CHAPTER 2

1 Differentiate between an analog and a digital electromagnetic signal.

A continuous or analog signal is one in which the signal intensity varies in a smooth fashion over time
while a discrete or digital signal is one in which the signal intensity maintains one of a finite number of
constant levels for some period of time and then changes to another constant level.

2.2 What are three important characteristics of a periodic signal?

Amplitude, frequency, and phase are three important characteristics of a periodic signal

2.3 How many radians are there in a complete circle of 360 degrees?

2π radians.

2.4 What is the relationship between the wavelength and frequency of a sine wave?

The relationship is λf = v, where λ is the wavelength, f is the frequency, and v is the speed at which the
signal is traveling

2.5 What is the relationship between a signal's spectrum and its bandwidth?

The spectrum of a signal consists of the frequencies it contains; the bandwidth of a signal is the width of
the spectrum

2.6 What is attenuation?

Attenuation is the gradual weakening of a signal over distance.

2.7 Define channel capacity.

The rate at which data can be transmitted over a given communication path, or channel, under given
conditions, is referred to as the channel capacity

2.8 What key factors affect channel capacity?

Bandwidth, noise, and error rate affect channel capacity.

2.9 Differentiate between guided media and unguided media.

With guided media, the electromagnetic waves are guided along an enclosed physical path, whereas
unguided media provide a means for transmitting electromagnetic waves through space, air, or water,
but do not guide them.

2.10 What are some major advantages and disadvantages of microwave transmission?

Point-to-point microwave transmission has a high data rate and less attenuation than twisted pair or
coaxial cable. It is affected by rainfall, however, especially above 10 GHz. It is also requires line of sight
and is subject to interference from other microwave transmission, which can be intense in some places.

2.11 What is direct broadcast satellite (DBS)?Direct broadcast transmission is a technique in which
satellite video signals are transmitted directly to the home for continuous operation
2.12 Why must a satellite have distinct uplink and downlink frequencies?

A satellite must use different uplink and downlink frequencies for continuous operation in order to avoid
interference

2.13 Indicate some significant differences between broadcast radio and microwave.

Broadcast is omnidirectional, does not require dish shaped antennas, and the antennas do not have to
be rigidly mounted in precise alignment.

2.14 Why is multiplexing so cost-effective?

Multiplexing is cost-effective because the higher the data rate, the more cost-effective the transmission
facility.

2.15 How is interference avoided by using frequency division multiplexing?

Interference is avoided under frequency division multiplexing by the use of guard bands, which are
unused portions of the frequency spectrum between sub channels.

2.16 Explain how synchronous time division multiplexing (TDM) works.

A synchronous time division multiplexer interleaves bits from each signal and takes turns transmitting
bits from each of the signals in a round-robin fashion.

NUMERICALS

2.1 A signal has a fundamental frequency of 1000 Hz. What is its period?

Period = 1/1000 = 0.001 s = 1 ms.

2.2 Express the following in the simplest form you can: a. sin(21Tft - 17) + sin(21Tft + 17) b. sin 217ft +
sin(21Tft - 17)

a. sin (2πft – π) + sin (2πft + π) = 2 sin (2πft + π) or 2 sin (2πft – π) or - 2 sin (2πft) b. sin (2πft) + sin (2πft
– π) = 0.

2.3 Sound may be modeled as sinusoidal functions. Compare the wavelength and relative frequency
of musical notes. Use 330 m/s as the speed of sound and the following frequencies for the musical
scale. Note C D E F G A B C Frequency 264 297 330 352 396 440 495 528

N C D E F G A B C
F 264 297 330 352 3.96 440 495 528
D 33 33 22 44 44 55 33
W 1.25 1.11 1 0.93 0.83 0.75 0.67 0.63
N = note; F = frequency (Hz); D = frequency difference; W = wavelength (m)

2.4 If the solid curve in Figure 2.14 represents sin(21Tt), what does the dotted curve represent? That
is, the dotted curve can be written in the form A sin(21Tft + ¢); what are A,f, and ¢?

2 sin(4πt + π); A = 2, f = 2, φ = π
2..5 Decompose the signal (1 + 0.1 cos 5t)cos lOOt into a linear combination ofsinusoidal function,
and find the amplitude, frequency, and phase of each component. Hint: Use the identity for cos a cos
b.

5 (1 + 0.1 cos 5t) cos 100t = cos 100t + 0.1 cos 5t cos 100t. From the trigonometric identity cos a cos b =
(1/2)(cos(a + b) + cos(a – b)), this equation can be rewritten as the linear combination of three sinusoids:
cos 100t + 0.05 cos 105t + 0.05 cos 95t

2.6 Find the period of the function f(t) = (10 cos tf

We have cos2x = cos x cos x = (1/2)(cos(2x) + cos(0)) = (1/2)(cos(2x) + 1). Then: f(t) = (10 cos t)2 = 100
cos2 t = 50 + 50 cos(2t). The period of cos(2t) is π and therefore the period of f(t) is π.

2.7 Consider two periodic functions A(t) and h(t), with periods 1i and 72, respectively. Is it always the
case that the function f(t) = A(t) + h(t) is periodic? Ifso, demonstrate this fact. If not, under what
conditions is f(t) periodic?

If f1(t) is periodic with period X, then f1(t) = f1(t +X) = f1(t +nX) where n is an integer and X is the
smallest value such that f1(t) = f1(t +X). Similarly, f2(t) = f2(t +Y) = f2(t + mY). We have f(t) = f1(t) + f2(t).
If f(t) is periodic with period Z, then f(t) = f(t + Z). Therefore f1(t) + f2(t) = f1(t + Z) + f2(t + Z). This last
equation is satisfied if f1(t) = f1(t + Z) and f2(t) = f2(t + Z). This leads to the condition Z = nX = mY for
some integers n and m. We can rewrite this last as (n/m) = (Y/X). We can therefore conclude that if the
ratio (Y/X) is a rational number, then f(t) is periodic.

2.8 Figure 2.5 shows the effect of eliminating higher-harmonic components of a square wave and
retaining only a few lower harmonic components. What would the signal look like in the opposite
case; that is, retaining all higher harmonics and eliminating a few lower harmonics?

The signal would be a low-amplitude, rapidly changing waveform.

2.9 What is the channel capacity for a teleprinter channel with a 300-Hz bandwidth and a signal-to-
noise ratio of 3 dB?

Using Shannon's equation: C = B log2 (1 + SNR) We have W = 300 Hz (SNR)dB = 3 Therefore, SNR = 100.3
-7- C = 300 log2 (1 + 100.3) = 300 log2 (2.995) = 474 bps

2.10 A digital signaling system is required to operate at 9600 bps. a. If a signal element encodes a 4-bit
word, what is the minimum required bandwidth of the channel? b. Repeat part (a) for the case of 8-bit
words.

Using Nyquist's equation: C = 2B log2M We have C = 9600 bps a. log2M = 4, because a signal element
encodes a 4-bit word Therefore, C = 9600 = 2B × 4, and B = 1200 Hz b. 9600 = 2B × 8, and B = 600 Hz
2.11 Study the works of Shannon and Nyquist on channel capacity. Each places an upper limit on the
bit rate of a channel based on two different approaches. How are the two related?

Nyquist analyzed the theoretical capacity of a noiseless channel; therefore, in that case, the signaling
rate is limited solely by channel bandwidth. Shannon addressed the question of what signaling rate can
be achieved over a channel with a given bandwidth, a given signal power, and in the presence of noise.

2.12 Given the narrow (usable) audio bandwidth of a telephone transmission facility, a nominal SNR
of 56dB (400,000), and a distortion level of a. What is the theoretical maximum channel capacity
(Kbps) of traditional telephone lines? b. What is the actual maximum channel capacity?

a. Using Shannon’s formula: C = 3000 log2 (1+400000) = 56 Kbps b. Due to the fact there is a distortion
level (as well as other potentially detrimental impacts to the rated capacity, the actual maximum will be
somewhat degraded from the theoretical maximum. A discussion of these relevant impacts should be
included and a qualitative value discussed.

2.13 Given a channel with an intended capacity of 20 Mbps, the bandwidth of the channel is 3 MHz.
What signal-to-noise ratio is required to achieve this capacity?

C = B log2 (1 + SNR) 20 × 106 = 3 × 106 × log2(1 + SNR) log2(1 + SNR) = 6.67 1 + SNR = 102 SNR = 101

2.14 Show that doubling the transmission frequency or doubling the distance between transmitting
antenna and receiving antenna attenuates the power received by 6 dB.

From Equation 2.1, we have LdB = 20 log (4πd/λ) = 20 log (4πdf/v), where λf = v (see Question 2.4). If we
double either d or f, we add a term 20 log(2), which is approximately 6 dB.

2.15 Fill in the missing elements in the following table of approximate power ratios for various dB
levels. Decibels 1 2 3 4 5 6 7 8 9 10 Losses 0.5 0.1 Gains 2 10

Decibels 1 2 3 4 5 6 7 8 9 10
Losses 0.8 0.63 0.5 0.4 0.32 0.25 0.2 0.16 0.125 0.1
Gains 1.25 1.6 2 2.5 3.2 4.0 5.0 6.3 8.0 10

2.16 If an amplifier has a 30 dB voltage gain, what voltage ratio does the gain represent?

For a voltage ratio, we have NdB = 30 = 20 log(V2/V1) V2/V1 = 1030/20 = 101.5 = 31.6

2.17 An amplifier has an output of 20 W. What is its output in dBW?

Power (dBW) = 10 log (Power/1W) = 10 log20 = 13 dBW

 CHAPTER 5

5.1 What two functions are performed by an antenna?

The two functions of an antenna are: (1) For transmission of a signal, radiofrequency electrical energy
from the transmitter is converted into electromagnetic energy by the antenna and radiated into the
surrounding environment (atmosphere, space, water); (2) for reception of a signal, electromagnetic
energy impinging on the antenna is converted into radio-frequency electrical energy and fed into the
receiver.

5.2 What is an isotropic antenna?

An isotropic antenna is a point in space that radiates power in all directions equally.

5.3 What information is available from a radiation pattern?

A radiation pattern is a graphical representation of the radiation properties of an antenna as a function

of space coordinates.

5.4 What is the advantage of a parabolic reflective antenna?

A parabolic antenna creates, in theory, a parallel beam without dispersion. In practice, there will be
some beam spread. Nevertheless, it produces a highly focused, directional beam.

5.5 What factors determine antenna gain?

Effective area and wavelength.

5.6 What is the primary cause of signal loss in satellite communications?

6 Free space loss.

5.7 Name and briefly define four types of noise.

Thermal noise is due to thermal agitation of electrons. Intermodulation noise produces signals at a
frequency that is the sum or difference of the two original frequencies or multiples of those frequencies.
Crosstalk is the unwanted coupling between signal paths. Impulse noise is noncontinuous, consisting of
irregular pulses or noise spikes of short duration and of relatively high amplitude.

5.8 What is refraction?

Refraction is the bending of a radio beam caused by changes in the speed of propagation at a point of
change in the medium.

5.9 What is fading?

The term fading refers to the time variation of received signal power caused by changes in the
transmission medium or path(s).

5.10 What is the difference between diffraction and scattering?

Diffraction occurs at the edge of an impenetrable body that is large compared to the wavelength of the
radio wave. The edge in effect become a source and waves radiate in different directions from the edge,
allowing a beam to bend around an obstacle. If the size of an obstacle is on the order of the wavelength
of the signal or less, scattering occurs. An incoming signal is scattered into several weaker outgoing
signals in unpredictable directions.

5.11 What is the difference between fast and slow fading?

Fast fading refers to changes in signal strength between a transmitter and receiver as the distance
between the two changes by a small distance of about one-half a wavelength. Slow fading refers to
changes in signal strength between a transmitter and receiver as the distance between the two changes
by a larger distance, well in excess of a wavelength.

5.12 What is the difference between flat and selective fading?

Flat fading, or nonselective fading, is that type of fading in which all frequency components of the
received signal fluctuate in the same proportions simultaneously. Selective fading affects unequally the
different spectral components of a radio signal.

5.13 Name and briefly define three diversity techniques.

3 Space diversity involves the physical transmission path and typical refers to the use of multiple
transmitting or receiving antennas. With frequency diversity, the signal is spread out over a larger
frequency bandwidth or carried on multiple frequency carriers. Time diversity techniques aim to spread
the data out over time so that a noise burst affects fewer bits.

PROBLEMS

5.1 For radio transmission in free space, signal power is reduced in proportion to the square of the
distance from the source, whereas in wire transmission, the attenuation is a fixed number of dB per
kilometer. The following table is used to show the dB reduction relative to some reference for free
space radio and uniform wire. Fill in the missing numbers to complete the table. Distance (km) Radio
(dB) Wire (dB) 1 -6 -3 2 4 8 16

Distance (km) Radio (dB Wire (dB)

1 -6 -3
2 -12 -6
4 -18 -12
8 -24 -24
16 -30 -48

5.2 Find the optimum wavelength and frequency for a half-wave dipole of length 10 m.

The length of a half-wave dipole is one-half the wavelength of the signal that can be transmitted most
efficiently. Therefore, the optimum wavelength in this case is λ = 20 m. The optimum free space
frequency is f = c/λ = (3 × 108)/20 = 15 MHz.
5.3 It turns out that the depth in the ocean to which airborne electromagnetic signals can be detected
grows with the wavelength. Therefore, the military got the idea of using very long wavelengths
corresponding to about 30 Hz to communicate with submarines throughout the world. If we want to
have an antenna that is about one-half wavelength long, how long would that be?

We have λf = c; in this case λ × 30 = 3 × 108 m/sec, which yields a wavelength of 10,000 km. Half of that
is 5,000 km which is comparable to the east-to-west dimension of the continental U.S. While an antenna
this size is impractical, the U.S. Defense Department has considered using large parts of Wisconsin and
Michigan to make an antenna many kilometers in diameter.

5.4 The audio power of the human voice is concentrated at about 300 Hz. Antennas of the
appropriate size for this frequency are impracticably large, so that to send voice by radio the voice
signal must be used to modulate a higher (carrier) frequency for which the natural antenna size is
smaller. a. What is the length of an antenna one-half wavelength long for sending radio at 300 Hz? b.
An alternative is to use a modulation scheme, as described in Chapter 6, for transmitting the voice
signal by modulating a carrier frequency, so that the bandwidth of the signal is a narrow band
centered on the carrier frequency. Suppose we would like a half-wave antenna to have a length of 1
m. What carrier frequency would we use?

a. Using λf = c, we have λ = (3 × 108 m/sec)/(300 Hz) = 1,000 km, so that λ/2 = 500 km. b. The carrier
frequency corresponding to λ/2 = 1 m is given by: f = c/λ = (3 × 108 m/sec)/(2 m) = 150 MHz.

5.5 Stories abound of people who receive radio signals in fillings in their teeth. Suppose you have one
filling that is 2.5 mm (0.0025 m) long that acts as a radio antenna. That is, it is equal in length to one-
half the wavelength. What frequency do you receive?

λ = 2 × 2.5 × 10–3 m = 5 × 10–3 m f = c/λ = (3 × 108 m/sec)/( 5 × 10-3 m) = 6 × 1010 Hz = 60 GHz

5.6 Section 5.1 states that if a source of electromagnetic energy is placed at the focus of the
paraboloid, and if the paraboloid is a reflecting surface, then the wave will bounce back in lines
parallel to the axis of the paraboloid.To demonstrate this, consider the parabola y 2 = 2px shown in
Figure 5.17. Let P(XI' yd be a point on the parabola and PF be the line from P to the focus. Construct
the line L through P parallel to the x-axis and the line M tangent to the parabola at P. The angle
between Land M is {3, and the angle between PF and Mis a. The angle a is the angle at which a ray
from F strikes the parabola at P. Because the angle of incidence equals the angle of reflection, the ray
reflected from P must be at an angle a to M.Thus, if we can show that a = {3, we have demonstrated
that rays reflected from the parabola starting at F will be parallel to the x-axis. a. First show that tan
{3 = (PIYI)' Hint: Recall from trigonometry that the slope of a line is equal to the tangent of the angle
the line makes with the positive x direction. Also recall that the slope of the line tangent to a curve at
a given point is equal to the derivative of the curve at that point. b. Now show that tan a = (PIYI),
which demonstrates that a = {3. Hint: Recall from trigonometry that the formula for the tangent of the
difference between two angles al and a2 is tan(a2 - al) = (tan a2 - tan al)/(l + tan a2 X tan al)' M 5.6 /
KEY TERMS, REVIEW QL'ESTIONS. AND PROBLEMS 125 F(pl2,O) x ~_---1. L o y :Figurc 5.17 Parabolic
Reflection

a. First, take the derivative of both sides of the equation y2 = 2px: dy dx y 2 = dy dx (2 px); 2y dy dx = 2 p;
dy dx = p y Therefore tan β = (p/y1). b. The slope of PF is (y1 – 0)/(x1 – (p/2)). Therefore: -19- tanα = y1
x1 − p 2 − p y1 1 + y1 x1 − p 2 p y1 = y1 2 − px1 + 1 2 p2 x1y1 − 1 2 py1 + py1 Because y1 2 = 2 px1, this
simplifies to tan α = (p/y1).

5.7 For each of the antenna types listed in Table 5.2, what is the effective area and gain at a
wavelength of 30 cm? Repeat for a wavelen¥th of 3 mm. Assume that the actual area for the hom and
parabolic antennas is 7Tm .

Antenna λ = 30 cm λ = 3 mm
Effective area Gain Effective area Gain
(m2) (m2)
Isotropic 0.007 1 7.2 × 10^7 1
Infinitesimal dipole or loop 0.011 1.5 1.1 × 10^6 1.5
Half-wave dipole 0.012 1.64 1.2 × 10^6 1.64
Horn 2.54 349 2.54 3.5 × 10^6
Parabolic 1.76 244 1.76 2.4 × 10^6
Turnstile 0.008 1.15 8.2 × 10^7 1.15

5.8 It is often more convenient to express distance in km rather than m and frequency in MHz rather
than Hz. Rewrite Equation (5.2) using these dimensions.

LdB = 20 log(f MHz) + 120 +20 log (dkm) + 60 – 147.56 = 20 log(f MHz) +20 log (dkm) + 32.44

5.9 Assume that two antennas are half-wave dipoles and each has a directive gain of 3 dB. If the
transmitted power is 1 Wand the two antennas are separated by a distance of 10 km, what is the
received power? Assume that the antennas are aligned so that the directive gain numbers are correct
and that the frequency used is 100 MHz.

We have Pr = [(Pt ) (Gt ) (Gr ) (c)2]/(4πfd)2 = [(1) (2) (2) (3 × 108)2]/[(16) (π)2 (3 × 108)2 (104) 2] = 0.76 ×
10–9 W Source: [THUR00]

5.10 Suppose a transmitter produces 50 W of power. a. Express the transmit power in units of dBm
and dBW. h. If the transmitter's power is applied to a unity gain antenna with a 900-MHz carrier
frequency, what is the received power in dBm at a free space distance of 100 m? c. Repeat (b) for a
distance of 10 km. d. Repeat (c) but assume a receiver antenna gain of 2.

a. From Appendix 2A, PowerdBW = 10 log (PowerW) = 10 log (50) = 17 dBW PowerdBm = 10 log
(PowermW) = 10 log (50,000) = 47 dBm b. Using Equation (5.2), LdB = 20 log(900 × 106) +20 log (100) –
147.56 = 120 + 59.08 +40 – 147.56 = 71.52 Therefore, received power in dBm = 47 – 71.52 = –24.52 dBm
c LdB = 120 + 59.08 +80 – 147.56 =111.52; Pr,dBm = 47 – 111.52 = –64.52 dBm d The antenna gain
results in an increase of 3 dB, so that Pr,dBm = –61.52 dBm Source: [RAPP02]

5.11 A microwave transmitter has an output of 0.1 W at 2 GHz. Assume that this transmitter is used in
a microwave communication system where the transmitting and receiving antennas are parabolas,
each 1.2 m in diameter. a. What is the gain of each antenna in decibels? b. Taking into account
antenna gain, what is the effective radiated power of the transmitted signal? c. If the receiving
antenna is located 24 km from the transmitting antenna over a free space path,find the available
signal power out of the receiving antenna in dBm units.
a. From Table 5.2, G = 7A/λ2 = 7Af2/c2 = (7×π×(0.6)2×(2×109) 2]/(3×108) 2 = 351.85 GdB = 25.46 dB b.
0.1 W x 351.85 = 35.185 W c. Use LdB = 20 log (4π) + 20 log (d) + 20 log (f) – 20 log (c) – 10 log(Gr ) – 10
log (Gt ) LdB = 21.98 + 87.6 + 186.02 – 169.54 – 25.46 – 25.46 = 75.14 dB The transmitter power, in dBm
is 10 log (100) = 20. The available received signal power is 20 – 75.14 = –55.14 dBm

5.12 Show that doubling the transmission frequency or doubling the distance between transmitting
antenna and receiving antenna attenuates the power received by 6 dB.

From Equation 2.2, the ratio of transmitted power to received power is Pt /Pr = (4πd/λ)2 If we double
the frequency, we halve λ, or if we double the distance, we double d, so the new ratio for either of these
events is: Pt /Pr2 = (8πd/λ)2 Therefore: 10 log (Pr/Pr2) = 10 log (22) = 6 dB

5.13 Section 5.2 states that with no intervening obstacles, the optical line of sight can be expressed as
d = 3.57Vh, where d is the distance between an antenna and the horizon in kilometers and h is the
antenna height in meters. Using a value for the earth's 126 CHAPTER 5 I ANTENNAS AND
PROPAGATION radius of 6370 km, derive this equation. Hint: Assume that the antenna is
perpendicular to the earth's surface, and note that the line from the top of the antenna to the horizon
forms a tangent to the earth's surface at the horizon. Draw a picture showing the antenna, the line of
sight, and the earth's radius to help visualize the problem.

By the Pythagorean theorem: d2 + r2 = (r + h) 2 Or, d2 = 2rh + h2. The h2 term is negligible with respect
to 2rh, so we use d2 = 2rh. Then, dkm = 2rkm hkm = 2rkmhm /1000 = 2 × 6.37 × hm = 3.57 hm

5.14 Determine the height of an antenna for a TV station that must be able to reach customers up to
80 km away.

For radio line of sight, we use d = 3.57 Kh , with K = 4/3, we have 802 = (3.57)2 × 1.33 × h. Solving for h,
we get h = 378 m.

5.15 What is the thermal noise level of a channel with a bandwidth of 10 kHz carrying 1000 watts of
power operating at 50°C? Compare the noise level to the operating power.

N = –228.6 dBW + 10 log T + 10 log B We have T = 273.15 + 50 = 323.15 K, and B = 10,000 N = –228.6
dBW + 25.09 +40 = –163.51 dBW Converting to watts, NW = 10N/10 = 4 × 10–17 W
 5.16 The square wave of Figure 2.5c, with T = 1 ms, is passed through a low-pass filter that passes
frequencies up to 8 kHz with no attenuation. a. Find the power in the output waveform. b. Assuming
that at the filter input there is a thermal nOlse voltage with No = 0.1 f.LW/Hz, find the output signal to
noise ratio in dB.

a. Output waveform: sin (2πf1t) + 1/3 sin (2π(3f1)t) + 1/5 sin (2π(5f1)t) + 1/7 sin (2π (7f1)t) where f1 =
1/T = 1 kHz Output power = 1/2 (1 + 1/9 + 1/25 + 1/49) = 0.586 watt b. Output noise power = 8 kHz × 0.1
µWatt/Hz = 0.8 mWatt SNR = 0.586/0.0008 = 732.5 (SNR)dB = 28.65

5.17 If the received signal level for a particular digital system is -151 dBW and the receiver system
effective noise temperature is 1500 K, what is Eb/No for a link transmitting 2400 bps?

7 (Eb/N0) = –151 dBW – 10 log 2400 – 10 log 1500 + 228.6 dBW = 12 dBW Source: [FREE98a]

5.18 Suppose a ray of visible light passes from the atmosphere into water at an angle to the
horizontal of 30°. What is the angle of the ray in the water? Note: At standard atmospheric conditions
at the earth's surface, a reasonable value for refractive index is 1.0003. A typical value of refractive
index for water is 4/3.

Let RI = refractive index, α = angle of incidence, β = angle of refraction (sin α)/sin β) = RIair/RIwater =
1.0003/(4/3) = 0.75 sin β = 0.5/0.75 = 0.66; β = 41.8°
 CHAPTER 9

9.1 List three different ways of categorizing communications satellites.

Coverage area: global, regional, or national. The larger the area of coverage, the more satellites must be
involved in a single networked system. Service type: fixed service satellite (FSS), broadcast service
satellite (BSS), and mobile service satellite (MSS). This chapter is concerned with FSS and BSS types.
General usage: commercial, military, amateur, experimental.

9.2 What are some key differences between satellite-based and terrestrial wireless communications?

(1) The area of coverage of a satellite system far exceeds that of a terrestrial system. In the case of a
geostationary satellite, a single antenna is visible to about one-fourth of the earth's surface. (2)
Spacecraft power and allocated bandwidth are limited resources that call for careful tradeoffs in earth
station/satellite design parameters. (3) Conditions between communicating satellites are more time
invariant that those between satellite and earth station or between two terrestrial wireless antennas.
Thus, satellite-to-satellite communication links can be designed with great precision. (4) Transmission
cost is independent of distance, within the satellite's area of coverage. (5) Broadcast, multicast, and
point-to-point applications are readily accommodated. (6) Very high bandwidths or data rates are
available to the user. (7) Although satellite links are subject to short-term outages or degradations, the
quality of transmission is normally extremely high. (8) For a geostationary satellite, there is an earth-
satellite-earth propagation delay of about one-fourth of a second. (9) A transmitting earth station can in
many cases receive its own transmission.

9.3 List three different ways of classifying satellite orbits.

(1) The orbit may be circular, with the center of the circle at the center of the earth, or elliptical, with
the earth's center at one of the two foci of the ellipse. (2) A satellite may orbit around the earth in
different planes. An equatorial orbit is directly above the earth's equator. A polar orbit passes over both
poles. Other orbits are referred to as inclined orbits. (3) The altitude of communications satellites is
classified as geostationary orbit (GEO), medium earth orbit (MEO), and low earth orbit (LEO).

9.4 Explain what GEO, LEO, and MEO satellites are (including what the acronyms stand for). Compare
the three types with respect to factors such as size and shape of orbits, signal power, frequency reuse,
propagation delay, number of satellites for global coverage, and handoff frequency.

4LEO, GEO and HEO stand for low earth orbit, geostationary (or geosynchronous) orbit, and highly
elliptical orbit, respectively. The traditional GEO satellite is in a circular orbit in an equatorial plane such
that the satellite rotates about the earth at the same angular velocity that the earth spins on its axis. To
accomplish this the satellite must be approximately 35,838 km above the earth’s surface at the equator.
LEO satellites are satellites with much lower orbits, on the order of 700 to 1,400 km high. Finally, HEO
satellites are characterized by an orbit that is an ellipse with one axis very substantially larger than the
other. The height of the orbit can vary; it is the shape of the orbit that characterizes this type of satellite.
Because of the high altitude of the GEO satellite the signal strength is relatively weak compared to LEOs.
Frequency reuse is more difficult because the antenna beam (all other things being equal) covers a
much greater area from a GEO than from a LEO. The propagation delay for a GEO satellite is about 1/4th
of second; that of a LEO satellite is much less. Because the GEO satellite must be over the CHAPTER 9
SATELLITE COMMUNICATIONS -45- equator, the coverage near the north and south poles is inadequate.
For these regions, better communication can be achieved by LEO or HEO satellites. HEO satellites have
the additional advantage that they spend most of their time at the “high” part of their orbit so that you
get the most coverage for the longest time for this type of satellite. On the other hand, tracking and
handoff is not necessary for GEO satellites because they appear stationary relative to the earth. LEO
satellites, since they are so low travel very much faster, and cover less area than GEO so that tracking is
more difficulty and passing off is frequent. HEOs require tracking and handoffs, as well. However, if the
HEOs have high orbits the handoff frequency can be much less and the tracking easier than for LEOs.

9.5 Under what circumstances would you use GEO, LEO, and MEO satellites, respectively?

You would use GEOs when the earth stations are not near the poles, when there is a premium on not
having to steer the earth station antennas, and when broad earth coverage is important, for television
broadcasting for instance. HEOs are primarily of use when coverage of areas near one of the poles is
essential, such as the use of the Molniya satellites to cover the northern parts of the former Soviet
Union. LEOs are useful for point-to-point communication, and for extensive frequency reuse. Since LEOs
have much less propagation delay they are useful for interactive data services. They also can cover polar
regions. Finally, while you need many more LEOs for broad coverage, each satellite is much less
expensive than a GEO.

9.6 What are three key factors related to satellite communications performance?

(1) Distance between earth station antenna and satellite antenna. (2) In the case of the downlink,
terrestrial distance between earth station antenna and the "aim point" of the satellite. (1) Atmospheric
attenuation.

9.7 What are the primary causes of atmospheric attenuation for satellite communications?

Oxygen and water.

9.8 What are three factors that limit the number of subchannels that can be provided within a
satellite channel via FDMA?

Thermal noise, intermodulation noise, and crosstalk

Problems

9.1 Using Kepler's laws of planetary motion, we can derive the following expression for a circular
orbit: 9.5/ KEY TERMS, REVIEW QlJESTIONS. AND PROBLEMS 263 where T = orbital period a = orbital
radius in km = distance from the center ofthe earth to the orbit jJ- = Kepler's constant = 3.986004418 X
105 km3 /s2 The earth rotates once per sidereal day of 23 h 56 min 4.09 s. a. Determine the orbital
radius of a GEO satellite. b. Assuming an earth radius of 6370 km, what is the orbit height h (Figure
9.1) of a GEO satellite. Note: Your answer should differ slightly from the figure used in the chapter.
Different sources in the literature give slightly different values.

a. Rearranging the equation, we have a3 = T2µ/(4π2). One sidereal day is T = 86,164.1 s a3 = (86,164.1)2
× (3.986004418 × 105)/(4π2) = 7.496020251 × 1013 km3 a = 42,164 km b. h = 35,794 km

9.2 The Space Shuttle is an example of a LEO satellite. Sometimes, it orbits at an altitude of 250 km. a.
Using a mean earth radius of 6378.14 km, calculate the period of the shuttle orbit. b. Determine the
linear velocity of the shuttle along this orbit.

a. a = 6378.14 + 250 = 6628.14 km T2 = (4π2a3)/µ = (4π2) × (6628.14)3/(3.986004418 × 105) =

2.88401145 × 107 s2 T = 5370.3 s = 89 min 30.3 s b. The linear velocity is the circumference divided by
the period (2πa)/T = (41645.83)/(5370.3) = 7.775 km/s

9.3 You are communicating between two satellites.The transmission obeys the free space law. The
signal is too weak. Your vendor offers you two options. The vendor can use a higher frequency that is
twice the current frequency or can double the effective area of both of the antennas. Which will offer
you more received power or will both offer the same improvement, all other factors remaining equal?
How much improvement in the received power do you obtain from the best option?

The received signal is, essentially, the same. The received power will increase by a factor of 4

9.4 A satellite at a distance of 40,000 km from a point on the earth's surface radiates a power of 10 W
from an antenna with a gain of 17 dB in the direction of the observer. The satellite operates at a
frequency of 11 GHz. The receiving antenna has a gain of 52.3 dB. Find the received power.

received_power = transmitted_power + transmitted_gain + received_gain – path_loss From Equation

(2.2): path_loss = 20 log (4πd/λ) = 20 log [4πd/(c/f)] -46- = 20 log [(4π × 4 × 107)/(2.727 x 10–2) = –205.3
dB received_power = 10 + 17 + 52.3 – 205.3 = –126 dBW

9.5 For the transponder scheme of Figure 9.10, what is the percentage bandwidth used for the
guardbands?

The total bandwidth is 500 MHz. The channel bandwidth is 12 × 36 = 432 MHz. So the overhead is ((500 -
432)/500) × 100% = 13.6%
 9.6 For the TDMA frame of Figure 9.13, assume the following parameters.The frame length is 2 ms.
Data are encoded using QPSK with a signal element rate of 60.136 Mbaud. All traffic bursts are of
equal length of 16,512 bits.The reference burst has length 576 bits, the preamble is 560 bits, and the
guard interval is 24 bits. Assume that there are two participating reference stations, so that two
reference bursts are required. a. Determine the maximum number of earth stations the system can
serve. h. What is the frame efficiency (fraction of frame carrying user data rather than overhead bits)?

a. The data rate R = 2 × QPSK baud rate = 120.272 Mbps The frame duration T = 0.002 s The number of
frame bits bF = R × T = 1.20272 × 108 × 2 × 10–3 = 240544 bits Overhead calculation: Overhead bits bo =
NRbR + NTbp + (NR + NT)bG where NR = number of participating reference stations = 2 bR = number of
bits in reference burst = 576 NT = number of participating traffic stations bp = number of preamble bits
=560 bG = number guard bits = 24 bo = (2)(576) + 560 NT + (2)(24) + 24 NT = 1200 + 584 NT bF – bo =
240544 – (1200 + 584 NT) = 239344 – 584 NT NT = (bF – bo)/16512 = (239344 – 584 NT)/16512 (16512 +
584) NT = 239344 Therefore, NT = 14 b. (bF – bo)/bF = (239344 – 584 × 14)/240544 = 0.96 Source:
[GLOV98]

9.7 A TDMA network of 5 earth stations shares a single transponder equally. The frame duration is 2
ms, the preamble time per station is 20 jJ-s, and guard bands of 5 jJ-s are used between bursts.
Transmission bursts are QPSK at 30 Mbaud. a. Calculate the number of 64-kbps voice channels that
each TDMA earth station can transmit. b. If the earth stations send data rather than digital speech,
what is the transmission rate of each earth station? c. What is the efficiency of the TDMA system
expressed as Efficiency = (message bits sent)/ (maximum number of possible bits that could have be
sent)?

a. The time, Td, available in each station burst for transmission of data bits is Td = [Tframe – N(tg +
tpre)]/N That is, take the frame time, subtract out all the guard and preamble times, and divide by the
number of stations N. Td = [2000 – 5(5 + 20)]/5 = 375 µs A burst transmission rate of 30 Mbaud is 30
million signal elements per second and QPSK signal elements carry 2 bits, so the transmitted bit rate in
each burst is Rb = 60 Mbps The capacity of each earth station, Cb, is the number of data bits transmitted
in one burst divided by the frame time: Cb = (375 × 60)/2000 = 11.25 Mbps The number of 64-kbps
channels that can be carried is: (11.25 × 106)/(64,000) = 175 b. 11.25 Mbps c. The total available
capacity is 60 Mbps. -47- The total data transmission rate is 5 × 11.25 = 56.25 Mbps Efficiency =
56.25/60 = 0.9375

9.8 Three identical large earth stations access a 36-MHz bandwidth transponder using TDMA, with a
frame length of 1 ms, a preamble tie of 10 jJ-s, and a guard time of 2 jJ-S. There is no reference burst
in the TDMA frame. The signals are transmitted using QPSK, and within the earth stations, the bit
rates of the signals are Station A: R = 15 Mbps Station B: R = 10 Mbps Station C: R = 5 Mbps Calculate
the burst duration and symbol rate for each earth station.

The transponder must carry a total data bit rate of 15 + 10 + 5 = 30 Mbps. Thus, each frame carries 30
Mbps × 0.001 s = 30 kb The three preamble and guard times take up 3 × (10 + 2) = 36 µs in each frame,
leaving 1000 – 36 = 964 µs for transmission of the data. Therefore, the burst bit rate is Rbit = 30 kb/964
µs = 31.12 Mbps For QPSK, the symbol rate is half the bit rate = 15.56 Mbaud
 CHAPTER 10

10.1 What geometric shape is used in cellular system design?

Hexagon

10.2 What is the principle of frequency reuse in the context of a cellular network?

For frequency reuse in a cellular system, the same set of frequencies are used in multiple cells, with
these cells separated from one another by enough distance to avoid interference.

10.3 List five ways of increasing the capacity of a cellular system.

Adding new channels: Typically, when a system is set up in a region, not all of the channels are used, and
growth and expansion can be managed in an orderly fashion by adding new channels. Frequency
borrowing: In the simplest case, frequencies are taken from adjacent cells by congested cells. The
frequencies can also be assigned to cells dynamically. Cell splitting: In practice, the distribution of traffic
and topographic features is not uniform, and this presents opportunities of capacity increase. Cells in
areas of high usage can be split into smaller cells. Cell sectoring: With cell sectoring, a cell is divided into
a number of wedge-shaped sectors, each with its own set of channels, typically 3 or 6 sectors per cell.
Each sector is assigned a separate subset of the cell's channels, and directional antennas at the base
station are used to focus on each sector. Microcells: As cells become smaller, antennas move from the
tops of tall buildings or hills, to the tops of small buildings or the sides of large buildings, and finally to
lamp posts, where they form microcells. Each decrease in cell size is accompanied by a reduction in the
radiated power levels from the base stations and the mobile units. Microcells are useful in city streets in
congested areas, along highways, and inside large public buildings.

10.4 Explain the paging function of a cellular system.

To complete a call to a mobile unit, the base stations in a number of cells will send out a page signal in
an attempt to find the mobile unit and make the connection.

10.5 List and briefly define different performance metrics that may be used to make the handoff
decision.

Cell blocking probability: the probability of a new call being blocked, due to heavy load on the BS traffic
capacity. In this case, the mobile unit is handed off to a neighboring cell based not on signal quality but
on traffic capacity. Call dropping probability: the probability that, due to a handoff, a call is terminated.
Call completion probability: the probability that an admitted call is not dropped before it terminates.
Probability of unsuccessful handoff: the probability that a handoff is executed while the reception
conditions are inadequate. Handoff blocking probability: the probability that a handoff cannot be
successfully completed. Handoff probability: the probability that a handoff occurs before call
termination. Rate of handoff: the number of handoffs per unit time. Interruption duration: the duration
of time during a handoff in which a mobile is not connected to either base station. Handoff delay: the
distance the mobile moves from the point at which the handoff should occur to the point at which it
does occur.
10.6 As a mobile unit in communication with a base station moves, what factors determine the need
for power control and the amount of power adjustment?

As the mobile unit moves away from the transmitter, the received power declines due to normal
attenuation. In addition, the effects of reflection, diffraction, and CHAPTER 10 CELLULAR WIRELESS
NETWORKS -49- scattering can cause rapid changes in received power levels over small distances. This is
because the power level is the sum from signals coming from a number of different paths and the
phases of those paths are random, sometimes adding and sometimes subtracting. As the mobile unit
moves, the contributions along various paths change.

10.7 Explain the difference between open-loop and closed-loop power control.

Open-loop power control depends solely on the mobile unit, with no feedback from the BS, and is used
in some SS systems. Closed loop power control adjusts signal strength in the reverse (mobile to BS)
channel based on some metric of performance in that reverse channel, such as received signal power
level, received signal-to-noise ratio, or received bit error rate.

10.8 What is the difference between traffic intensity and the mean rate of calls in a system?

The mean rate of calls is the number of calls attempted in a unit time, so its dimensions are calls per
second or a similar dimension. Traffic intensity is a normalized version of mean rate of calls, and equals
the average number of calls arriving during the average holding period. Thus, traffic intensity is
dimensionless.

10.9 What are the key differences between first- and second-generation cellular systems?

Digital traffic channels: The most notable difference between the two generations is that first
generation systems are almost purely analog, where as second generation systems are digital. In
particular, the first generation systems are designed to support voice channels using FM; digital traffic is
supported only by the use of a modem that converts the digital data into analog form. Second
generation systems provide digital traffic channels. These readily support digital data; voice traffic is first
encoded in digital form before transmitting. Of course, for second-generation systems, the user traffic
(data or digitized voice) must be converted to an analog signal for transmission between the mobile unit
and the base station. Encryption: Because all of the user traffic, as well as control traffic, is digitized in
second-generation systems, it is a relatively simple matter to encrypt all of the traffic to prevent
eavesdropping. All second-generation systems provide this capability, whereas first generation systems
send user traffic in the clear, providing no security. Error detection and correction: The digital traffic
stream of second-generation systems also lends itself to the use of error detection and correction
techniques. The result can be very clear voice reception. Channel access: In first generation systems,
each cell supports a number of channels. At any given time a channel is allocated to only one user.
Second generation systems also provide multiple channels per cell, but each channel is dynamically
shared by a number of users using time division multiple access (TDMA) or code division multiple access
(CDMA).
10.10 What are the advantages of using CDMA for a cellular network?

Frequency diversity: Because the transmission is spread out over a larger bandwidth, frequency-
dependent transmission impairments, such as noise bursts and selective fading, have less effect on the
signal. Multipath resistance: The chipping codes used for CDMA not only exhibit low cross-correlation
but also low autocorrelation. Therefore, a version of the signal that is delayed by more than one chip
interval does not interfere with the dominant signal as much as in other multipath environments.
Privacy: Because spread spectrum is obtained by the use of noise-like signals, where each user has a
unique code, privacy is inherent. Graceful degradation: With FDMA or TDMA, a fixed number of users
can simultaneously access the system. However, with CDMA, as more users simultaneously access the
system, the noise level and hence the error rate -50- increases; only gradually does the system degrade
to the point of an unacceptable error rate.

10.11 What are the disadvantages of using CDMA for a cellular network?

Self-jamming: Unless all of the mobile users are perfectly synchronized, the arriving transmissions from
multiple users will not be perfectly aligned on chip boundaries. Thus the spreading sequences of the
different users are not orthogonal and there is some level of cross-correlation. This is distinct from
either TDMA or FDMA, in which for reasonable time or frequency guardbands, respectively, the received
signals are orthogonal or nearly so. Near-far problem: Signals closer to the receiver are received with
less attenuation than signals farther away. Given the lack of complete orthogonality, the transmissions
from the more remote mobile units may be more difficult to recover. Thus, power control techniques
are very important in a CDMA system. Soft handoff: A smooth handoff from one cell to the next requires
that the mobile acquire the new cell before it relinquishes the old. This is referred to as a soft handoff,
and is more complex than the hard handoff used in FDMA and TDMA schemes.

10.12 Explain the difference between hard and soft handoff.

Hard handoff: When the signal strength of a neighboring cell exceeds that of the current cell, plus a
threshold, the mobile station is instructed to switch to a new frequency band that is within the
allocation of the new cell. Soft handoff: a mobile station is temporarily connected to more than one
base station simultaneously. A mobile unit may start out assigned to a single cell. If the unit enters a
region in which the transmissions from two base stations are comparable (within some threshold of
each other), the mobile unit enters the soft handoff state in which it is connected to the two base
stations. The mobile unit remains in this state until one base station clearly predominates, at which time
it is assigned exclusively to that cell.

10.13 What are some key characteristics that distinguish third-generation cellular systems from
second-generation cellular systems?

Voice quality comparable to the public switched telephone network; 144 kbps data rate available to
users in high-speed motor vehicles over large areas; 384 kbps available to pedestrians standing or
moving slowly over small areas; Support (to be phased in) for 2.048 Mbps for office use; Symmetrical
and asymmetrical data transmission rates; Support for both packet switched and circuit switched data
services; An adaptive interface to the Internet to reflect efficiently the common asymmetry between
inbound and outbound traffic; More efficient use of the available spectrum in general; Support for a
wide variety of mobile equipment; Flexibility to allow the introduction of new services and technologies
Problems

10.1 Consider four different cellular systems that share the following characteristics. The frequency
bands are 825 to 845 MHz for mobile unit transmission and 870 to 890 MHz for base station
transmission. A duplex circuit consists of one 3D-kHz channel in each direction. The systems are
distinguished by the reuse factor, which is 4, 7,12, and 19, respectively.

a. Suppose that in each of the systems, the cluster of cells (4, 7, 12, 19) is duplicated 16 times. Find the
number of simultaneous communications that can be supported by each system.

b. Find the number of simultaneous communications that can be supported by a single cell in each
system.

c. What is the area covered, in cells, by each system?

d. Suppose the cell size is the same in all four systems and a fixed area of 100 cells is covered by each
system. Find the number of simultaneous communications that can be supported by each system.

a. We have the number of clusters M = 16; bandwidth assigned to cluster BCL = 40 MHz; bandwidth
required for each two-way channel bch = 60 kHz. The total number of simultaneous calls that can be
supported by the system is kSYS = MBCL/bch = 10,666 channels b. Total number of channels available is
K = BCL/bch = 666. For a frequency reuse factor N, each cell can use kCE = K/N channels. For N = 4, kCE =
166 channels For N = 7, kCE = 95 channels -51- For N = 12, kCE = 55 channels For N = 19, kCE = 35
channels c. For N = 4, area = 64 cells; For N = 7, area = 112 cells; For N = 12, area = 192 cells; For N = 19,
area = 304 cells. d. From part b, we know the number of channels that can be carried per cell for each
system. The total number of channels available is just 100 times that number, for a result of 16600,
9500, 5500, 3500, respectively. Source: [CARN99]

10.2 Describe a sequence of events similar to that of Figure 10.6 for a. a call from a mobile unit to a
fixed subscriber b. a call from a fixed subscriber to a mobile unit

a. Steps a and b are the same. The next step is placing the call over the ordinary public switched
telephone network (PSTN) to the called subscriber. Steps d, e, and f are the same except that only the
mobile unit can be involved in a handoff. b. Instead of steps a, b, and c, the process starts with a call
coming in from the PSTN to an MTSO. From there, steps c, d, e, and f are the same except that only the
mobile unit can be involved in a handoff.

10.3 In the discussion of the handoff procedure based on relative signal strength with threshold, it
was pointed out that if the threshold is set quite low, such as Th3, the mobile unit may move far into
the new cell (L4). This reduces the quality of the communication link and may result in a dropped call.
Can you suggest another drawback to this scheme?

This causes additional interference to co-channel users.

10.4 Hysteresis is a technique commonly used in control systems. As an example, describe the
hysteresis mechanism used in a household thermostat.

Suppose that a thermostat on a heating system is set to 20° C. Suppose the temperature in the room is
greater than 20° C and falling. The heating system may not click on until, say, 19° C. As the temperature
in the room rises, the thermostat may cause the heater to remain on until room temperature reaches
21° C.

10.5 A telephony connection has a duration of 23 minutes. This is the only connection made by this
caller during the course of an hour. How much is the amount of traffic, in Erlangs, of this connection?

A = λh = 1 × (23/60) = 0.383 Erlangs

10.6 Using Table 10.3, approximate the answers to the following. Also, in each case, give a description
in words of the general problem being solved. Hint: Straight-line interpolation is adequate.

a. Given N = 20, A = 10.5, find P. b. Given N = 20, P = 0.015,find A. c. Given P = 0.005, A = 6, find N.

a. For a given traffic level (A) and given capacity (N), what is the probability of blocking (P)? (10.5 –
10.07)/(11.1 – 10.07) = (P – 0.002)/(0.005 – 0.002); P = 0.00325 b. What traffic level can be supported
with a given capacity to achieve a given probability of blocking? (0.015 – 0.01)/(0.02 – 0.01) = (A –
12.03)/(13.19 – 12.03); A = 12.61 c. For a given traffic level, what capacity is needed to achieve a certain
upper bound on the probability of blocking ? (6 – 3.96)/(11.1 – 3.96) = (N – 10)/(20 – 10); N = 12.857

10.7 An analog cellular system has a total of 33 MHz of bandwidth and uses two 25-kHz simplex (one-
way) channels to provide full duplex voice and control channels. a. What is the number of channels
available per cell for a frequency reuse factor of (1) 4 cells, (2) 7 cells, and (3) 12 cells? b. Assume that
1 MHz is dedicated to control channels but that only one control channel is needed per cell.
Determine a reasonable distribution of control channels and voice channels in each cell for the three
frequency reuse factors of part (a).

a. The total number of available channels is K = 33000/50 = 660. For a frequency reuse factor N, each
cell can use kCE = K/N channels. For N = 4, kCE = 165 channels For N = 7, kCE = 94 channels For N = 12,
kCE = 55 channels b. 32 MHz is available for voice channels for a total of 640 channels. For N = 4, we can
have 160 voice channels and one control channel per cell For N = 7, we can have 4 cells with 91 voice
channels and 3 cells with 92 voice channels, and one control channel per cell. For N = 12, we can have 8
cells with 53 voice channels and 4 cells with 54 voice channels, and one control channel per cell. Source:
[RAPP96]

10.8 As was mentioned, the one-way bandwidth available to a single operator in the AMPS system is
12.5 MHz with a channel bandwidth of 30 kHz and 21 control channels. We would like to calculate the
efficiency with which this system utilizes bandwidth for a particular installation. Use the following
parameters: • Cell area = 8 km2 • Total coverage area = 4000 km2 • Frequency reuse factor = 7 •
Average number of calls per user during the busy hour = 1.2 • Average holding time of a call = 100 s •
Call blocking probability = 2% a. How many voice channels are there per cell? b. Use Table 10.3 and a
simple straight-line interpolation to determine the total traffic carried per cell, in Erlangs/cell. Then
convert that to Erlangs/km2 . c. Calculate the number of calls/hour/cell and the number of
calls/hour/km2 . d. Calculate the number of users/hour/cell and the number of users/hour/channel.
e. A common definition of spectral efficiency with respect to modulation, or modulation efficiency, in
ErlangsIMHz/km2 , is (Total traffic carried by the system) TJm = (Bandwidth)(Total coverage area) 316
CHAPTErz. 10 / CELLULAR WIRELESS NETWORKS Determine the modulation efficiency for this system.
a. Number of 30-kHz channels = 12500/30 = 416 Number of voice channels = 416 – 21 = 395 Number of
voice channels per cell = 395/7 = 56 -52- b. (56 – 40)/(70 – 40) = (A – 31)/(59.13 – 31); A = 46 Erlangs/cell
c. Number of calls/hour/cell = 46/(100/3600) = 1656 Number of calls/hour/km2 = 1656/8 = 207 d.
Number of users/hour/cell = 1656/1.2 = 1380 Number of users/hour/channel = 1380/56 = 24.6 e. The
total number of cells is 4000/8 = 500 η = (46 Erlangs/cell × 500 cells)/(12.5 MHz × 4000 km2) = 0.46
Erlangs/MHz/km2 Source: [GARG96]

10.9 A cellular system uses FDMA with a spectrum allocation of 12.5 MHz in each direction, a guard
band at the edge of the allocated spectrum of 10 kHz, and a channel bandwidth of 30 kHz. What is the
number of available channels?

(12.5 × 106 – 2(10 x 103)/(30 × 103) = 416

10.10 If 8 speech channels are supported on a single radio channel, and if no guard band is assumed,
what is the number ofsimultaneous users that can be accommodated in GSM?

(25 × 106)/((200 x 103)/8) = 1000 Source: [RAPP96]

10.11 a. What is the duration of a bit in GSM?

b. If a user is allocated one time slot per frame, what is the delay between successive transmissions in
successive frames?

a. From Figure 10.14, we have 156.25 bits in 0.577 ms. Thus, bit duration is: (0.577 x 10–3)/156.25 =
3.6928 µs b. The delay is the duration of 1 frame, which is 4.615 ms

10.12 If we consider the trailing bits, stealing bits, guard bits, and training bits in a GSM frame as
overhead, and the rest of the bits as data, then what is the percentage overhead in a GSM frame?

Total bits in one timeslot = 156.25. Data bits = 114. Overhead = (156.25 – 114)/156.25 = 0.27

10.13 Using the definition of slow frequency hopping from Chapter 7, demonstrate that GSM uses
slow frequency hopping. BN

Slow FHSS = multiple signal elements per hop. In GSM, the frequency is changed one per frame, which is
many bits, so GSM uses slow frequency hopping.

10.14 For a cellular system, FDMA spectral efficiency is defined as 7Ja = T, where w Be = channel
bandwidth Bw = total bandwidth in one direction NT = total number of voice channels in the covered
area a. What is an upper bound on 7Ja? b. Determine 7Ja for the system of Problem 8.

a. The amount of bandwidth allocated to voice channels (Bc Nt ) must be no greater than the total
bandwidth (Bw). Therefore ηa ≤ 1. b. x = (30 × 103 × 395)/(12.5 × 106) = 0.948

10.15 Consider a 7-cell system covering an area of 3100 km2 . The traffic in the seven cells is as
follows: Cell number 1 2 3 4 5 6· 7 Traffic (Erlangs) 30.8 66.7 48.6 33.2 38.2 37.8 32.6 Each user
generates an average of 0.03 Erlangs of traffic per hour, with a mean holding time of 120 s. The
system consists of a total of 395 channels and is designed for a grade of service of 0.02.

a. Determine the number of subscribers in each cell.

a. Number of subscribers = Traffic/0.03

Cell number 1 2 3 4 5 6 7
Subscriber 1026.7 2223.3 1620.0 1106.7 1273.3 1260.0 1086.7

b. Determine the number of calls per hour per subscriber.

Number of calls per hour per subscriber = λ = A/h = 0.03/(120/3600) = 0.9

c. Determine the number of calls per hour in each cell.

Multiply results of part (a) by 0.9

Cell number 1 2 3 4 5 6 7
Subscriber 924 2001 1458 996 1146 1134 978
√√√√

d. Determine the number of channels required in each cell. Hint: You will need to extrapolate using
Table 10.3.

The table in the problem statement gives the value of A. Use P = 0.02. Find N

Cell number 1 2 3 4 5 6 7
Subscriber 40 78 59 43 48 48 42

e. Determine the total number of subscribers.

Total number of subscribers = the sum of the values from part (a) = 9597

f. Determine the average number of subscribers per channel.

From (d), the total number of channels required = 358 Average number of subscribers per channel =
9597/358 = 26.8

g. Determine the subscriber density per km2 •

Subscriber density = 9597/3100 = 3.1 subscribers per km2

h. Determine the total traffic (total Erlangs).

. Total traffic = the sum of the values from table in the problem statement = 287.9

i. Determine the Erlangs per km2.

Erlangs per km2 = 287.9/3100 = 0.09

j. What is the radius of a cell?

The area of a hexagon of radius R is A = 1.5R^2 √ 3 . For A = 3100/7 = 442.86 km2 we have R = 13 km