Problems Dealing with Information Communication2.42001/08/212004/09/02 10:30:00 GMT-5DonJohnsondhj@rice.eduDonJohnsondhj@rice.edu(Blank Abstract)Noise in AM Systems
The signal
s_t
emerging from an AM communication system consists of two
parts: the message signal
st and additive noise. The plot shows the message
spectrum
Sf and noise power spectrum
PNf. The noise power spectrum lies completely within
the signal's band, and has a constant value there of
N02.
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?
No answer provided.Phase Modulation
A message signal
mtphase modulates a carrier if the transmitted signal
equals
xtA2fctφdmt
where
φd
is known as the phase deviation. In this problem, the phase
deviation is small. As with all analog modulation schemes,
assume that
mt1
, the message is bandlimited to
W Hz, and the carrier
frequency
fc
is much larger than W.
What is the transmission bandwidth?Find a receiver for this modulation scheme.What is the signal-to-noise ratio of the
received signal?
Hint: Use the facts that
x1
and
xx
for small x.
No answer provided.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
W Hz.
xtA1mlt2fctAmrt2fctFind the Fourier transform of
xt .
What is the transmission bandwidth and how does it compare with that
of standard AM?
Let's use a coherent demodulator at the receiver. The
block diagram for this receiver is shown below. 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.
Diagram for part (b)No answer provided.A Novel Communication System
A clever system designer claims that the following
transmitter has, despite its complexity, advantages over
the usual amplitude modulation system. The message
signal
mt
is bandlimited to W Hz, and
the carrier frequency
fc≫W
The channel attenuates the transmitted signal
xt
and adds white noise of spectral height
N02.
The transfer function
Hf
is given by
mtjf0jf0Find an expression for the spectrum of
xt. 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.
No answer provided.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 T-second
interval. For example, B bits would be transmitted
according to
xtAk0B1bk2k1f0t
for
0tT.
Here,
f0
is the frequency offset for each bit and it is harmonically
related to the bit interval T.
The value of
bk
is either 0 or 1.
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
TsTN). How should N be
related to B, 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?No answer provided.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. 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 d 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 d 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.No answer provided.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
s1t
and
s1t
and in data stream 2 by
s2t
and
s2t
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?No answer provided.Mixed Analog and Digital Transmission
A signal
mt
is transmitted using amplitude modulation in the usual way.
The signal has bandwidth W Hz,
and the carrier frequency is
fc. 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
dt representing the text is
transmitted as the same time as the analog signal
mt. The transmission signal spectrum is as shown,
and has a total bandwidth B.
Write an expression for the time-domain version of the
transmitted signal in terms of
mt
and the digital signal
dt.
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.No answer provided.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
pe
that depends on signal-to-noise ratio
EbN0. 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
pe?
Compute and plot the received signal's signal-to-noise
ratio for the two transmission schemes for a few values of
channel signal-to-noise ratios.
Compare and evaluate these systems.No answer provided.Source Compression
Consider the following 5-letter source.
Letter
Probability
a
.5
b
0.25
c
0.125
d
0.0625
e
0.0625
Find this source's entropyShow 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?No answer provided.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?No answer provided.Speech Compression
When we sample a signal, such as speech, we quantize the
signal's amplitude to a set of integers. For a b-bit
converter, signal amplitudes are represented by
2b 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].
Find the relative frequency of occurrence of quantized
amplitude values. The following Matlab program computes
the number of times each quantized value occurs.
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?No answer provided.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
10-3? If the receiver moves twice as far from the
transmitter (relative to the distance at which the
10-3
error rate was obtained), how does the performance change? No answer provided.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 d, 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
pe?
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?No answer provided.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 G and parity-check
matrix H 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)?No answer provided.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.
c1d1d2c2d2d3c3d1d3c4d1d2d3c5d1d2c6d1d2d3Find the generator matrix G and parity-check matrix H
for this code.Find a 3 x 6 matrix that recovers the data bits from
the codeword.What is the error correcting capability of the
code?No answer provided.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
G1000010000100001111001111011.
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.No answer provided.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 and 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?No answer provided.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.
No answer provided.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 H of the sampled
message signal depends on how many bits b are used in the A/D
converter (see figure below).
The signal is to be sent through a noisy channel having a
bandwidth of 25 kHz channel centered at 2MHz and a
signal-to-noise ratio 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 specifcations be met? Justify your answer.
No answer provided.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 1035 x 1840
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?No answer provided.Optimial Ethernet Random Access Protocols
Assume a population of N
computers want to transmit information on a random
access channel. The access algorithm works as follows.
Before transmitting, flip a coin that has
probability p of
coming up heads
If only one of the N
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
N 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
p be to maximize the
probability that exactly one computer
transmits?
What is the probability of one computer transmitting
when this optimal value of
p 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?No answer provided.