# 5.13: Exercises

- Page ID
- 41227

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vectorC}[1]{\textbf{#1}} \)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Research at Bell Labs in the 1960s showed that the minimum acceptable SIR for voice communications is \(17\text{ dB}\). This applies to analog modulated signals, but not digitally modulated signals, where BER is important. Consider a sevencell cluster. If the power falls off as \(1/d^{3}\), where \(d\) is distance, determine the worst possible SIR considering only interference from other radios. The worst situation will be when a mobile handset is at the edge of its cell. To do this you need to estimate the distance from the handset to the other basestations (in neighboring clusters) that are operating at the same power levels. Consider the cells to be hexagons. Develop a symbolic expression for the total interference signal level at the handset, assuming that all basestations are radiating at the same power level, \(P\). You can use approximate distances. For example, each distance can be expressed in terms of integer multiples of cell radii, \(R\). Is the \(17\text{ dB}\) SIR achieved using a \(7\)-cell cluster?
- Describe the following concepts.
- Clusters in a cellular phone system.
- Multipath effects in a central city area compared to multipath effects in a desert.

- Describe the concept of clusters in a cellular phone system in four lines.
- Short answer questions on modulation and spectral efficiency.
- What is the PMEPR of a phase modulated signal?
- In five lines explain your understanding of spectral efficiency as it relates to bits per hertz. That is, how can you have a spectral efficiency of \(n\text{ bit/s/Hz}\) where \(n\) is more than \(1\)? [Note that sometimes this is expressed as \(\text{bit/s/Hz}\) as well as \(\text{bit/s/Hz}\).]
- What is the modulation efficiency of a QPSK-modulated signal? Ignore the impact of the number of cells in a cluster.

- A cellular communication system uses a frequency reuse plan with seven cells per cluster to obtain the required minimum SIR. If a QPSK system is used, what is the radio spectrum efficiency in terms of \(\text{bit/s/Hz/cell}\) if all transitions on the constellation diagram are allowable? Assume that there is no coding.
- A cellular communication system uses GMSK modulation and a frequency reuse plan with three cells per cluster. What is the radio spectrum efficiency in terms of \(\text{bit/s/Hz/cell}\)? Assume that there is no coding.
- A cellular communication system uses \(\pi /4\)-DQPSK modulation and a frequency reuse plan with five cells per cluster. What is the radio spectrum efficiency in terms of \(\text{bit/s/Hz/cell}\)? Assume that there is no coding.
- A frequency reuse plan has three cells per cluster. If ideal 16-QAM modulation is used, what is the system spectral efficiency (in \(\text{bits/s/Hz/cell}\))?
- A 2G cellular communication system uses a frequency reuse plan with seven cells per cluster. If ideal QPSK modulation is used, what is the system spectral efficiency (in \(\text{bits/s/Hz/cell}\))?
- A 2G system has three cells per cluster. If \(3π/8\)-8PSK? modulation is used, what is the system spectral efficiency (in \(\text{bits/s/Hz/cell}\))?
- A cellular communication system uses \(3π/8\)-8PSK modulation and a frequency reuse plan with three cells per cluster. What is the radio spectrum efficiency in terms of \(\text{bit/s/Hz/cell}\)? Assume that there is no coding.
- A communication system uses a modulation with a modulation efficiency of \(5\text{ bit/s/Hz}\). Ignore coding so that \(R_{b} = R_{c}\). What is the radio spectral efficiency in terms of \(\text{bit/s/Hz/cell}\) if there are three cells per cluster?
- A proposed modulation format has a modulation efficiency of \(3.5\text{ bit/s/Hz}\). Antenna sectoring and required SNR lead to a system with seven cells per cluster. You can ignore the impact of coding so you can assume that \(R_{b} = R_{c}\). What is the radio spectral efficiency in terms of \(\text{bit/s/Hz/cell}\) modulated signal?
- A cellular radio system uses a frequency reuse plan with \(12\) cells per cluster. If ideal 8-PSK modulation is used, what is the system spectral efficiency in terms of \(\text{bit/s/Hz/cell}\)?
- Consider a cellular system having a frequency reuse plan with seven cells per cluster to obtain the minimum signal-to-interference ratio. If ideal QPSK modulation is used, what is the system spectral efficiency in terms of bits per second per hertz per cell?
- A monostatic free-space \(10\text{ GHz}\) pulsed radar system is used to detect a fighter plane having a radar cross section, \(\sigma\), of \(5\text{ m}^{2}\). The antenna gain is \(30\text{ dB}\) and the transmitted power is \(1\text{ kW}\). If the minimum detectable received signal is \(−120\text{ dBm}\), what is the detection range?
- An antenna with a gain of \(10\text{ dB}\) presents an RF signal with a power of \(5\text{ dBm}\) to a low-noise amplifier along with noise of \(1\text{ mW}\) and an interfering signal of \(2\text{ mW}\).
- What is the RF SIR? Include both noise and the interfering signal in your calculation. Express your answer in decibels.
- The modulation format and coding scheme used have a processing gain, \(G_{P}\), of \(7\text{ dB}\). The modulation scheme has four states. What is the ratio of the energy per bit to the noise per bit, that is, what is the effective \(E_{b}/N_{o}\) after despreading?

- The receiver in a digital radio system receives a \(100\text{ pW}\) signal and the interference from other radios at the input of the receiver is \(20\text{ pW}\). The receiver has an overall gain of \(40\text{ dB}\) and the noise added by the receiver, referred to the output of the receiver, is \(100\text{ nW}\).
- What is the RF SIR at the output of the receiver?
- If 16-QAM modulation with a modulation efficiency of \(2.98\text{ bit/s/Hz}\) is used and the processing gain is \(30\text{ dB}\), what is the effective SIR after despreading, i.e. what is \(E_{b,\text{ eff}}/N_{o, b}\)?

- A new communication system is being investigated for sending data to a printer. The system will use GMSK modulation and a channel with \(25\text{ MHz}\) bandwidth and an information bit rate of \(10\text{ Mbit/s}\). The modulation format will result in a spectrum that distributes power almost uniformly over the \(25\text{ MHz}\) bandwidth. [Parallels Example 5.6.2]
- What is the processing gain?
- If the received RF SIR is \(6\text{ dB}\), what is the effective system SIR (or \(E_{b, i}/N_{o, i}\)) after DSP? Express your answer in decibels.

- A \(4\text{ kHz}\) bandwidth voice signal is coded by a vocodor as an \(8\text{ kbit/s}\) data stream. Coding increases the data stream to \(64\text{ kbit/s}\). What is the processing gain that can be achieved at the receiver if QPSK modulation is used with a modulation efficiency of \(1.4\text{ bit/s/Hz}\)?
- A receive antenna with a gain of \(10\text{ dB}\) presents a signal with a power of \(5\text{ dBm}\) to a low-noise amplifier along with noise of \(1\text{ mW}\) and an interfering signal of \(2\text{ mW}\).
- What is the SIR at the input to the amplifier? Express your answer in decibels.
- BPSK modulation is used and coding results in a processing gain, \(G_{P}\), of \(7\text{ dB}\). What is the ratio of the energy per bit to the noise power per bit (i.e., what is \(E_{b}/N_{o}\)) after despreading?

- If a received RF signal has an SIR of \(−5\text{ dB}\) and the processing gain (calculated from bit rates) that can be achieved for the modulation and coding used is \(15\text{ dB}\), what is the \(E_{b}/N_{0}\) after processing? There are \(4\text{ bits}\) per symbol.
- A signal with a power of \(13\text{ dBm}\) is input to a low-noise amplifier along with noise of \(1\text{ mW}\) and an interfering signal of \(2\text{ mW}\).
- What is the SIR at the input to the amplifier? Express your answer in decibels.
- QPSK modulation is used and coding results in a processing gain, \(G_{P}\), of \(13\text{ dB}\). What is the ratio of the effective energy per bit to the noise power per bit (i.e., what is \(E_{b}/N_{o}\)) after despreading?

- Short answer questions. Each part requires a short paragraph of five lines and a figure.
- Explain how OFDM reduces the impact of multiple paths in a wireless communication system.
- Explain how MIMO exploits multipath to enhance the capacity of a digital communication system.

- A coding rate of \(2/3\) is required to manage transmission errors in a \(54\text{ Mbit/s}\) data link. That is, the information bit rate is \(54\text{ Mbit/s}\). What is the total bit rate required (including data and coding bits)?
- The channel bandwidth in the GSM cellular phone system is \(200\text{ kHz}\) and the GMSK modulation scheme used has a spectral efficiency of \(1.354\text{ bit/s/Hz}\).
- What is the data rate of one frequency channel?
- A time slot is \(577\:\mu\text{s}\) long. How many bits are there in one (i.e. a duration of \(8.25\text{ bits}\)). How many data bits are there in a GSM time slot?
- A GSM frame duration is \(4.615\text{ ms}\) long and has eight time slots and a voice user has one time slot every frame. How many data bits per second are available to a single user?

- Consider an OFDM system with \(48\) subcarriers carrying data and which uses 16-QAM modulation of each subcarrier and a coding rate of \(2/3\). There are also four pilot subcarriers that are used for frequency and phase reference, to ensure that spectral lines are not created, and to facilitate carrier recovery. The pilot carriers can be ignored in this problem so consider \(48\) subcarriers. The modulation efficiency achieved for this particular implementation of 16-QAM is \(2.98\text{ bit/s/Hz}\).
- How many symbols are there for each subcarrier? That is, how many points are there in the constellation diagram for one subcarrier?
- How many coded bits are there on each subcarrier? That is, how many bits per symbol are there for each subcarrier?
- Considering all of the data subcarriers, how many coded bits are there per OFDM symbol? [Hint, there are \(16\) subcarriers, so for each OFDM symbol there will be \(16\) subcarrier symbols.]
- Considering the coding rate, determine the number of data bits per OFDM symbol. That is, ignore coding bits.

- Consider an OFDM system with \(12\) subcarriers carrying data and which uses 8-PSK modulation of each subcarrier and a coding rate of \(3/4\). Pilot subcarriers they can be ignored in this problem so consider all \(12\) subcarriers.
- How many symbols are there for each subcarrier? That is, how many points are there in the constellation diagram for one subcarrier?
- How many coded bits (code + data) are there on each subcarrier? That is, how many bits per symbol are there for each subcarrier?
- Considering all of the data subcarriers, how many coded bits are there per OFDM symbol? [Hint, there are \(12\) subcarriers, so for each OFDM symbol there will be \(12\) subcarrier symbols.]
- Considering the coding rate, determine the number of data bits per OFDM symbol. That is, ignore coding bits.

- A digital radio system transmits a baseband digital signal of \(100\text{ Mbit/s}\) over a channel that is \(300\text{ MHz}\) wide. The digital modulation scheme effectively fills the \(300\text{ MHz}\) channel with uniform power.
- What is the processing gain that can be achieved with this system?
- Consider that the signal received and delivered to the input of the receiver front end is \(100\text{ pW}\) and the interference from other radios delivered to the receiver front-end is \(20\text{ pW}\). What is the SIR at the input to the receiver electronics?

- The L1 band of the global positioning system (GPS), is centered at \(1.57542\text{ GHz}\) and has two overlapping spread-spectrum encoded signals. The stronger of these is the coarse acquisition (C/A) signal with an information bit rate of \(50\text{ bits/s}\) and a transmission rate of \(1.023\) million chips per second using BPSK modulation with an RF bandwidth of \(2.046\text{ MHz}\). In ideal conditions the C/A signal received has a power of \(−130\text{ dBm}\). A GPS receiver has an antenna noise temperature is of \(290\text{ K}\).
- What is the processing gain?
- What is the noise in \(\text{dBm}\) received in the \(2.046\text{ MHz}\) bandwidth?
- What is the SNR in decibels?
- If a C/A signal is received from each of \(10\) satellites (so there are \(9\) interfering signals), what is the total interference power for one satellite’s C/A signal?
- With respect to just one of the C/A signals, what is the SINR (signal to interference plus noise ratio) at the receiver?
- If the receiver does not contribute noise, what is the effective SNR of the despread bitstream from each satellite?
- If the required minimum effective SNR is \(6\text{ dB}\), what is the minimum acceptable power, in \(\text{dBm}\), of the GPS signal received from one satellite?

- GLONASS is the Russian satellite navigation system with one of two open signals called the L1OF band at \(1600.995\text{ MHz}\). The system uses DSSS encoding and BPSK modulation and each GLONASS satellite transmits on a different frequency. The symbol rate is \(511,000\text{ chips/s}\), the bandwith of the transmitted signal is approximately \(540\text{ kHz}\), and there are \(50\) information bits per second.
- What is the system’s processing gain?
- What is the noise in \(\text{dBm}\) received in the \(540\text{ kHz}\) bandwidth?
- If the required system minimum effective SNR is \(6\text{ dB}\), what is the minimum acceptable power, in \(\text{dBm}\), of the received signal? Assume that the receiver is noiseless.

- A new communication system uses a channel that is \(100\text{ MHz}\) wide and uses direct sequence CDMA to efficiently spread a \(5\text{ Mbit/s}\) digital signal over the full channel.
- What is the processing gain that can be achieved with the received signal?
- The analog signal at the output of the receive antenna has an SIR of \(1\text{ dB}\), what is energy per bit divided by the noise per bit?

- A deep-space communication system will use direct sequence spread spectrum to code a data stream of \(10\text{ kbit/s}\) then modulate the signal to transmit over a \(5\text{ GHz}\) link to a ground station. Since the propagation loss is very high it has been determined that the processing gain must be \(50\text{ dB}\). If the link has a bandwidth of \(1\text{ MHz}\), what is the maximum baseband bit rate (in \(\text{bit/s}\)) that can be supported?
- A direct sequence spread spectrum code of \(10\text{ Mbit/s}\) is used to code a \(4\text{ kbit/s}\) data steam that is modulated using \(3π/8\)-8PSK modulation to produce an RF signal at \(1900\text{ MHz}\). The modulation efficiency of \(3π/8\)-8PSK modulation is \(2.7\text{ bit/s/Hz}\).
- What is the bandwidth of the RF signal?
- What processing gain can be achieved in the receiver?

- An OFDM system with \(12\) data subcarriers, uses a coding rate of \(3/4\), and each subcarrier uses 16-QAM modulation (with a modulation efficiency of \(2.7\text{ bit/s/Hz}\)) with a bandwidth of \(250\text{ kHz}\). What is the maximum data rate supported?
- An OFDM system with \(48\) subcarriers carrying data uses 16-QAM modulation of each subcarrier and a coding rate of \(2/3\). The actual modulation efficiency for the 16-QAM system here is \(2.7\text{ bit/s/Hz}\). What is the maximum data rate supported in \(\text{Mbit/s}\) when the bandwidth of each modulated subcarrier is \(312\text{ kHz}\)?
- Explain using sentences and a diagram how OFDM reduces the impact of multiple paths in a wireless communication system.
- Explain using sentences and a diagram how MIMO exploits multipath to enhance the capacity of a digital communication system.
- A free-space \(2\text{ GHz}\) pulsed monostatic radar system transmits a \(2\text{ kW}\) pulse and has a minimum detectable received signal power of \(−90\text{ dBm}\). What is the antenna gain required to be able to detect a target with a radar cross section of \(10\text{ m}^{2}\) at \(10\text{ km}\)?
- A \(10\text{ GHz}\) bistatic radar has a minimum detectable received signal power of \(−150\text{ dBm}\), an antenna gain of \(26\text{ dB}\), and a required range of \(100\text{ km}\). What is the transmitted pulse power in \(\text{dBm}\) needed to detect a
- conventional fighter aircraft having an RCS of \(5\text{ m}^{2}\)?
- a stealth aircraft with an RCS of \(0.05\text{ m}^{2}\)?

- The L5 band is a new public band of the GPS system and is centered at \(1.176.5\text{ GHz}\). The coarse acquisition (C/A) signal has an information bit rate of \(50\text{ bits/s}\) and a spread-spectrum transmission rate of \(10.23\cdot 10^{6}\text{ chips per second}\) using BPSK modulation and occupying an RF bandwidth of \(20.46\text{ MHz}\). The noiseless GPS receiver has an omnidirectional antenna with a noise temperature is \(290\text{ K}\).
- What is the system’s processing gain?
- What is the noise in \(\text{dBm}\) in the \(20.46\text{ MHz}\) bandwidth?
- If overlapping C/A signals are received from \(10\) satellites, what is the total interference power for the signal from one satellite? The power of a C/A signal is \(S\).
- If the required system minimum effective SNR is \(6\text{ dB}\), what is the minimum acceptable received power, in \(\text{dBm}\), of the signal from one satellite?

## 5.16.1 Exercises By Section

\(†\)challenging, \(‡\)very challenging

\(§5.3 1‡, 2†, 3†, 4‡, 5†, 6†, 7‡\)

\(§5.5 8†, 9, 10, 11†, 12, 13, 14†, 15†, 16†, 17†, 18†, 19, 20, 21\)

\(§5.6 22, 23, 24, 25‡, 26†, 27†, 28†, 29†\)

\(§5.8 30†, 31†, 32†, 33‡, 34‡, 41† \)

\(§5.10 35, 36, 37, 38\)

\(§5.13 39, 40\)

## 5.16.2 Answers to Selected Exercises

- \(12.5\text{ dB}\)

- \(0.45\text{ bit/s/Hz/cell}\)

- (d) \(3\text{ bit/s/Hz}\)

- \(0.1\)

- (b) \(7.228\text{ dB}\)

- \(81\text{ Mbit/s}\)

- (d) \(128\)
- (d) \(27\)
- (b) \(6.99\text{ dB}\)

- (b) \(14\text{ dB}\)

- (a) \(81\text{ Mbit/s}\)