# 7.10: Exercises

- Page ID
- 46148

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

- The first stage of a two-stage amplifier has a linear gain of \(16\text{ dB}\) and output \(1\text{ dB}\) gain compression, \(P_{1o,1\text{ dB}} = −20\text{ dBm}\). For the second stage the linear gain is \(30\text{ dB}\) and \(P_{2o,1\text{ dB}} = 0\text{ dBm}\).
- Determine the input-referred gain compression, \(P_{2i,1\text{ dB}}\) of stage \(2\).
- Compare \(P_{2i,1\text{ dB}}\) and \(P_{1o,1\text{ dB}}\). Which stage dominates gain compression?
- What is the amplifier’s output gain compression level, \(P^{T}_{o,1\text{ dB}}\) considering only compression from the dominant stage.
- Calculate \(P^{T}_{o,1\text{ dB}}\) using the method described in Section 7.2.1.
- Compare \(P^{T}_{o,1\text{ dB}}\) calculated in (c) and (d) and briefly discuss any discrepancy.

- An amplifier has two cascaded stages with linear gains of \(G_{1} = 20\text{ dB}\) and \(G_{2} = 30\text{ dB}\), and output-referred third-order intercepts of \(\text{OIP3}_{1} = 0\text{ dBm}\) and \(\text{OIP3}_{2} = 20\text{ dBm}\), respectively. What is \(\text{IIP3}\) of the amplifier? Use the organized cascade intercept method.
- A single-stage amplifier has a linear gain of \(16\text{ dB}\) and an output \(1\text{ dB}\) gain compression point of \(10\text{ dBm}\). A communication signal with a PMEPR of \(6\text{ dB}\) is used. What is the maximum average power of the input signal before the output suffers significant compression? This is defined at the point at which the peak signal is compressed by \(1\text{ dB}\).
- The first stage of a two-stage amplifier has a linear gain \(G_{1} = 30\text{ dB}\) and an output \(1\text{ dB}\) gain compression point \(P_{1o,1\text{ dB}} = −10\text{ dBm}\). The second stage has a linear gain \(G_{2} = 20\text{ dB}\) and an output \(1\text{ dB}\) gain compression point \(P_{2o,1\text{ dB}} = 10\text{ dBm}\). What is the output-referred \(1\text{ dB}\) gain compression point of the cascade amplifier? [Parallels Example 7.2.1]
- An amplifier consists of two cascaded stages. The first stage has a linear gain \(G_{1} = 30\text{ dB}\) and an output \(1\text{ dB}\) gain compression point \(P_{1o,1\text{ dB}} = 0.1\text{ dBm}\). The second stage has a linear gain \(G_{2} = 20\text{ dB}\) and an output \(1\text{ dB}\) gain compression point \(P_{2o,1\text{ dB}} = 1\text{ dBm}\). What is the input-referred \(1\text{ dB}\) gain compression point of the cascade amplifier? [Parallels Example 7.2.1]
- The stages of a two-stage amplifier have linear gains of \(G_{1}\) and \(G_{2}\), and output \(1\text{ dB}\) gain compression powers of \(P_{1o,1\text{ dB}}\) and \(P_{2o,1\text{ dB}}\), respectively. Develop a symbolic expression for the input-referred \(1\text{ dB}\) gain compression point of the cascade amplifier.
- An amplifier has two stages with linear gains of \(G_{1} = 20\text{ dB}\) and \(G_{2} = 30\text{ dB}\), and output \(1\text{ dB}\) gain compression powers of \(P_{1o,1\text{ dB}} = 0.1\text{ dBm}\) and \(P_{2o,1\text{ dB}} = 1\text{ dBm}\), respectively. What is the input-referred \(1\text{ dB}\) gain compression power of the amplifier?
- The first stage of a two-stage amplifier has a linear power gain of \(26\text{ dB}\) and an output \(1\text{ dB}\) gain compression power of \(10\text{ dBm}\). The corresponding parameters of the second stage are \(10\text{ dB}\) and \(13\text{ dBm}\).
- What is the linear power gain of the two-stage amplifier?
- What is the output \(1\text{-dB}\) gain compression power of the amplifier for a sinusoidal RF input signal?
- What is the maximum average output RF power of the \(\text{64-QAM-}\)modulated signal (with a PMEPR of \(7.8\text{ dB}\)) for an undistorted output (as defined by \(1\text{-dB}\) gain compression)?

- The stages of a three-stage amplifier have linear gains of \(10\text{ dB},\: 20\text{ dB}\), and \(20\text{ dB}\) respectively, and \(1\text{ dB}\) output gain compression levels of \(−60\text{ dBm},\: −40\text{ dBm}\), and \(−20\text{ dBm}\) respectively. What is the output power when the gain of the amplifier is compressed by \(1\text{ dB}\)?
- The first stage of a two-stage amplifier has a linear power gain of \(26\text{ dB}\), an output \(1\text{ dB}\) gain compression power of \(10\text{ dBm}\), and an output-referred third-order intercept point \(\text{OIP3} = 26\text{ dBm}\). The second stage has a linear power gain of \(10\text{ dB}\), an output \(1\text{ dB}\) gain compression point of \(13\text{ dBm}\), and an output-referred third-order intercept point \(\text{OIP3} = 33\text{ dBm}\).
- What is the linear power gain of the two-stage amplifier?
- What is the output \(1\text{-dB}\) gain compression power of the two-stage amplifier for a sinusoidal RF input signal?
- What is the \(\text{OIP3}\) of the two-stage amplifier?
- What is the input-referred third-order intercept point, \(\text{IIP3}\)?

- The final RF output of a cell phone has a driver amplifier followed by a power amplifier. The driver amplifier has a linear gain of \(30\text{ dB}\) and an output-referred third-order intercept point, \(\text{OIP3}\), of \(50\text{ dBm}\). The power amplifier has a linear gain of \(12\text{ dB}\) and an output-referred third-order intercept point, \(\text{OIP3}\), of \(55\text{ dBm}\). What is the \(\text{OIP3}\) of the driver-power amplifier cascade?
- A two-stage amplifier has a linear power gain of \(20\text{ dB}\), an output \(1\text{ dB}\) gain compression point of \(30\text{ dBm}\), and an output-referred third-order intercept point \(\text{OIP3} = 53\text{ dBm}\).
- What is the power of the maximum input signal when the gain of the amplifier is compressed by \(1\text{ dB}\)?
- What is the input-referred third-order intercept point, \(\text{IIP3}\)?

- The first stage of a two-stage amplifier has a linear power gain of \(23\text{ dB}\), an output \(1\text{ dB}\) gain compression power of \(1\text{ dBm}\), and an output-referred third-order intercept point \(\text{OIP3} = 20\text{ dBm}\). The second stage has a linear power gain of \(10\text{ dB}\), an output \(1\text{-dB}\) gain compression point of \(10\text{ dBm}\), and an output-referred third-order intercept point, \(\text{OIP3} = 30\text{ dBm}\). The signal applied to the amplifier uses QPSK modulation with a PMEPR of \(3\text{ dB}\).
- What is the linear power gain in decibels of the two-stage amplifier?
- What is the output \(1\text{-dB}\) gain compression power, in \(\text{dBm}\), of the two-stage amplifier?
- What is the \(\text{OIP3}\), in \(\text{dBm}\), of the two-stage amplifier?
- What is the input-referred third-order intercept point, \(\text{IIP3}_{\text{m}}\)?
- What is the single-tone output power at \(1\text{ dB}\) gain compression?
- What is the maximum output RF power of the QPSK-modulated signal for an undistorted output?

- An amplifier has two cascaded stages. The stages have linear gains of \(G_{1}\) and \(G_{2}\), and output-referred third-order intercepts of \(\text{OIP3}_{1}\) and \(\text{OIP3 dBm}\), respectively. What is \(\text{IIP3}\) of the amplifier?
- An amplifier has two cascaded stages with linear gains of \(G_{1} = 20\text{ dB}\) and \(G_{2} = 30\text{ dB}\), and output-referred third-order intercepts of \(\text{OIP3}_{1} = 0\text{ dBm}\) and \(\text{OIP3}_{2} = 20\text{ dBm}\), respectively. What is \(\text{IIP3}\) of the amplifier? Use the unorganized cascade intercept method.
- The first stage of a room-temperature two-stage amplifier with a \(100\text{ MHz}\) bandwidth has a linear power gain of \(26\text{ dB}\), an output \(1\text{ dB}\) gain compression power of \(10\text{ dBm}\), and an output-referred third-order intercept point \(\text{OIP3} = 26\text{ dBm}\). The second stage has a linear power gain of \(6\text{ dB}\), an output \(1\text{ dB}\) gain compression point of \(13\text{ dBm}\), and an \(\text{OIP3}\) of \(33\text{ dBm}\). The noise figure of the first stage is \(3\text{ dB}\) and the noise figure of the second stage is \(6\text{ dB}\). The minimum acceptable SNR, SNR\(_{\text{min}}\), at the output of the amplifier is \(16\text{ dB}\).
- What is the linear power gain of the two-stage amplifier?
- What is the output \(1\text{-dB}\) gain compression power of the two-stage amplifier for a sinusoidal RF input signal?
- What is the \(\text{OIP3}\) of the two-stage amplifier?
- What is the noise figure of the two-stage amplifier?
- What is the noise, in \(\text{dBm}\), applied to the input of the two-stage amplifier in a \(100\text{ MHz}\) bandwidth is the source has a Thevenin resistor at room temperature?
- What is the power of the noise, in \(\text{dBm}\), in a \(100\text{ MHz}\) bandwidth at the output of the two-stage amplifier?
- What is the output-referred spurious free dynamic range of the two-stage amplifier in decibels?
- What is the output-referred dynamic range of the two-stage amplifier in decibels?

## 7.10.1 Exercises By Section

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

\(§7.2\: 1†, 2, 3, 4, 5†, 6†, 7†, 8, 9, 10, 11†, 12†, 13†, 14, 15\)

\(§7.7\: 16†\)

## 7.10.2 Answers to Selected Exercises

- \(-11\text{ dBm}\)

- (e) \(5.2\text{ dBm}\)