A modulated signal needs to be sent over a transmission
line having a characteristic impedance of
. So that the signal does not interfere with
signals others may be transmitting, it must be bandpass
filtered so that its bandwidth is 1 MHz and centered at
3.5 MHz. The filter's gain should be one in
magnitude. An op-amp filter is proposed.
What is the transfer function between the input
voltage and the voltage across the transmission line?
Find values for the resistors and capacitors so
that design goals are met.
Noise in AM Systems
The signal
emerging from an AM communication system consists of two parts:
the message signal,
, and additive noise. The plot shows the message spectrum
and noise power spectrum
. The noise power spectrum lies completely
within the signal's band, and has a constant value there
of
.
What is the message signal's power? What is the
signal-to-noise ratio?
Because the power in the message decreases with
frequency, the signal-to-noise ratio is not constant
within subbands. What is the signal-to-noise ratio in
the upper half of the frequency band?
A clever 241 student suggests filtering the
message before the transmitter modulates it so that
the signal spectrum is balanced
(constant) across frequency. Realizing that this
filtering affects the message signal, the student
realizes that the receiver must also compensate for
the message to arrive intact. Draw a block diagram of
this communication system. How does this system's
signal-to-noise ratio compare with that of the usual
AM radio?
Complementary Filters
Complementary filters usually have
“opposite” filtering characteristics (like a
lowpass and a highpass) and have transfer functions that
add to one. Mathematically,
and
are complementary if
We can use complementary filters to separate a signal
into two parts by passing it through each filter. Each
output can then be transmitted separately and the
original signal reconstructed at the receiver. Let's
assume the message is bandlimited to
and that
.
What circuits would be used to produce the
complementary filters?
Sketch a block diagram for a communication system
(transmitter and receiver) that employs complementary
signal transmission to send a message
.
What is the receiver's signal-to-noise ratio? How
does it compare to the standard system that sends the
signal by simple amplitude modulation?
Phase Modulation
A message signal
phase modulates a carrier if the
transmitted signal equals
where
is known as the phase deviation. In this problem, the
phase deviation is small. As with all analog modulation
schemes, assume that
, the message is bandlimited to
Hz, and the carrier
frequency
is much larger than .
What is the transmission bandwidth?
Find a receiver for this modulation
scheme.
What is the signal-to-noise ratio of the
received signal?
Use the facts that
and
for small .
Digital Amplitude Modulation
Two ELEC 241 students disagree about a homework
problem. The issue concerns the discrete-time signal
, where the signal
has no special characteristics and the
modulation frequency
is known. Sammy says that he can recover
from its amplitude-modulated version by the
same approach used in analog communications. Samantha
says that approach won't work.
What is the spectrum of the modulated signal?
Who is correct? Why?
The teaching assistant does not want to take
sides. He tells them that if
and
were both available,
can be recovered. What does he have in mind?
Anti-Jamming
One way for someone to keep people from receiving an AM
transmission is to transmit noise at the same carrier
frequency. Thus, if the carrier frequency is
so that the transmitted signal is
the jammer would transmit
.
The noise
has a constant power density spectrum over the bandwidth
of the message
.
The channel adds white noise of spectral height
.
What would be the output of a traditional AM
receiver tuned to the carrier frequency
?
RU Electronics proposes to counteract jamming by
using a different modulation scheme. The scheme's
transmitted signal has the form
where
is a periodic carrier signal (period
) having the
indicated waveform . What is the spectrum
of the transmitted signal with the proposed scheme?
Assume the message bandwidth
is much less than
the fundamental carrier frequency
.
The jammer, unaware of the change, is transmitting
with a carrier frequency of
,
while the receiver tunes a standard AM receiver to a
harmonic of the carrier frequency. What is the
signal-to-noise ratio of the receiver tuned to the
harmonic having the largest power that does not
contain the jammer?
Secret Comunications
A system for hiding AM transmissions has the transmitter
randomly switching between two carrier frequencies
and
. "Random switching" means that one carrier
frequency is used for some period of time, switches to
the other for some other period of time, back to the
first, etc. The receiver knows what the carrier
frequencies are but not when carrier frequency switches
occur. Consequently, the receiver must be designed to
receive the transmissions regardless of which carrier
frequency is used. Assume the message signal has
bandwidth . The channel
adds white noise of spectral height
.
How different should the carrier frequencies be so
that the message could be received?
What receiver would you design?
What signal-to-noise ratio for the demodulated
signal does your receiver yield?
AM Stereo
Stereophonic radio transmits two signals simultaneously
that correspond to what comes out of the left and right
speakers of the receiving radio. While FM stereo is
commonplace, AM stereo is not, but is much simpler to
understand and analyze. An amazing aspect of AM stereo
is that both signals are transmitted within the same
bandwidth as used to transmit just one. Assume the left
and right signals are bandlimited to
Hz.
Find the Fourier transform of
. What is the transmission bandwidth and
how does it compare with that of standard AM?
Let us use a coherent demodulator as the
receiver, shown in [link]. Show that this receiver
indeed works: It produces the left and right signals
separately.
Assume the channel adds white noise to the
transmitted signal. Find the signal-to-noise ratio
of each signal.
A Novel Communication System
A clever system designer claims that the depicted
transmitter has, despite its complexity,
advantages over the usual amplitude modulation system.
The message signal
is bandlimited to Hz, and
the carrier frequency
. The channel attenuates the transmitted signal
and adds white noise of spectral height
.
The transfer function
is given by
Find an expression for the spectrum of
.
Sketch your answer.
Show that the usual coherent receiver
demodulates this signal.
Find the signal-to-noise ratio that results
when this receiver is used.
Find a superior receiver (one that yields a
better signal-to-noise ratio), and analyze its
performance.
Multi-Tone Digital Communication
In a so-called multi-tone system, several bits are
gathered together and transmitted simultaneously on
different carrier frequencies during a
second interval. For
example, bits would be
transmitted according to
Here,
is the frequency offset for each bit and it is
harmonically related to the bit interval
. The value of
is either
or
.
Find a receiver for this transmission scheme.
An ELEC 241 almuni likes digital systems so much
that he decides to produce a discrete-time version. He
samples the received signal (sampling interval
). How should
be related to
, the number of
simultaneously transmitted bits?
The alumni wants to find a simple form for the
receiver so that his software implementation runs as
efficiently as possible. How
would you recommend he implement the
receiver?
City Radio Channels
In addition to additive white noise, metropolitan
cellular radio channels also contain multipath: the
attenuated signal and a delayed, further attenuated
signal are received superimposed. As shown in
[link], multipath occurs
because the buildings reflect the signal and the
reflected path length between transmitter and receiver
is longer than the direct path.
Assume that the length of the direct path is
meters and the
reflected path is 1.5 times as long. What is the
model for the channel, including the multipath and the
additive noise?
Assume is 1 km.
Find and sketch the magnitude of the transfer function
for the multipath component of the channel. How would
you characterize this transfer function?
Would the multipath affect AM radio? If not, why
not; if so, how so? Would analog cellular telephone,
which operates at much higher carrier frequencies (800
MHz vs. 1 MHz for radio), be affected or not? Analog
cellular telephone uses amplitude modulation to
transmit voice.
How would the usual AM receiver be modified to
minimize multipath effects? Express your modified
receiver as a block diagram.
Downlink Signal Sets
In digital cellular telephone systems, the base station
(transmitter) needs to relay different voice signals to
several telephones at the same time. Rather than send
signals at different frequencies, a clever Rice engineer
suggests using a different signal set for each data
stream. For example, for two simultaneous data streams,
she suggests BPSK signal sets that have the depicted basic
signals.
Thus, bits are represented in data stream 1 by
and
and in data stream 2 by
and
, each of which are modulated by 900 MHz
carrier. The transmitter sends the two data streams so
that their bit intervals align. Each receiver uses a
matched filter for its receiver. The requirement is
that each receiver not receive the
other's bit stream.
What is the block diagram describing the proposed
system?
What is the transmission bandwidth required by
the proposed system?
Will the proposal work? Does the fact that the
two data streams are transmitted in the same
bandwidth at the same time mean that each receiver's
performance is affected? Can each bit stream be
received without interference from the other?
Mixed Analog and Digital Transmission
A signal
is transmitted using amplitude modulation in the usual
way. The signal has bandwidth
Hz, and the carrier
frequency is
. In addition to sending this analog signal,
the transmitter also wants to send ASCII text in an
auxiliary band that lies slightly above the
analog transmission band. Using an 8-bit representation
of the characters and a simple baseband BPSK signal set
(the constant signal +1 corresponds to a 0, the constant
-1 to a 1), the data signal
representing the text is transmitted as the
same time as the analog signal
. The transmission signal spectrum is as shown, and has a
total bandwidth .
Write an expression for the time-domain version
of the transmitted signal in terms of
and the digital signal
.
What is the maximum datarate the scheme can
provide in terms of the available bandwidth?
Find a receiver that yields both the analog
signal and the bit stream.
Digital Stereo
Just as with analog communication, it should be possible
to send two signals simultaneously over a digital
channel. Assume you have two CD-quality signals (each
sampled at 44.1 kHz with 16 bits/sample). One
suggested transmission scheme is to use a quadrature
BPSK scheme. If
and
each represent a bit stream, the transmitted
signal has the form
where
is a unit-amplitude pulse having duration
and
,
equal either +1 or -1 according to the bit being
transmitted for each signal. The channel adds white
noise and attenuates the transmitted signal.
What value would you choose for the carrier frequency
?
What is the transmission bandwidth?
What receiver would you design that would yield
both bit streams?
Digital and Analog Speech Communication
Suppose we transmit speech signals over comparable
digital and analog channels. We want to compare the
resulting quality of the received signals. Assume the
transmitters use the same power, and the channels
introduce the same attenuation and additive white noise.
Assume the speech signal has a 4 kHz bandwidth and, in
the digital case, is sampled at an 8 kHz rate with
eight-bit A/D conversion. Assume simple binary source
coding and a modulated BPSK transmission scheme.
What is the transmission bandwidth of the analog
(AM) and digital schemes?
Assume the speech signal's amplitude has a magnitude
less than one. What is maximum amplitude
quantization error introduced by the A/D
converter?
In the digital case, each bit in quantized speech
sample is received in error with probability
that depends on signal-to-noise ratio
. However, errors in each bit have a
different impact on the error in the reconstructed
speech sample. Find the mean-squared error between
the transmitted and received amplitude.
In the digital case, the recovered speech signal can
be considered to have two noise sources added to each
sample's true value: One is the A/D amplitude quantization
noise and the second is due to channel errors. Because
these are separate, the total noise power equals the sum
of these two. What is the signal-to-noise ratio of the
received speech signal as a function of
?
Compute and plot the received signal's
signal-to-noise ratio for the two transmission
schemes as a function of channel signal-to-noise
ratio.
Compare and evaluate these systems.
Source Compression
Consider the following 5-letter source.
Letter
Probability
a
0.5
b
0.25
c
0.125
d
0.0625
e
0.0625
Find this source's entropy.
Show that the simple binary coding is
inefficient.
Find an unequal-length codebook for this sequence
that satisfies the Source Coding Theorem. Does your
code achieve the entropy limit?
How much more efficient is this code than the
simple binary code?
Source Compression
Consider the following 5-letter source.
Letter
Probability
a
0.4
b
0.2
c
0.15
d
0.15
e
0.1
Find this source's entropy.
Show that the simple binary coding is
inefficient.
Find the Huffman code for this source.
What is its average code length?
Speech Compression
When we sample a signal, such as
speech, we quantize the signal's amplitude to a set of
integers. For a -bit
converter, signal amplitudes are represented by
integers. Although these integers could be represented
by a binary code for digital transmission, we should
consider whether a Huffman coding would be more
efficient.
Load into Matlab the segment of speech contained
in y.mat. Its sampled values lie in the
interval (-1, 1). To simulate a 3-bit converter, we use
Matlab's round function to create quantized amplitudes
corresponding to the integers [0 1 2 3 4 5 6
7].
y_quant = round(3.5*y + 3.5);
Find the relative frequency of occurrence of quantized
amplitude values. The following Matlab program computes
the number of times each quantized value occurs.
for n=0:7;
count(n+1) = sum(y_quant == n);
end;
Find the entropy of this source.
Find the Huffman code for this source. How
would you characterize this source code in
words?
How many fewer bits would be used in
transmitting this speech segment with your Huffman
code in comparison to simple binary coding?
Digital Communication
In a digital cellular system, a signal bandlimited to 5
kHz is sampled with a two-bit A/D converter at its
Nyquist frequency. The sample values are found to have
the shown relative frequencies.
Sample
Value
Probability
0
0.15
1
0.35
2
0.3
3
0.2
We send the bit stream consisting of
Huffman-coded samples using one of the two depicted signal sets.
What is the datarate of the compressed
source?
Which choice of signal set maximizes the
communication system's performance?
With no error-correcting coding, what
signal-to-noise ratio would be needed for your chosen
signal set to guarantee that the bit error probability
will not exceed
?
If the receiver moves twice as far from the transmitter
(relative to the distance at which the
error rate was obtained), how does the performance change?
Signal Compression
Letters drawn from a four-symbol alphabet have the
indicated probabilities.
Letter
Probability
a
1/3
b
1/3
c
1/4
d
1/12
What is the average number of bits necessary to
represent this alphabet?
Using a simple binary code for this alphabet, a
two-bit block of data bits naturally emerges. Find an
error correcting code for two-bit data blocks that
corrects all single-bit errors.
How would you modify your code so that the
probability of the letter
being confused with the
letter is minimized? If
so, what is your new code; if not, demonstrate that this
goal cannot be achieved.
Universal Product Code
The Universal Product Code (UPC), often known as a bar
code, labels virtually every sold good. An
example of a
portion of the code is shown.
Here a sequence of black and white bars, each having width
, presents an 11-digit number
(consisting of decimal digits) that uniquely identifies the
product. In retail stores, laser scanners read this code,
and after accessing a database of prices, enter the price
into the cash register.
How many bars must be used to represent a single
digit?
A complication of the laser scanning system is that
the bar code must be read either forwards or backwards.
Now how many bars are needed to represent each
digit?
What is the probability that the 11-digit code is
read correctly if the probability of reading a single
bit incorrectly is
?
How many error correcting bars would need to be
present so that any single bar error occurring in the
11-digit code can be corrected?
Error Correcting Codes
A code maps pairs of information bits into codewords of
length 5 as follows.
Data
Codeword
00
00000
01
01101
10
10111
11
11010
What is this code's efficiency?
Find the generator matrix and parity-check
matrix
for this code.
Give the decoding table for this code. How many
patterns of 1, 2, and 3 errors are correctly
decoded?
What is the block error
probability (the probability of any number of errors
occurring in the decoded codeword)?
Digital Communication
A digital source produces sequences of nine letters with the following probabilities.
letter
a
b
c
d
e
f
g
h
i
probability
Find a Huffman code that compresses this source.
How does the resulting code compare with the best possible code?
A clever engineer proposes the following (6,3) code to correct errors after transmission through a digital channel.
What is the error correction capability of this code?
The channel's bit error probability is 1/8.
What kind of code should be used to transmit data over this channel?
Overly Designed Error Correction Codes
An Aggie engineer wants not only to have codewords for his
data, but also to hide the information from Rice engineers
(no fear of the UT engineers). He decides to represent
3-bit data with 6-bit codewords in which none of the data
bits appear explicitly.
Find the generator matrix and parity-check matrix
for this
code.
Find a
matrix that recovers the data bits from the
codeword.
What is the error correcting capability of the
code?
Error Correction?
It is important to realize that when more transmission
errors than can be corrected, error correction algorithms
believe that a smaller number of errors have occurred and
correct accordingly. For example, consider a (7,4) Hamming
code having the generator matrix
This code corrects all single-bit error, but if a double bit error
occurs, it corrects using a single-bit error correction approach.
How many double-bit errors can occur in a
codeword?
For each double-bit error pattern, what is the
result of channel decoding? Express your result as a
binary error sequence for the data bits.
Selective Error Correction
We have found that digital transmission errors occur with a
probability that remains constant no matter how "important"
the bit may be. For example, in transmitting digitized
signals, errors occur as frequently for the most significant
bit as they do for the least significant bit. Yet, the
former errors have a much larger impact on the overall
signal-to-noise ratio than the latter. Rather than applying
error correction to each sample value, why not concentrate
the error correction on the most important bits? Assume
that we sample an 8 kHz signal with an 8-bit A/D converter.
We use single-bit error correction on the most significant
four bits and none on the least significant four. Bits are
transmitted using a modulated BPSK signal set over an
additive white noise channel.
How many error correction bits must be added to
provide single-bit error correction on the most
significant bits?
How large must the signal-to-noise ratio of the
received signal be to insure reliable
communication?
Assume that once error correction is applied, only
the least significant 4 bits can be received in error.
How much would the output signal-to-noise ratio improve
using this error correction scheme?
Compact Disk
Errors occur in reading audio compact disks. Very few
errors are due to noise in the compact disk player; most
occur because of dust and scratches on the disk surface.
Because scratches span several bits, a single-bit error is
rare; several consecutive bits in error
are much more common. Assume that scratch and dust-induced
errors are four or fewer consecutive bits long. The audio
CD standard requires 16-bit, 44.1 kHz analog-to-digital
conversion of each channel of the stereo analog signal.
How many error-correction bits are required to
correct scratch-induced errors for each 16-bit
sample?
Rather than use a code that can correct several
errors in a codeword, a clever 241 engineer proposes
interleaving consecutive coded
samples. As the cartoon shows, the bits representing
coded samples are interpersed before they are written on
the CD. The CD player de-interleaves the coded data,
then performs error-correction. Now, evaluate this
proposed scheme with respect to the non-interleaved
one.
Communication System Design
RU Communication Systems has been asked to design a
communication system that meets the following requirements.
The baseband message signal has a bandwidth of
10 kHz.
The RUCS engineers find that the entropy
of the sampled message
signal depends on how many bits
are used in the A/D
converter (see table below).
The signal is to be sent through a noisy channel
having a bandwidth of 25 kHz channel centered at 2 MHz
and a signal-to-noise ration within that band of 10
dB.
Once received, the message signal must have a
signal-to-noise ratio of at least 20 dB.
b
H
3
2.19
4
3.25
5
4.28
6
5.35
Can these specifications be met? Justify your answer.
HDTV
As HDTV (high-definition television) was being developed,
the FCC restricted this digital system to use in the same
bandwidth (6 MHz) as its analog (AM) counterpart. HDTV
video is sampled on a
raster at 30 images per second for each of the
three colors. The least-acceptable picture received by
television sets located at an analog station's broadcast
perimeter has a signal-to-noise ratio of about 10 dB.
Using signal-to-noise ratio as the criterion, how
many bits per sample must be used to guarantee that a
high-quality picture, which achieves a signal-to-noise
ratio of 20 dB, can be received by any HDTV set within
the same broadcast region?
Assuming the digital television channel has the same
characteristics as an analog one, how much compression
must HDTV systems employ?
Digital Cellular Telephones
In designing a digital version of a wireless telephone, you
must first consider certain fundamentals. First of all, the
quality of the received signal, as measured by the
signal-to-noise ratio, must be at least as good as that
provided by wireline telephones (30 dB) and the message
bandwidth must be the same as wireline telephone. The
signal-to-noise ratio of the allocated wirelss channel,
which has a 5 kHz bandwidth, measured 100 meters from the
tower is 70 dB. The desired range for a cell is 1 km. Can
a digital cellphone system be designed according to these
criteria?
Optimal Ethernet Random Access Protocols
Assume a population of
computers want to transmit information on a random access
channel. The access algorithm works as follows.
Before transmitting, flip a coin that has
probability of coming up
heads
If only one of the
computer's coins comes up heads, its transmission occurs
successfully, and the others must wait until that
transmission is complete and then resume the
algorithm.
If none or more than one head comes up, the
computers will either
remain silent (no heads) or a collision will occur (more
than one head). This unsuccessful transmission
situation will be detected by all computers once the
signals have propagated the length of the cable, and the
algorithm resumes (return to the beginning).
What is the optimal probability to use for flipping
the coin? In other words, what should
be to maximize the
probability that exactly one computer transmits?
What is the probability of one computer
transmitting when this optimal value of
is used as the number of
computers grows to infinity?
Using this
optimal probability, what is the average number of coin
flips that will be necessary to resolve the access so
that one computer successfully transmits?
Evaluate this algorithm. Is it realistic? Is it
efficient?
Repeaters
Because signals
attenuate with distance from the transmitter,
repeaters are frequently employed for both analog
and digital communication. For example, let's assume that the
transmitter and receiver are
m apart, and a repeater is
positioned halfway between them ([link]). What the repater does is amplify its
received signal to exactly cancel the attenuation
encountered along the first leg and to re-transmit the
signal to the ultimate receiver. However, the signal the
repeater receives contains white noise as well as the
transmitted signal. The receiver experiences the same amount
of white noise as the repeater.
What is the block diagram for this system?
For an amplitude-modulation communication system, what
is the signal-to-noise ratio of the demodulated signal at
the receiver? Is this better or worse than the
signal-to-noise ratio when no repeater is present?
For digital communication, we must consider the
system's capacity. Is the capacity larger with the
repeater system than without it? If so, when; if not, why
not?
Designing a Speech Communication System
We want to examine both analog and digital communication alternatives for a dedicated speech transmission system.
Assume the speech signal has a 5 kHz bandwidth.
The wireless link between transmitter and receiver is such that 200 watts of power can be received at a pre-assigned carrier frequency.
We have some latitude in choosing the transmission bandwidth, but the noise power added by the channel increases with bandwidth with a proportionality constant of 0.1 watt/kHz.
Design an analog system for sending speech under this scenario.
What is the received signal-to-noise ratio under these design constraints?
How many bits must be used in the A/D converter to achieve the same signal-to-noise ratio?
Is the bandwidth required by the digital channel to send the samples without error greater or smaller than the analog bandwidth?
Digital vs. Analog
You are the Chairman/Chairwoman of the FCC.
The frequency band 3 MHz to 3.5 MHz has been allocated for a new “high-quality” AM band.
Each station licensed for this band will transmit signals having a bandwidth of 10 kHz, twice the message bandwidth of what current stations can send.
How many stations can be allocated to this band and with what carrier frequencies?
Looking ahead, conversion to digital transmission is not far in the future.
The characteristics of the new digital radio system need to be established and you are the boss!
Detail the characteristics of the analog-to-digital converter that must be used to prevent aliasing and ensure a signal-to-noise ratio of 25 dB.
Without employing compression, how many digital radio stations could be allocated to the band if each station used BPSK modulation?
Evaluate this design approach.