inspection of the output spectrum, Xδo(f), shows that the spectrum of the original signal is
reproduced symmetrically about each frequency harmonic of the sampling signal. The original
spectrum can be recovered from Xδo(f), through low-pass filtering, as shown in Fig. 4.5(a). This is
true as long as neighboring replicas of the input spectrum do not overlap. In order to avoid the
following conditions must be met:
1. - The input, Xi(f), must have no frequency components outside the frequency interval, - fb to +
fb, and
2. - The sampling frequency, l/Ts , must be greater than or equal to twice the band-limit of the
input signal 2fb.
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These two conditions form a restatement of the sampling theorem. A complete block diagram of an
ideal sampling system is shown in Fig. 4.5 (b).
Fig.4.5. Recovery of original signal from sampled signal
3. - Effect of Finite Duration Sampling Pulses
A sampling composed of a set of infinitely short pulses cannot be realized in practice, so the
sampling operation just described must be modified to take the finite duration of the sampling
pulses into account. Consider the sampling pulse train shown in Fig. 4.6 (a). The spectrum of this
pulse train is shown in Fig. 4.6(b).
Fig.4.6. Finite width pulse amplitude spectrum
Note: The value of τp should be evenly divisible by Ts.
When the spectrum shown in Fig.4.6 (b) is convolved with Xi(f) shown in Fig.4.4 (a), the output
spectrum Xo(f), shown in Fig.4.7 results.
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Fig.4.7. Amplitude spectra of X0(f): in a system with a finite width sampling pulse, and the
amplitude transfer characteristics of the low-pass filter used to recover the original signal
from the sampled signal.
The original signal, xi(t), can still be recovered from the new output signal, xo(t) by low-pass
filtering as an examination of Fig. 4.7 reveals. The effect of the finite width of the sampling pulse is
to reduce the gain of the sampling system. In the ideal case
Whereas the finite-width sampling pulse gives
Note: The gain of the low-pass filter is assumed to be one
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4. - Practical Sampling Circuit
The sampling and reconstruction circuit used in this experiment is displayed in Fig.4.8.
Fig.4.8. Practical sampling and reconstruction circuit
The LF398 in Fig.4.8 is a sample-and-hold chip. With a short circuit from pin 6 to ground, the hold
is disabled and the chip becomes a sampler.
The MF4cn-100 is a fourth-order, switched-capacitor, Butterworth lowpass filter that will
reconstruct the sampled signal. The input signal X100c(t) is a TTL (0-5 V) square wave. The cut off
frequency of the filter can be selected by choosing the frequency of X100c(t) , which will be 100
times larger than the cut off frequency of the filter. The maximum cut-off frequency of MF4cn-100
is 10 KHz. (refer data sheet).
The four 10µF capacitors will cut out the noise on the power line. The signals related to the circuit
in Fig.4.8 are
xi(t) : input signal
xs(t) : sampling signal
xo(t) : sampled signal with finite pulse width
x100c(t) : cut off frequency control signal (will be 100 times larger than the cut off frequency of
the filter)
xLP (t) : Reconstructed signal
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PRE LAB
Complete all the Pre Lab exercises before coming to lab. Show all your work
1.- Given the circuit in Fig. 4.8 such that:
xi(t) = 500mVpp, 10kHz, sine wave, no DC component;
xs(t) = 0V- 5V, 40kHz, 50% duty cycle, pulse wave; and
x100c(t) = 0V- 5V, 50% duty cycle, must be set so that fc is 1/2 fs OR will be 100 times larger
than the cut off frequency of the filter, whichever one is larger.
Sketch: xo(t) and xLP(t)
Find: Xo(f) and XLP(f) up to 200kHz
2.- Given the circuit in Fig. 4.8 such that:
xi(t) = 500mVpp, 10kHz, sine wave, no DC component;
xs(t) = 0V- 5V, 15kHz, 50% duty cycle, pulse wave; and
x100c(t) =0V- 5V, 50% duty cycle, must be set so that fc is 1/2 fs OR will be 100 times larger
than the cut off frequency of the filter, whichever one is larger.
Sketch: xo(t) and xLP(t)
Find: Xo(f) and XLP(f) up to 100kHz
3.- Given the circuit in Fig. 4.8 such that:
xi(t) = 500mVpp, 10kHz, sine wave, no DC component;
xs(t) = waveform of Fig. 4.9; and
x100c(t) = 0V- 5V, 50% duty cycle, must be set so that fc is 1/2 fs OR will be 100 times larger
than the cut off frequency of the filter, whichever one is larger.
Sketch: xo(t) and xLP(t)
Find: Xo(f) and XLP(f) up to 200kHz
Fig. 4.9 Sampling pulse train
4.- Given the circuit in Fig. 4.8 such that:
xs(t) = 0V- 5V, 40kHz, 50 % duty cycle, pulse wave
x100c(t) = 0V- 5V, 50% duty cycle,must be set so that fc is 1/2 fs OR will be 100 times larger
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than the cut off frequency of the filter, whichever one is larger.
a) xi(t) = 500mVpp, 2kHz square wave, no DC component
Sketch: xo(t) and xLP(t)
Find: Xo(f) and XLP(f) up to 200 kHz
b) xi(t) = 500mVpp, 5kHz square wave, no DC component
Sketch: xo(t) and xLP(t)
Find: Xo(f) and XLP(f) up to 200 kHz
c) xi(t) = 500mVpp, 2 kHz triangle wave, no DC component
Sketch: xo(t) and xLP(t)
Find: Xo(f) and XLP(f) up to 200 kHz
d) xi(t) = 500mVpp, 5kHz triangle wave, no DC component
Sketch: xo(t) and xLP(t)
Find: Xo(f) and XLP(f) up to 200 kHz
5.- Questions
5.1. Which square wave should be reconstructed more accurately? Why?
5.2. Which triangle wave should be reconstructed more accurately? Why?
5.3. Which wave form will be easier to sample and reconstruct triangle wave or square wave? Why?
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PROCEDURE
1.- Prepare: Circuit of Fig.4.8:
xi(t) = 500mVpp, 10kHz, sine wave, no DC component;
xs(t) = 0V- 5 V, 40kHz, 50% duty cycle, pulse wave; and
x100c(t) =0V- 5V, 50% duty cycle, must be set so that fc is 1/2 fs OR will be 100 times
larger than the cut off frequency of the filter, whichever one is larger.
Sketch: xo(t) and xLP(t)
Measure: Xo(f) and XLP(f) up to 200kHz
2.- Prepare: Circuit of Fig. 4.8:
xi(t) = 500mVpp, 10kHz, sine wave, no DC component;
xs(t) = 0V- 5 V, 15kHz, 50% duty cycle, pulse wave; and
x100c(t) = 0V- 5V, 50% duty cycle,must be set so that fc is 1/2 fs OR will be 100 times larger
than the cut off frequency of the filter, whichever one is larger.
Sketch: xo(t) and xLP(t)
Measure: Xo(f) and XLP(f) up to 100kHz
3.- Prepare: Circuit of Fig. 4.8:
xi(t) = 500mVpp, 10kHz, sine wave, no DC component;
xs(t) = waveform of Fig.4.9 ; and
x100c(t) = 0V- 5V, 50% duty cycle,must be set so that fc is 1/2 fs OR will be 100 times larger
than the cut off frequency of the filter, whichever one is larger.
Sketch: xo(t) and xLP(t)
Measure: Xo(f) and XLP(f) up to 200 kHz
4.- Prepare: Circuit of Fig. 4.8:
xs(t) = 0V- 5 V, 40kHz, 50 % duty cycle, pulse wave
x100c(t) =0V- 5V, 50% duty cycle, must be set so that fc is 1/2 fs OR will be 100 times larger
than the cut off frequency of the filter, whichever one is larger.
a) xi(t) = 500mVpp, 2kHz square wave, no DC component
Sketch: xo(t) and xLP(t)
Measure: Xo(f) and XLP(f) up to 200 kHz
b) xi(t) = 500mVpp, 5kHz square wave, no DC component
Sketch: xo(t) and xLP(t)
Measure: Xo(f) and XLP(f) up to 200 kHz
c) xi(t) = 500mVpp, 2kHz triangle wave, no DC component
Sketch: xo(t) and xLP(t)
Measure: Xo(f) and XLP(f) up to 200 kHz
d) xi(t) = 500mVpp, 5kHz triangle wave, no DC component
Sketch: xo(t) and xLP(t)
Measure: Xo(f) and XLP(f) up to 200 kHz
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5. - Questions
5.1. Which square wave should be reconstructed more accurately? Why?
5.2. Which triangle wave should be reconstructed more accurately? Why?
5.3. Which wave form will be easier to reconstruct, triangle wave or square wave? Why?
5.4. What was the effect of not having the cut-off frequency of the LPF = ½ f (or 100 times
s
larger than the cut off frequency of the filter, whichever one is larger.)
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POST LAB
Before plotting any measured amplitude spectrum, convert all dBm values into Vp values. All
graphs and plots must be on graph paper or done by a graphing program with voltage and frequency
scales displayed.
An amplitude line spectrum of a periodic signal should be a set of vertical lines. Each line is drawn
from the 0volts axis to the amplitude of the harmonic represented by that line. The horizontal
placement of the line is the frequency of that particular harmonic.
An amplitude or phase response plot of a circuit consists of a line or curve fitted to a set of points,
which represent amplitude or phase response at discrete values of frequency.
1. Display: Sketch of x0(t) from Pre-lab exercise 1 and Procedure 1 together. Do the same for xLP(t).
Compare: The theoretical sketches and the experimental sketches.
Plot: Amplitude spectra of Xo(f) from Pre-lab exercise 1 and Procedure 1 together. Do the same
for XLP(f).
Compare: The theoretical plots and the experimental plots.
2. Display: Sketch of x0(t) from Pre-lab exercise 2 and Procedure 2 together. Do the same for x LP(t).
Compare: Theoretical sketches with the experimental sketches.
Plot: Amplitude spectra of Xo(f) from Pre-lab exercise 2 and Procedure 2 together. Do the same
for XLP(f).
Compare: The theoretical plots and the experimental plots.
3. Display: Sketches of x0(t) from Pre-lab exercise 3 and Procedure 3 together. Do the same for
xLP(t).
Compare: Theoretical sketches with the experimental sketches.
Plot: The amplitude spectra of Xo(f) from Pre-lab exercise 3 and Procedure 3 together. Do the
same for XLP(f).
Compare: The theoretical plot and the experimental plot.
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4. Display: Sketches of x0(t) from Pre-lab exercise 4 and Procedure 4 together. Do the same for
xLP(t).
Compare: Theoretical sketches with the experimental sketches.
Plot: The amplitude spectra of Xo(f) from Pre-lab exercise 4 and Procedure 4 together. Do the
same for XLP(f).
Compare: The theoretical plot and the experimental plot.
5. - Do the questions for Pre-lab exercise 5 back up the findings in Procedure 4?
Post Experiment (Report) Requirements:
1- Every student must have his own individual lab report.
2- The report should include the following:
a) Results with detailed explanations are needed.
b) Answer the questions if there are any.
c) Conclusion - what did you learn in this experiment? Please write only a few lines.
3- All reports should be word processed and should also have the assigned cover page.
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