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.

Figure 1

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 .

Figure 2

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 .

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.

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.

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 .

Figure 3

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 .

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

Figure 4

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 .

Figure 5

The transfer function Hf H f is given by Hf=ⅈiff<0-ⅈiff>0 H f f 0 f 0

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

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.

Figure 6

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.

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.

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.

Figure 8

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.

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.

Source Compression

Consider the following 5-letter source.

Source Compression

Consider the following 5-letter source.

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.

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.

We send the bit stream consisting of Huffman-coded samples using one of the two depicted signal sets.

Figure 9

Signal Compression

Letters drawn from a four-symbol alphabet have the indicated probabilities.

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.

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.

Error Correcting Codes

A code maps pairs of information bits into codewords of length 5 as follows.

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

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.

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.

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.

Figure 11

Communication System Design

RU Communication Systems has been asked to design a communication system that meets the following requirements.

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 1035×1840 1035 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.

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?

Optimial Ethernet Random Access Protocols

Assume a population of NN computers want to transmit information on a random access channel. The access algorithm works as follows.

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 D D m apart, and a repeater is positioned halfway between them (Figure 12). 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.

Figure 12