Chapter 2 Assigned Problems

Answer Key

 

4. If a binary signal is sent over a 3-kHz channel whose signal-to-noise ratio is 20dB, what is the maximum achievable data rate?

A signal-to-noise ratio of 20 dB means S/N = 100. Since log2 of 101 is about 6.658, the Shannon limit is about 19.975 kbps. The Nyquist limit is 6 kbps. The bottleneck is therefore the Nyquist limit, giving a maximum channel capacity of 6 kbps.

6. What is the difference between a passive star and an active repeater in a fiber network?

A passive star has no electronics. The light from one fiber illuminates a number of others. An active repeater converts the optical signal to an electrical one for further processing.

7. How much bandwidth is there in 0.1 micron of spectrum at a wavelength of 1 micron?

Use delta f = c Δλ /λ² with Δλ = 10^-7 meters and λ = 10^-6 meters. This gives a bandwidth (Δf) of 30,000 GHz.

9. Is the Nyquist theorem true for optical fiber or only for copper wire?

The Nyquist theorem is a property of mathematics and has nothing to do with technology. It says that if you have a function whose Fourier spectrum does not contain any sines or cosines above f, then by sampling the function at a frequency of 2f you capture all the information there is. Thus, the Nyquist theorem is true for all media.

11. Radio antennas oftern work best when the diameter of the antenna is equal to the wavelength of the radio wave. Reasonable antennas range from 1cm to 5 meters in diameter. What frequency range does this cover?

Start with λf = c. We know that c is 3 × 10⁸ m/s. For λ = 1 cm, we get 30 GHz. For λ = 5 m, we get 60 MHz. Thus, the band covered is 60 MHz to 30 GHz.

14. 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?

With 66/6 or 11 satellites per necklace, every 90 minutes 11 satellites pass overhead. This means there is a transit every 491 seconds. Thus, there will be a handoff about every 8 minutes and 11 seconds.

20. Is an oil pipelinea simplex system, a half-duplex system, a full-duplex system, or none of the above?

Like a single railroad track, it is half duplex. Oil can flow in either direction, but not both ways at once.

29. Why has the PCM sampling time been set at 125 micro seconds?

A sampling time of 125 µsec corresponds to 8000 samples per second. According to the Nyquist theorem, this is the sampling frequency needed to capture all the information in a 4 kHz channel, such as a telephone channel. (Actually the nominal bandwidth is somewhat less, but the cutoff is not sharp.)

30. 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?

The end users get 7 × 24 = 168 of the 193 bits in a frame. The overhead is therefore 25/193 = 13%.

32. If a T1 carrier system slips and loses track of where it is, it tries to resynch. using the 1st bit in each frame. How many frames will have to be inspected on average to resynch with a probability of 0.001 of being wrong?

Ten frames. The probability of some random pattern being 0101010101 (on a digital channel) is 1/1024.

39. What is the essential difference between message switching and packet switching?

Message switching sends data units that can be arbitrarily long. Packet switching has a maximum packet size. Any message longer than that is split up into multiple packets.

56. 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 2Mbps downstream bandwidth to each house at any time. Describe what the cable company needs to do to provide this guarantee.

A 2-Mbps downstream bandwidth guarantee to each house implies at most 50 houses per coaxial cable. Thus, the cable company will need to split up the existing cable into 100 coaxial cables and connect each of them directly to a fiber node.