2.8 SummaryThe physical layer is the basis of all networks. Nature imposes two fundamental limits on all channels, and these determine their bandwidth. These limits are the Nyquist limit, which deals with noiseless channels, and the Shannon limit, which deals with noisy channels.Transmission media can be guided or unguided. The principal guided media are twisted pair, coaxial cable, and fiber optics. Unguided media include radio, microwaves, infrared, and lasers through the air. An up-and-coming transmission system is satellite communication, especially LEO systems.A key element in most wide area networks is the telephone system. Its main components are the local loops, trunks, and switches. Local loops are analog, twisted pair circuits, which require modems for transmitting digital data. ADSL offers speeds up to 50 Mbps by dividing the local loop into many virtual channels and modulating each one separately. Wireless local loops are another new development to watch, especially LMDS.Trunks are digital, and can be multiplexed in several ways, including FDM, TDM, and WDM. Both circuit switching and packet switching are important.For mobile applications, the fixed telephone system is not suitable. Mobile phones are currently in widespread use for voice and will soon be in widespread use for data. The first generation was analog, dominated by AMPS. The second generation was digital, with D-AMPS, GSM, and CDMA the major options. The third generation will be digital and based on broadband CDMA.An alternative system for network access is the cable television system, which has gradually evolved from a community antenna to hybrid fiber coax. Potentially, it offers very high bandwidth, but the actual bandwidth available in practice depends heavily on the number of other users currently active and what they are doing.
ProblemsCompute the Fourier coefficients for the function f(t) = t (0 t 1).A noiseless 4-kHz channel is sampled every 1 msec. What is the maximum data rate?Television channels are 6 MHz wide. How many bits/sec can be sent if four-level digital signals are used? Assume a noiseless channel.If a binary signal is sent over a 3-kHz channel whose signal-to-noise ratio is 20 dB, what is the maximum achievable data rate?What signal-to-noise ratio is needed to put a T1 carrier on a 50-kHz line?What is the difference between a passive star and an active repeater in a fiber network?How much bandwidth is there in 0.1 micron of spectrum at a wavelength of 1 micron?It is desired to send a sequence of computer screen images over an optical fiber. The screen is 480 x 640 pixels, each pixel being 24 bits. There are 60 screen images per second. How much bandwidth is needed, and how many microns of wavelength are needed for this band at 1.30 microns?Is the Nyquist theorem true for optical fiber or only for copper wire?In Fig. 2-6 the lefthand band is narrower than the others. Why?Radio antennas often work best when the diameter of the antenna is equal to the wavelength of the radio wave. Reasonable antennas range from 1 cm to 5 meters in diameter. What frequency range does this cover?Multipath fading is maximized when the two beams arrive 180 degrees out of phase. How much of a path difference is required to maximize the fading for a 50-km-long 1-GHz microwave link?A laser beam 1 mm wide is aimed at a detector 1 mm wide 100 m away on the roof of a building. How much of an angular diversion (in degrees) does the laser have to have before it misses the detector?The 66 low-orbit satellites in the Iridium project are divided into six necklaces around the earth. At the altitude they are using, the period is 90 minutes. What is the average interval for handoffs for a stationary transmitter?Consider a satellite at the altitude of geostationary satellites but whose orbital plane is inclined to the equatorial plane by an angle . To a stationary user on the earth's surface at north latitude , does this satellite appear motionless in the sky? If not, describe its motion.How many end office codes were there pre-1984, when each end office was named by its three-digit area code and the first three digits of the local number? Area codes started with a digit in the range 29, had a 0 or 1 as the second digit, and ended with any digit. The first two digits of a local number were always in the range 29. The third digit could be any digit.Using only the data given in the text, what is the maximum number of telephones that the existing U.S. system can support without changing the numbering plan or adding additional equipment? Could this number of telephones actually be achieved? For purposes of this problem, a computer or fax machine counts as a telephone. Assume there is only one device per subscriber line.A simple telephone system consists of two end offices and a single toll office to which each end office is connected by a 1-MHz full-duplex trunk. The average telephone is used to make four calls per 8-hour workday. The mean call duration is 6 min. Ten percent of the calls are long-distance (i.e., pass through the toll office). What is the maximum number of telephones an end office can support? (Assume 4 kHz per circuit.)A regional telephone company has 10 million subscribers. Each of their telephones is connected to a central office by a copper twisted pair. The average length of these twisted pairs is 10 km. How much is the copper in the local loops worth? Assume that the cross section of each strand is a circle 1 mm in diameter, the density of copper is 9.0 grams/cm3, and that copper sells for 3 dollars per kilogram.Is an oil pipeline a simplex system, a half-duplex system, a full-duplex system, or none of the above?The cost of a fast microprocessor has dropped to the point where it is now possible to put one in each modem. How does that affect the handling of telephone line errors?A modem constellation diagram similar to Fig. 2-25 has data points at the following coordinates: (1, 1), (1, -1), (-1, 1), and (-1, -1). How many bps can a modem with these parameters achieve at 1200 baud?A modem constellation diagram similar to Fig. 2-25 has data points at (0, 1) and (0, 2). Does the modem use phase modulation or amplitude modulation?In a constellation diagram, all the points lie on a circle centered on the origin. What kind of modulation is being used?How many frequencies does a full-duplex QAM-64 modem use?An ADSL system using DMT allocates 3/4 of the available data channels to the downstream link. It uses QAM-64 modulation on each channel. What is the capacity of the downstream link?In the four-sector LMDS example of Fig. 2-30, each sector has its own 36-Mbps channel. According to queueing theory, if the channel is 50% loaded, the queueing time will be equal to the download time. Under these conditions, how long does it take to download a 5-KB Web page? How long does it take to download the page over a 1-Mbps ADSL line? Over a 56-kbps modem?Ten signals, each requiring 4000 Hz, are multiplexed on to a single channel using FDM. How much minimum bandwidth is required for the multiplexed channel? Assume that the guard bands are 400 Hz wide.Why has the PCM sampling time been set at 125 µsec?What is the percent overhead on a T1 carrier; that is, what percent of the 1.544 Mbps are not delivered to the end user?Compare the maximum data rate of a noiseless 4-kHz channel using(a) Analog encoding (e.g., QPSK) with 2 bits per sample.(b) The T1 PCM system.If a T1 carrier system slips and loses track of where it is, it tries to resynchronize using the 1st bit in each frame. How many frames will have to be inspected on average to resynchronize with a probability of 0.001 of being wrong?What is the difference, if any, between the demodulator part of a modem and the coder part of a codec? (After all, both convert analog signals to digital ones.)A signal is transmitted digitally over a 4-kHz noiseless channel with one sample every 125 µsec. How many bits per second are actually sent for each of these encoding methods?(a) CCITT 2.048 Mbps standard.(b) DPCM with a 4-bit relative signal value.(c) Delta modulation.A pure sine wave of amplitude A is encoded using delta modulation, with x samples/sec. An output of +1 corresponds to a signal change of +A/8, and an output signal of -1 corresponds to a signal change of -A/8. What is the highest frequency that can be tracked without cumulative error?SONET clocks have a drift rate of about 1 part in 109. How long does it take for the drift to equal the width of 1 bit? What are the implications of this calculation?In Fig. 2-37, the user data rate for OC-3 is stated to be 148.608 Mbps. Show how this number can be derived from the SONET OC-3 parameters.To accommodate lower data rates than STS-1, SONET has a system of virtual tributaries (VT). A VT is a partial payload that can be inserted into an STS-1 frame and combined with other partial payloads to fill the data frame. VT1.5 uses 3 columns, VT2 uses 4 columns, VT3 uses 6 columns, and VT6 uses 12 columns of an STS-1 frame. Which VT can accommodate(a) A DS-1 service (1.544 Mbps)?(b) European CEPT-1 service (2.048 Mbps)?(c) A DS-2 service (6.312 Mbps)?What is the essential difference between message switching and packet switching?What is the available user bandwidth in an OC-12c connection?Three packet-switching networks each contain n nodes. The first network has a star topology with a central switch, the second is a (bidirectional) ring, and the third is fully interconnected, with a wire from every node to every other node. What are the best-, average-, and-worst case transmission paths in hops?Compare the delay in sending an x-bit message over a k-hop path in a circuit-switched network and in a (lightly loaded) packet-switched network. The circuit setup time is s sec, the propagation delay is d sec per hop, the packet size is p bits, and the data rate is b bps. Under what conditions does the packet network have a lower delay?Suppose that x bits of user data are to be transmitted over a k-hop path in a packet-switched network as a series of packets, each containing p data bits and h header bits, with x p + h. The bit rate of the lines is b bps and the propagation delay is negligible. What value of p minimizes the total delay?In a typical mobile phone system with hexagonal cells, it is forbidden to reuse a frequency band in an adjacent cell. If 840 frequencies are available, how many can be used in a given cell?The actual layout of cells is seldom as regular that as shown in Fig. 2-41. Even the shapes of individual cells are typically irregular. Give a possible reason why this might be.Make a rough estimate of the number of PCS microcells 100 m in diameter it would take to cover San Francisco (120 square km).Sometimes when a mobile user crosses the boundary from one cell to another, the current call is abruptly terminated, even though all transmitters and receivers are functioning perfectly. Why?D-AMPS has appreciably worse speech quality than GSM. Is this due to the requirement that D-AMPS be backward compatible with AMPS, whereas GSM had no such constraint? If not, what is the cause?Calculate the maximum number of users that D-AMPS can support simultaneously within a single cell. Do the same calculation for GSM. Explain the difference.Suppose that A, B, and C are simultaneously transmitting 0 bits, using a CDMA system with the chip sequences of Fig. 2-45(b). What is the resulting chip sequence?In the discussion about orthogonality of CDMA chip sequences, it was stated that if S•T = 0 then is also 0. Prove this.Consider a different way of looking at the orthogonality property of CDMA chip sequences. Each bit in a pair of sequences can match or not match. Express the orthogonality property in terms of matches and mismatches.A CDMA receiver gets the following chips: (-1 +1 -3 +1 -1 -3 +1 +1). Assuming the chip sequences defined in Fig. 2-45(b), which stations transmitted, and which bits did each one send?At the low end, the telephone system is star shaped, with all the local loops in a neighborhood converging on an end office. In contrast, cable television consists of a single long cable snaking its way past all the houses in the same neighborhood. Suppose that a future TV cable were 10 Gbps fiber instead of copper. Could it be used to simulate the telephone model of everybody having their own private line to the end office? If so, how many one-telephone houses could be hooked up to a single fiber?A cable TV system has 100 commercial channels, all of them alternating programs with advertising. Is this more like TDM or like FDM?A cable company decides to provide Internet access over cable in a neighborhood consisting of 5000 houses. The company uses a coaxial cable and spectrum allocation allowing 100 Mbps downstream bandwidth per cable. To attract customers, the company decides to guarantee at least 2 Mbps downstream bandwidth to each house at any time. Describe what the cable company needs to do to provide this guarantee.Using the spectral allocation shown in Fig. 2-48 and the information given in the text, how many Mbps does a cable system allocate to upstream and how many to downstream?How fast can a cable user receive data if the network is otherwise idle?Multiplexing STS-1 multiple data streams, called tributaries, plays an important role in SONET. A 3:1 multiplexer multiplexes three input STS-1 tributaries onto one output STS-3 stream. This multiplexing is done byte for byte, that is, the first three output bytes are the first bytes of tributaries 1, 2, and 3, respectively. The next three output bytes are the second bytes of tributaries 1, 2, and 3, respectively, and so on. Write a program that simulates this 3:1 multiplexer. Your program should consist of five processes. The main process creates four processes, one each for the three STS-1 tributaries and one for the multiplexer. Each tributary process reads in an STS-1 frame from an input file as a sequence of 810 bytes. They send their frames (byte by byte) to the multiplexer process. The multiplexer process receives these bytes and outputs an STS-3 frame (byte by byte) by writing it on standard output. Use pipes for communication among processes.