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Information Communication Problems

Module by: Don Johnson

Summary: Problems Dealing with Information Communication.

Problem Set
Problem 1:

Signals on Transmission Lines

A modulated signal needs to be sent over a transmission line having a characteristic impedance of Z 0 = 50 Ω Z 0 50 Ω . 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.
opamp20.png
Figure 1
  1. What is the transfer function between the input voltage and the voltage across the transmission line?
  2. Find values for the resistors and capacitors so that design goals are met.
Problem 2:

Noise in AM Systems

The signal s ^ t s ^ t emerging from an AM communication system consists of two parts: the message signal, st s t , and additive noise. The plot shows the message spectrum Sf S f and noise power spectrum P N f P N f . The noise power spectrum lies completely within the signal's band, and has a constant value there of N02 N0 2 .
spectrum15.png
Figure 2
  1. What is the message signal's power? What is the signal-to-noise ratio?
  2. 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?
  3. 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?
Problem 3:

Complementary Filters

Complementary filters usually have “opposite” filtering characteristics (like a lowpass and a highpass) and have transfer functions that add to one. Mathematically, H 1 f H 1 f and H 2 f H 2 f are complementary if H 1 f+ H 2 f=1 H 1 f H 2 f 1 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 mesage is bandlimited to WHz W Hz and that H 1 f=aa+ⅈ2πf H 1 f a a 2 f .
  1. What circuits would be used to produce the complementary filters?
  2. Sketch a block diagram for a communication system (transmitter and receiver) that employs complementary signal transmission to send a message mt m t .
  3. 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?
Problem 4:

Phase Modulation

A message signal mt m t phase modulates a carrier if the transmitted signal equals xt=Asin2π f c t+ φ d mt x t A 2 f c t φ d m t where φd φd is known as the phase deviation. In this problem, the phase deviation is small. As with all analog modulation schemes, assume that |mt|<1 m t 1 , the message is bandlimited to WW Hz, and the carrier frequency fc fc is much larger than WW.
  1. What is the transmission bandwidth?
  2. Find a receiver for this modulation scheme.
  3. What is the signal-to-noise ratio of the received signal?
hint: Use the facts that cosx≈1 x 1 and sinx≈x x x for small xx.
Problem 5:

Digital Amplitude Modulation

Two ELEC 241 students disagree about a homework problem. The issue concerns the discrete-time signal sncos2π f 0 n s n 2 f 0 n , where the signal sn s n has no special characteristics and the modulation frequency f 0 f 0 is known. Sammy says that he can recover sn s n from its amplitude-modulated version by the same approach used in analog communications. Samantha says that approach won't work.
  1. What is the spectrum of the modulated signal?
  2. Who is correct? Why?
  3. The teaching assistant does not want to take sides. He tells them that if sncos2π f 0 n s n 2 f 0 n and snsin2π f 0 n s n 2 f 0 n were both available, sn s n can be recovered. What does he have in mind?
Problem 6:

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 f c f c so that the transmitted signal is A T 1+mtsin2π f c t A T 1 m t 2 f c t the jammer would transmit A J ntsin2π f c t+φ A J n t 2 f c t φ . The noise nt n t has a constant power density spectrum over the bandwidth of the message mt m t . The channel adds white noise of spectral height N 0 2 N 0 2 .
  1. What would be the output of a traditional AM receiver tuned to the carrier frequency f c f c ?
  2. RU Electronics proposes to counteract jamming by using a different modulation scheme. The scheme's transmitted signal has the form A T 1+mtct A T 1 m t c t where ct c t is a periodic carrier signal (period 1 f c 1 f c ) having the indicated waveform . What is the spectrum of the transmitted signal with the proposed scheme? Assume the message bandwidth WW is much less than the fundamental carrier frequency f c f c .
  3. The jammer, unaware of the change, is transmitting with a carrier frequency of f c f c , 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?
sig46.png
Figure 3
Problem 7:

Secret Comunications

A system for hiding AM transmissions has the transmitter randomly switching between two carrier frequencies f 1 f 1 and f 2 f 2 . "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 WW. The channel adds white noise of spectral height N 0 2 N 0 2 .
  1. How different should the carrier frequencies be so that the message could be received?
  2. What receiver would you design?
  3. What signal-to-noise ratio for the demodulated signal does your receiver yield?
Problem 8:

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 WW Hz. xt=A1+mltcos2πfct+Amrtsin2πfct x t A 1 ml t 2 fc t A mr t 2 fc t
  1. Find the Fourier transform of xt x t . What is the transmission bandwidth and how does it compare with that of standard AM?
  2. Let us use a coherent demodulator as the receiver, shown in Figure 4. Show that this receiver indeed works: It produces the left and right signals separately.
  3. Assume the channel adds white noise to the transmitted signal. Find the signal-to-noise ratio of each signal.
sys4.png
Figure 4
Problem 9:

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 mt m t is bandlimited to WW Hz, and the carrier frequency fc ≫W ≫ fc W . The channel attenuates the transmitted signal xt x t and adds white noise of spectral height N02 N0 2 .
sys5.png
Figure 5
The transfer function Hf H f is given by Hf=ⅈiff<0-ⅈiff>0 H f f 0 f 0
  1. Find an expression for the spectrum of xt x t . Sketch your answer.
  2. Show that the usual coherent receiver demodulates this signal.
  3. Find the signal-to-noise ratio that results when this receiver is used.
  4. Find a superior receiver (one that yields a better signal-to-noise ratio), and analyze its performance.
Problem 10:

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 TT second interval. For example, BB bits would be transmitted according to
∀t,0≤t<T:xt=A∑k=0B-1bksin2πk+1f0t t 0 t T x t A k 0 B 1 bk 2 k 1 f0 t (1)
Here, f0 f0 is the frequency offset for each bit and it is harmonically related to the bit interval TT. The value of bk bk is either -11 or +11.
  1. Find a receiver for this transmission scheme.
  2. 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 Ts=TN Ts T N ). How should NN be related to BB, the number of simultaneously transmitted bits?
  3. 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?
Problem 11:

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 Figure 6, multipath occurs because the buildings reflect the signal and the reflected path length between transmitter and receiver is longer than the direct path.
sys6.png
Figure 6
  1. Assume that the length of the direct path is dd 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?
  2. Assume dd 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?
  3. 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.
  4. How would the usual AM receiver be modified to minimize multipath effects? Express your modified receiver as a block diagram.
Problem 12:

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.
sig12.png
Figure 7
Thus, bits are represented in data stream 1 by s1t s1 t and -s1 t s1 t and in data stream 2 by s2t s2 t and -s2 t s2 t , 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.
  1. What is the block diagram describing the proposed system?
  2. What is the transmission bandwidth required by the proposed system?
  3. 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?
Problem 13:

Mixed Analog and Digital Transmission

A signal mt m t is transmitted using amplitude modulation in the usual way. The signal has bandwidth WW Hz, and the carrier frequency is fc 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 d t representing the text is transmitted as the same time as the analog signal mt m t . The transmission signal spectrum is as shown, and has a total bandwidth BB.
spectrum14.png
Figure 8
  1. Write an expression for the time-domain version of the transmitted signal in terms of mt m t and the digital signal dt d t .
  2. What is the maximum datarate the scheme can provide in terms of the available bandwidth?
  3. Find a receiver that yields both the analog signal and the bit stream.
Problem 14:

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 b 1 n b 1 n and b 2 n b 2 n each represent a bit stream, the transmitted signal has the form xt=A∑ b 1 nsin2π f c t-nTpt-nT+ b 2 ncos2π f c t-nTpt-nT x t A n b 1 n 2 f c t n T p t n T b 2 n 2 f c t n T p t n T where pt p t is a unit-amplitude pulse having duration TT and b 1 n b 1 n , b 2 n b 2 n equal either +1 or -1 according to the bit being transmitted for each signal. The channel adds white noise and attenuates the transmitted signal.
  1. What value would you choose for the carrier frequency f c f c ?
  2. What is the transmission bandwidth?
  3. What receiver would you design that would yield both bit streams?
Problem 15:

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.
  1. What is the transmission bandwidth of the analog (AM) and digital schemes?
  2. Assume the speech signal's amplitude has a magnitude less than one. What is maximum amplitude quantization error introduced by the A/D converter?
  3. In the digital case, each bit in quantized speech sample is received in error with probability pe pe that depends on signal-to-noise ratio EbN0 Eb N0 . 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.
  4. 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 pe?
  5. 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.
  6. Compare and evaluate these systems.
Problem 16:

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
  1. Find this source's entropy.
  2. Show that the simple binary coding is inefficient.
  3. Find an unequal-length codebook for this sequence that satisfies the Source Coding Theorem. Does your code achieve the entropy limit?
  4. How much more efficient is this code than the simple binary code?
Problem 17:

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
  1. Find this source's entropy.
  2. Show that the simple binary coding is inefficient.
  3. Find the Huffman code for this source. What is its average code length?
Problem 18:

Speech Compression

When we sample a signal, such as speech, we quantize the signal's amplitude to a set of integers. For a bb-bit converter, signal amplitudes are represented by 2b 2 b 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.
  1. 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.
  2. Find the Huffman code for this source. How would you characterize this source code in words?
  3. How many fewer bits would be used in transmitting this speech segment with your Huffman code in comparison to simple binary coding?
Problem 19:

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.
sig11.png
Figure 9
  1. What is the datarate of the compressed source?
  2. Which choice of signal set maximizes the communication system's performance?
  3. 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 10 -3 ? If the receiver moves twice as far from the transmitter (relative to the distance at which the 10-3 10 -3 error rate was obtained), how does the performance change?
Problem 20:

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
  1. What is the average number of bits necessary to represent this alphabet?
  2. 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.
  3. How would you modify your code so that the probability of the letter aa being confused with the letter dd is minimized? If so, what is your new code; if not, demonstrate that this goal cannot be achieved.
Problem 21:

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.
sig37.png
Figure 10
Here a sequence of black and white bars, each having width dd, 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.
  1. How many bars must be used to represent a single digit?
  2. 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?
  3. What is the probability that the 11-digit code is read correctly if the probability of reading a single bit incorrectly is pe pe?
  4. 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?
Problem 22:

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
  1. What is this code's efficiency?
  2. Find the generator matrix GG and parity-check matrix HH for this code.
  3. Give the decoding table for this code. How many patterns of 1, 2, and 3 errors are correctly decoded?
  4. What is the block error probability (the probability of any number of errors occurring in the decoded codeword)?
Problem 23:

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. c1=d1⊕d2 c1 d1 d2 c2=d2⊕d3 c2 d2 d3 c3=d1⊕d3 c3 d1 d3 c4=d1⊕d2⊕d3 c4 d1 d2 d3 c5=d1⊕d2 c5 d1 d2 c6=d1⊕d2⊕d3 c6 d1 d2 d3
  1. Find the generator matrix GG and parity-check matrix HH for this code.
  2. Find a 3 × 6 3×6 matrix that recovers the data bits from the codeword.
  3. What is the error correcting capability of the code?
Problem 24:

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 G=1000010000100001111001111011 G 1000 0100 0010 0001 1110 0111 1011 This code corrects all single-bit error, but if a double bit error occurs, it corrects using a single-bit error correction approach.
  1. How many double-bit errors can occur in a codeword?
  2. 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.
Problem 25:

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.
  1. How many error correction bits must be added to provide single-bit error correction on the most significant bits?
  2. How large must the signal-to-noise ratio of the received signal be to insure reliable communication?
  3. 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?
Problem 26:

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.
  1. How many error-correction bits are required to correct scratch-induced errors for each 16-bit sample?
  2. 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.
sig43.png
Figure 11
Problem 27:

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 HH of the sampled message signal depends on how many bits bb 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.
Problem 28:

HDTV

As HDTV (high-definition television) was being developed, the FCC restricted this digital system to use in the same