# 2.11: Exercises

- Page ID
- 41183

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Develop a formula for the average power of a signal \(x(t)\). Consider \(x(t)\) to be a voltage across a \(1\:\Omega\) resistor.
- What is the PAPR of a \(5\)-tone signal when the amplitude of each tone is the same?
- What is the PMEPR of a \(10\)-tone signal when the amplitude of each tone is the same?
- Consider two uncorrelated analog signals combined together. One signal is denoted \(x(t)\) and the other \(y(t)\), where \(x(t)=0.1 \sin (10^{9}t)\) and \(y(t)=0.05 \sin (1.01\cdot 10^{9}t)\). The combined signal is \(z(t) = x(t) + y(t)\). [Parallels Example 2.2.3]
- What is the PAPR of \(x(t)\) in decibels?
- What is the PAPR of \(z(t)\) in decibels?
- What is the PMEPR of \(x(t)\) in decibels?
- Is it possible to calculate the PMEPR of \(z(t)\)? If so, what is it?

- Consider two uncorrelated analog signals combined together. One signal is denoted \(x(t)\) and the other \(y(t)\), where \(x(t)=0.1 \sin (10^{8}t)\) and \(y(t)=0.05 \sin (1.01 \cdot 10^{8}t)\). What is the PMEPR of this combined signal? Express PMEPR in decibels. [Parallels Example 2.2.3]
- What is PMEPR of a three-tone signal when the amplitude of each tone is the same?
- What is PMEPR of a four-tone signal when the amplitude of each tone is the same?
- A tone \(x_{1}(t)=0.12 \cos(\omega_{1}t)\) is added to two other tones \(x_{2}(t)=0.14 \cos(\omega_{2}t)\) and \(x_{3}(t) = 0.1 \cos(\omega_{3}t)\) to produce a signal \(y(t) = x_{1}(t) + x_{2}(t) + x_{3}(t),\) where \(y(t),\: x_{1}(t),\: x_{2}(t)\) and \(x_{3}(t)\) are voltages across a \(100\:\Omega\) resistor. Consider that \(\omega_{1},\:\omega_{2},\) and \(\omega_{3}\) are \(10\%\) apart and that the signals at these frequencies are uncorrelated.
- What is the PMEPR of \(x_{1}(t)\)? Express your answer in decibels.
- Sketch \(y(t)\).
- The combined signal appears as a pseudocarrier with a time-varying envelope. What is the power of the largest single cycle of the pseudo-carrier?
- What is the average power of \(y(t)\)?
- What is the PMEPR of \(y(t)\)? Express your answer in decibels.

- Consider two uncorrelated analog signals summed together. One signal is denoted \(x(t)\) and the other \(y(t)\), where \(x(t) = \sin (10^{9}t)\) and \(y(t) = 2 \sin (1.01\cdot 10^{9}t)\) so that the total signal is \(z(t) = x(t) + y(t)\). What is the PMEPR of \(z(t)\) in decibels? [Parallels Example 2.2.3]
- What is the PMEPR of an FM signal at \(1\text{ GHz}\) with a maximum modulated frequency deviation of \(±10\text{ kHz}\)?
- What is the PMEPR of a two-tone signal (consisting of two sinewaves at different frequencies that are, say, \(1\%\) apart)? First, use a symbolic expression, then consider the special case when the two amplitudes are equal. Consider that the two tones are close in frequency.
- What is the PMEPR of a three-tone signal (consisting of three equal-amplitude sinewaves, say, \(1\%\) apart in frequency)?
- A phase modulated tone \(x_{1}(t) = A_{1} \cos[\omega_{1}t + \phi_{1}(t)\). What is the PMEPR of \(x_{1}(t)\)? Express your answer in decibels.
- What is the PMEPR of an AM signal with \(75\%\) amplitude modulation?
- Two FM voltage signals \(x_{1}(t)\) and \(x_{2}(t)\) are added together and then amplified by an ideal linear amplifier terminated in \(50\:\Omega\) with a gain of \(10\text{ dB}\) and the output voltage of the amplifier is \(y(t) =\sqrt{10} [x_{1}(t) + x_{2}(t)]\).
- What is the PMEPR of \(x_{1}(t)\)? Express your answer in decibels?
- What effect does the amplifier have on the PMEPR of the signal?
- If \(x_{1}(t) = A_{1} \cos[\omega_{1}(t)t]\) and \(x_{2}(t) = A_{2} \cos[\omega_{2}(t)t]\), what is the PMEPR of the output of the amplifier, \(y(t)\)? Express PMEPR in decibels. Consider that \(\omega_{1}(t)\) and \(\omega_{2}(t)\) are within \(0.1\%\) of each other.

- An FM signal has a maximum frequency deviation of \(20\text{ kHz}\) and a modulating signal between \(300\text{ Hz}\) and \(5\text{ kHz}\). What is the bandwidth required to transmit the modulated RF signal when the carrier is \(200\text{ MHz}\)? Is this considered to be narrowband FM or wideband FM?
- A high-fidelity stereo audio signal has a frequency content ranging from \(50\text{ Hz}\) to \(20\text{ kHz}\). If the signal is to be modulated on an FM carrier at \(100\text{ MHz}\), what is the bandwidth required for the modulated RF signal? The maximum frequency deviation is \(5\text{ kHz}\) when the modulating signal is at its peak value.
- Consider FM signals close in frequency but whose spectra do not overlap. [Parallels Example 2.4.1]
- What is the PMEPR of just one PM signal? Express your answer in decibels.
- What is the PMEPR of a signal comprised of two uncorrelated narrowband PM signals with the same average power?

- Consider two nonoverlapping equal amplitude FM signals having center frequencies within \(1\%\).
- What is the PMEPR in \(\text{dB}\) of just one FM modulated signal?
- What is the PMEPR in \(\text{dB}\) of a signal comprising two FM signals of the same power?

- Consider a signal \(x(t)\) that is the sum of two uncorrelated signals, a narrowband AM signal with \(50\%\) modulation, \(y(t)\), and a narrow-band FM signal, \(z(t)\). The center frequencies of \(y(t)\) and \(z(t)\) are within \(1\%\). The carriers have equal amplitude. Express answers in \(\text{dB}\).
- What is the PAPR of the AM signal \(x(t)\)?
- What is the PAPR of the FM signal \(z(t)\)?
- What is the PAPR of \(x(t)\)?
- What is the PMEPR of the AM signal \(x(t)\)?
- What is the PMEPR of the FM signal \(z(t)\)?
- What is the PMEPR of \(x(t)\)?

- Two phase modulated tones \(x_{1}(t) = A_{1} \cos[\omega_{1}t+\phi_{1}(t)\) and \(x_{2}(t) = A_{2} \cos[\omega_{2}t+\phi_{2}(t)\) are added together as \(y(t) = x_{1}(t) +x_{2}(t)\). What is the PMEPR of \(y(t)\) in decibels. Consider that \(\omega_{1}\) and \(\omega_{2}\) are within \(0.1\%\) of each other.
- A radio uses a channel with a bandwidth of \(25\text{ kHz}\) and a modulation scheme with a gross bit rate of \(100\text{ kbits/s}\) that is made of an information bit rate of \(60\text{ kbits/s}\) and a code bit rate of \(40\text{ kbits/s}\).
- What is the modulation efficiency in \(\text{bits/s/Hz}\)?
- What is the spectral efficiency in \(\text{bits/s/Hz}\)?

- A cellular communication system uses \(π/4\)- DQPSK modulation with a modulation efficiency of \(1.63\text{ bits/s/Hz}\) to transmit data at the rate of \(30\text{ kbits/s}\). This would be the spectral efficiency in the absence of coding. However, \(25\%\) of the transmitted bits are used to implement a forward error correction code.
- What is the gross bit rate?
- What is the information bit rate?
- What is the bandwidth required to transmit the information and code bits?
- What is the spectral efficiency in \(\text{bits/s/Hz}\)?

- A radio uses a channel with a \(5\text{ MHz}\) bandwidth and uses 256-QAM modulation with a modulation efficiency of \(6.33\text{ bits/s/Hz}\). The coding rate is \(3/4\) (i.e. of every \(4\text{ bits}\) sent \(3\) are data bits and the other is an error correction bit).
- What is gross bit rate in \(\text{Mbits/s}\)?
- What is information rate in \(\text{Mbits/s}\)?
- What is the spectral efficiency in \(\text{bits/s/Hz}\)?

- The following sequence of bits \(\mathsf{0100110111}\) is to be transmitted using QPSK modulation. Take these data in pairs, that is, as \(\mathsf{01 00 11 01 11}\). These pairs, one bit at a time, drive the \(I\) and \(Q\) channels. Show the transitions on a constellation diagram. [Parallels Example 2.8.1]
- The following sequence of bits \(\mathsf{0100110111}\) is to be transmitted using \(π/4\)-DQPSK modulation. Take these data in pairs, that is, as \(\mathsf{01 00 11 01 11}\). These pairs, one bit at a time, drive the \(I\) and \(Q\) channels. Use five constellation diagrams, with each diagram showing one transition or symbol. [Parallels Example 2.7.1]
- The following sequence of bits \(\mathsf{0100110111}\) is transmitted using OQPSK modulation. Take these data in pairs, that is, as \(\mathsf{01 00 11 01 11}\). These pairs, one bit at a time, drive the \(I\) and \(Q\) channels. Show the transitions on a constellation diagram.
- Draw the constellation diagram of OQPSK.
- Draw the constellation diagrams of \(3π/8\)- 8DPSK and explain the operation of this system and describe its advantages.
- How many bits per symbol can be sent using \(3π/8\)-8PSK?
- How many bits per symbol can be sent using 8- PSK?
- How many bits per symbol can be sent using 16- QAM?
- Draw the constellation diagram of OQPSK modulation showing all possible transitions. You may want to use two diagrams.
- What is the PMEPR of a \(5\)-tone signal when the amplitude of each tone is the same?
- Draw the constellation diagram of 64QAM.
- How many bits per symbol can be sent using 32QAM?
- How many bits per symbol can be sent using 16QAM?
- How many bits per symbol can be sent using 2048QAM?
- Consider a two-tone signal and describe intermodulation distortion in a short paragraph and include a diagram.
- A 16-QAM modulated signal has a maximum RF phasor amplitude of \(5\text{ V}\). If the noise on the signal has an rms value of \(0.2\text{ V}\), what is the EVM of the modulated signal? [Parallels Example 2.11.1]
- Consider a digitally modulated signal and describe the impact of a nonlinear amplifier on the signal. Include several negative effects.
- A carrier with an amplitude of \(3\text{ V}\) is modulated using 8-PSK modulation. If the noise on the modulated signal has an rms value of \(0.1\text{ V}\), what is the EVM of the modulated signal? [Parallels Example 2.11.1]
- Consider a 32-QAM modulated signal which has a maximum \(I\) component, and a maximum \(Q\) component, of the RF phasor of \(5\text{ V}\). If the noise on the signal has an RMS value of \(0.1\text{ V}\), what is the modulation error ratio of the modulated signal in decibels? Refer to Figure 2.8.21(b). [Parallels Example 2.11.1]

## 2.14.1 Exercises By Section

\(§12.2 1, 2, 3, 4, 5, 6, 7, 8, 9 10, 11, 12\)

\(§12.4 13, 14, 15, 16, 17, 18, 19 20, 21\)

\(§12.5 22, 23, 24\)

\(§12.8 25, 26, 27, 28, 29, 30, 31\)

\(§12.9 32, 33, 34, 35, 36, 37, 38\)

\(§12.11 39, 40, 41, 42, 43\)

## 2.14.2 Answers to Selected Exercises

- \(2.55\text{ dB}\)

- (e) \(3.78\text{ dB}\)
- \(0.00022\text{ W}\)

- no effect

- (a) \(6\text{ dB}\)

- (e) \(0\text{ dB}\)

- (a) \(4\text{ bits/s/Hz}\)

- \(36.02\text{ dB}\)