Ee463 communications 2 - lab 1 - loren schwappach

CTU: EE 463 – Communications 2: Lab 1: MATLAB Project – Sampling 1

Colorado Technical University
EE 463 – Communications 2
Lab 1: MATLAB Project – Sampling
October 2010
Loren K. Schwappach

ABSTRACT: This lab report was completed as a course requirement to obtain full course credit in EE463,
Communications 2 at Colorado Technical University. This Lab investigates the concepts of sampling, aliasing, and recovery of
signals after sampling. In addition this lab explores the similarities between amplitude modulated signals and sampling. All of
the code mentioned in this lab report was saved as a MATLAB m-file for convenience, quick reproduction, and troubleshooting
of the code. All of the code below can also be found at the end of the report as an attachment, as well as all figures.
If you have any questions or concerns in regards to this laboratory assignment, this laboratory report, the process used in
designing the indicated circuitry, or the final conclusions and recommendations derived, please send an email to
LSchwappach@yahoo.com. All computer drawn figures and pictures used in this report are of original and authentic content.

IV. PROCEDURE / RESULTS
I. INTRODUCTION
MATLAB is a powerful program and is helpful in the The procedures used in this lab are illustrated by the
visualization of applied mathematics, physics, and practical included MATLAB code (not applicable for this lab
engineering. In this lab assignment MATLAB’s Simulink assignment) and Simulink diagrams (applicable) in this report.
tools are used to explore the generation, recovery, and aliasing This Simulink diagrams can also be found at the end of this
of a sampled signal. report as attachments for easier visibility.

1. Part A – Generating A Sampled Message
II. OBJECTIVES For the first part of this lab assignment (Part A),
Simulink is used to demonstrate the concept of sampling.
In this communications 2 lab exercise MATLAB will
be used to accomplish the following objectives: Sampling of the frequency domain is accomplished
by multiplying a message signal with a non-zero average value
1. Demonstrate the concepts of sampling, aliasing and the sampling signal (pulse) at a sampling frequency that at the
recovery of signals from a sampled signal. very least exceeds the Nyquist frequency.

2. Demonstrate the relationship between amplitude For this part of the lab assignment a 10 Hertz Sine
modulation and sampling. wave message and 10 Hertz Pulse message are independently
sampled.
3. Apply the concept of sampling to envelope detection.
The effects of biasing of a sinusoidal message are
also explored as well as the effect of using a zero average
sampling pulse over a non-zero average pulse, however the
III. EQUIPMENT effects are not apparent until message recovery is completed in
Part B.
The following tools and or equipment were used for
this lab assignment: First this lab will examine the effects of using a 10
Hertz biased sine wave (Figure 1).
1. MATLAB version 2009b to 2010a.

2. Simulink (Part of MATLAB)

3. Communications Toolbox (Part of Simulink)


Figure 3: sampled(t).

Figure 4 below shows the biased 10 Hertz Sine wave
in the frequency domain and its corresponding frequencies (10
Hertz, -10 Hertz, and DC 0 Hertz bias.)
Figure 1: Lab 1 Part A Sine Wave Input.

Figure 1 above was designed in Simulink to allow
either a 10 Hertz sine wave or 10 Hertz pulse. The 10 Hertz
sine wave is chosen first and the results displaying the correct
frequency results and DC biasing are shown by Figure 2.

Figure 4: M(f) biased message.

Figure 5 below shows the sampled biased sine wave
in the frequency domain. In the frequency domain you can
see several duplicates of the message signal as well as the
Figure 2: m(t) biased message.
original message signal. The duplicates (samples) of the
message signal are located (centered) at harmonics (multiples)
Figure 3 below shows the time domain results of
of the sampling frequency.
sampling the biased sine wave with a Sampling pulse
generator that is sampling at a frequency slightly higher than
It can already be seen that by biasing the message
the Nyquist frequency. In this case 20.2 Hertz is chosen as the
signal prior to sampling we achieve what looks to be several
sampling frequency which exceeds the Nyquist theorem (A
amplitude modulated (AM) signals in the frequency domain.
signal must be sampled by a sampling frequency that is at least
If we were to filter out one of these sampling harmonics we
twice as high as the highest message frequency.)
would be left with a result very similar to an AM wave. This
concept will be explored later in this lab.


Figure 7: m(t) unbiased message.

Figure 5: Sampled(f)

As observed by Figure 5 above sampling results in
frequencies at 0 Hz, +-10Hz (original signal (fm)) as well as
several clones of the sampled signal centered at the sampling
frequency (fs) and its harmonics (n*fs). So the sampled
spectrum contains n*fs+-fm.

Next an analysis of biasing the message signal is
further explored by un-biasing the 10 Hertz sine wave as
shown by Figure 6.


The results of Figures 7 and 8 are as expected as well
as the spectrum (frequency domain) results shown by Figure 9
and 10.

It can already be seen that by un-biasing the message
signal prior to sampling we achieve what looks to be several
double sideband suppressed carrier (DSB-SC) signals in the
frequency domain. If we were to filter out one of these
sampling harmonics we would be left with a result very
similar to a DSB-SC wave. This concept will be explored
later in this lab.

Figure 6: Lab 1 Part A Sine Zero Bias.

The unbiased sine wave transient analysis (time
domain) results can be observed by Figure 7. And the
corresponding sampling results can be seen in Figure 8.

Figure 9: M(f) unbiased.


Figure 12: pulse m(t).

Figure 10: Sampled(f).

Next, our sine wave source is replaced with a 10
Hertz pulse wave as shown by Figure 11. A pulse in the time
domain results in a sinc function in the frequency domain.
Thus, in-order to accurately represent a pulse wave you need
to capture at least ten harmonic frequencies created as a result
of the sinc function. Since our pulse is a 10 hertz we should
have an accurate representation of the pulse by grabbing
frequencies up to 11* pulse frequency. Now letting this high
frequency become the highest frequency of the pulse wave our
sampling rate should be slightly higher than twice the highest Figure 13: sampled(t).
message frequency. This ensures a sampling rate of at least
222.2 Hertz and is the frequency used by our sampling pulse Figure 14 illustrates the 10 Hertz pulse in the
generator in Figure 11. frequency domain (representing the magnitude of a sinc) and
Figure 15 illustrates the sampled pulse in the frequency
domain.

As observed by Figure 15 the sampling results in
frequencies of the original signal as well as several clones of
the sampled signal centered at the sampling frequencies (fs)
and its harmonics (n*fs). So the sampled spectrum contains
n*fs+-fm.

Figure 11: Lab 1 Part A Pulse.

Figure 12 illustrates the 10 Hertz pulse in the time
domain and Figure 13 illustrates the sampled pulse in the time
domain. Figure 14: M(f).


Figure 17: pulse m(t).


For the last part of Part A, a constant (-5) DC signal
is added to our sampling pulse making the sampling pulse
have a zero-average value. This has the effect of canceling
out our DC pulse components and will cause the signal to be
unrecoverable in Part B of this lab report. This is done in
Simulink as shown by Figure 16.


The results of using the zero average sampler are
shown by Figures 17 and 18 in the time domain and by
Figures 19 and 20 in the frequency domain.

As expected and shown by Figure 20 by using a zero
average sampling function we have eliminated several critical
pieces of our message signal (effectively canceled out the DC
pulse components.) Without these components recovery of
our message signal will be impossible. Thus the sampling
function must have a non-zero average value. This concept
will gain further validity after we attempt message recovery in
Part 2.
Figure 16: Lab 1 Part A Pulse (Zero Avg. Sampler).


First we will attempt recovery of our original
sampled 10 Hertz sine wave using a sampling rate only
slightly higher than the Nyquist rate and a 5th order
Butterworth low-pass (LP) filter as shown by Figure 21.

Figure 19: M(f).

Figure 21: Lab 1 Part B Sine. Nyquist Sampled.

As shown by figure 22, using a sampling rate only
slightly (1%) higher than the Nyquist is insufficient for
message recovery since an ideal LP filter is unrealistic. The
frequency components displayed by Figure 23 contain more
than the original message (although hard to observe).

Thus we need to increase our sampling rate in order
to decrease our LP filter approximation.

Figure 20: Sampled(f). Sampled using Zero Average
value pulse. Notice the missing DC components.

2. Part 2 – Recovery of a Sampled Message

For the second part of this lab assignment (Part B),
Simulink is used to demonstrate the concept of message
recovery from a sampled message.
Figure 22: recovered(t) recovered poorly using Nyquist
Since sampling results in the original message Sampling Rate.
frequencies as well as several clones of the original message
frequencies centered at the sampling frequency (fs) and its
harmonics (n*fs). Recovery of the message should be
possible by low pass filtering the original message from the
sampled harmonics.

This should be possible so long as the sampling
frequency is at least twice as high as the highest frequency
component of the message frequency.

Furthermore, since ideal LP filters are hard to come
by we should further increase our chances of successful
message recovery by a further boost to our sampling
frequency.


Figure 25: recovered(t) recovered nicely by over sampling.

Figure 23: Recovered(f) recovered poorly using Nyquist
sampling rate.

Next the sampling rate is adjusted to a rate much
higher than the Nyquist (fm*5 versus fm*2). This should
ensure greater probability of message recovery. This was
accomplished using the Simulink model displayed by Figure
24.


Next our input sources are changed and an attempt at
recovering our original 10 Hertz pulse using a sampling rate
much higher than the Nyquist (fm*5 versus fm*2) is
attempted. This was accomplished using the model illustrated
by Figure 27.

Figure 24: Lab 1 Part B Sine. Over Sampled.

Figure 25 shows that our original sine wave was
successfully recovered using the higher sampling rate of 5*fm.
This allowed the 5th order Butterworth LP filter to effectively
eliminate the harmonic sampling components created through
the sampling process. Figure 26 shows the successful
recovery of the message in the frequency domain.

Figure 27: Lab 1 Part B Pulse. Over sampled and
recovered.

It has already been shown that a pulse in the time
domain results in a sync in the frequency domain, and we can


approximate a pulse by capturing as many harmonics in the
frequency domain as possible.

However, to reduce bandwidth we can achieve a
close approximation to our pulse by limiting the pulse to
approximately 10 harmonics. Thus, for this lab assignment fm
* 11 was chosen as the highest frequency component of the
pulse.

As Figure 28 illustrates successful recovery of the
pulse was accomplished by using a high sampling rate and
capturing LP filtering out the sampling resultant frequencies.

If the sampling had been too low aliasing could have
occurred making message recovery impossible. This concept
is explored further in Part C. Figure 30: Lab 1 Part B Pulse Recovery with Zero Avg.
Sampling.

As a result of using a zero average value function
versus a non-zero average value function we effectively
eliminate our DC message components when sampling and
make message recovery very improbable. The time domain
results of this concept are shown by Figure 31 which looks
nothing like our original pulse! The frequency domain results
shown by Figure 32 show the elimination of the messages DC
components.


Figure 31: recovered(t). Recovery was not successful due
to Zero Avg. Sampling frequency.

Figure 29: Sampled(f)

Next a final proof of the effects of using a zero
average value sampling function is explored by again adding a
constant (-0.5) to our sampling pulse as shown by Figure 30.


Thus even if we were to use an ideal low pass filter
capable of capturing all of the original message frequencies
(frequencies < 10 Hertz), we would still capture an extra
frequency due to the aliasing caused by sampling lower than
the Nyquist frequency as shown by equation 4:

et et et
As you can see 5 Hertz is less than our 10 Hertz
message frequency so our resultant low pass filter would
output both frequencies (5 Hertz and 10 Hertz). This is not the
recovered output we expected and thus aliasing has made
recovery of this message signal impossible.

To further explore this concept graphically the
following Simulink model was build using our previous
Figure 32: Sampled(f). Message unrecovered due to Zero
MATLAB concepts (10 Hertz sine message with a sampling
Avg. value of sampling pulse.
of 15 Hertz) as shown by Figure 33.
3. Part 3 – Aliasing

Aliasing refers to an effect which causes different
signals to become indistinguishable due to sampling. The
aliasing effect can make recover of a message signal
impossible when the sampling rate is too low and effectively
causes samples to shift into the original message.

Thus, aliasing can easily be caused and observed by
sampling at a rate that is lower than the Nyquist rate.

To demonstrate aliasing we can use our original 10
Hertz sine wave message and modify the sampling rate to be
lower than the Nyquist rate. For this lab assignment
demonstration a sampling rate of 1.5 times the message was
chosen. Figure 33: Lab 1 Part C Aliasing.

Mathematically aliasing will result in the following As illustrated by Figure 34 the resultant recovered
frequency components: message m(t) is no longer composed of a single sine wave but
seems to be a composite sinusoidal. Figure 35 further
confirms this and shows the additional frequency (5 Hertz)
M
component in our frequency spectrum.
ulse
ulse M S mple
So suppose we use a 10 Hertz sine wave message. If
we sample below the Nyquist minimum frequency at say 1.5
times the message, the sampled message would be composed
of the message frequencies and the addition of the samples
centered above and below the sampling frequency and its
multiples.

If the sampling frequency (fs) is 1.5 times the
message frequency then the sampling frequency must be 15
Hertz.

The samples appear both above and below this
Figure 34: recovered(t). Unrecovered message caused by
sampling frequency as described by equation 3:
aliasing. Sampling rate < Nyquist rate.


Figure 37: m(t). Biased.

Figure 38 shows the sampled message in the time
Figure 35: Recovered(f). Unrecovered message caused by domain. However we still need to filter out the extra
aliasing. Sampling rate < Nyquist rate. frequency components adding to the pulse form of the output.
By using a band-pass filter with a lower pass-band fs-fm and
4. Part 4 – AM & DSB-SC Wave creation upper pass-band fs+fm we get the correct AM result as shown
through sampling. by Figure 39.
An amplitude modulated wave is represented in the
time domain mathematically by:

A
Looking at this equation we can observe that a AM
wave is produced by multiplying a zero average sampling
function (like a sine or cosine) with a biased message signal
and then filtering (Band Pass) out the AM wave (remember
(Figure 36) sampling will make numerous copies each
centered at the sampling frequency) which contains a
duplicate of the message above the sampling frequency and a
duplicate of the message reflected below the message
frequency. So long as the message sinusoidal is biased Figure 38: sampled(t).
(contains a DC value) the filtered result is an AM wave. If the
message sinusoidal is unbiased the result will be a DSB-SC
wave.

Figure 39: AM filtered(t). After filtering result is AM
wave.

Figure 36: Lab 1 Part D Amplitude Modulated Signal.
Biased Sine and Zero Avg. Sampling.


Figure 40: M(f). Figure 42: m(t).

Figure 44 below correctly demonstrates an AM
(DSB-LC) wave created using a biased message, zero average
sampling, and band pass filtering.


Figure 44: AM Filtered(f).

Next to demonstrate a DSB-SC wave we need to
unbias the sinusoidal message. This will remove the DC
(carrier) of the DSB-LC (AM) wave. This was accomplished
in Simulink using Figure 41.

Figure 44: AM filtered(t).

The spectrum results required by a DSB-SC wave are
correctly illustrated by Figure 47. Thus we have proved that
you can create an AM wave and a DSB-SC wave using
bandpass filtering and zero averaged sampling.

Figure 41: Lab 1 Part D DSB-SC. Unbiased m(t).


V. CONCLUSIONS

In conclusion this lab has demonstrated the concepts
of sampling (Part A), aliasing (Part C) and the recovery (Part
B) of signals from a sampled signal. Finally this assignment
has applied the concept of sampling to amplitude modulation
and DSB-SC modulation.

It was demonstrated that by using the Nyquist
frequency as the sampling frequency we can mathematically
avoid aliasing and sample the signal in such a way as to allow
signal recovery through low-pass filtering. However since
Figure 45: M(f). ideal LP filters are very improbable and a using a higher
sampling rate decreases the filter design constraints it is often
better to use a sampling rate higher than the Nyquist rate.

Finally by using a zero average sampling function
and band-pass filtering we can achieve what appears to be an
AM or DSB-SC signal in the frequency domain. Whether or
not the carrier is present (AM vs. DSB-SC) is dependent upon
the presence of a bias on the message signal.

REFERENCES
[1] ykin, S , “An log n Digit l Communi tions 2nd
Edition” John Wiley & Sons, boken, NJ, 2007.

Figure 47 below correctly illustrates a DSB-SC
spectrum.

Figure 47: DSB-SC Filtered(f).


Figure 48: Lab 1 Part A Sine Wave Input.

Figure 49: Lab 1 Part A Sine Zero Bias.


Figure 50: Lab 1 Part A Pulse.

Figure 51: Lab 1 Part A Pulse (Zero Avg. Sampler).


Figure 52: Lab 1 Part B Sine. Nyquist Sampled.

Figure 53: Lab 1 Part B Pulse Recovery with Zero Avg. Sampling.


Figure 54: Lab 1 Part C Aliasing.

Figure 55: Lab 1 Part D Amplitude Modulated Signal. Biased Sine and Zero Avg. Sampling.


Figure 56: Lab 1 Part D DSB-SC. Unbiased m(t).

Ee463 communications 2 - lab 1 - loren schwappach

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