Skip to main content
Engineering LibreTexts

4.11: Exercises

  1. Last updated
  2. Save as PDF
  • Page ID
    46116

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    \(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)
    1. An amplifier consists of three cascaded stages with the following characteristics:
        Stage 1 Stage 2 Stage 3
      Gain (\(\text{dB}\)) \(-3\) \(15\) \(5\)
      NF (\(\text{dB}\)) \(3\) \(2\) \(2\)
      Table \(\PageIndex{1}\)
      1. What is the overall gain of the amplifier?
      2. What is the overall noise figure of the amplifier?
    2. Briefly describe the effect of a lossy filter on SNR. Consider signals at the input and output of the filter.
    3. What is the available noise power of a \(50\:\Omega\) resistor in a \(10\text{ MHz}\) bandwidth. The resistor is at standard temperature.
    4. A \(50\:\Omega\) resistor a \(20\:\Omega\) resistor are in shunt. If both resistors have a temperature of \(300\text{ K}\), what is the total available noise power spectral density of the shunt resistors?
    5. The thermal noise power at the output of a system is \(1\text{ fW}\) and the shot noise power is \(1\text{ fW}\). What is the available white noise power?
    6. A \(2\text{ GHz}\) amplifier in a \(50\:\Omega\) system has a bandwidth of \(10\text{ MHz}\), a gain of \(40\text{ dB}\), and a noise figure of \(3\text{ dB}\). The amplifier is driven by a circuit with a Thevenin equivalent resistance of \(50\:\Omega\) held at \(290\text{ K}\) (standard temperature). What is the available noise power at the output of the amplifier?
    7. A \(30\text{ dB}\) attenuator is terminated at Port \(\mathsf{2}\) in a matched resistor and both are at \(290\text{ K}\). What is the noise temperature at Port \(\mathsf{1}\) of the attenuator?
    8. A \(20\text{ dB}\) attenuator is terminated in a matched resistor and both are held at \(30^{\circ}\text{C}\). What is the noise temperature at the input of the attenuator in kelvin?
    9. The effective noise temperature at the coaxial output of an antenna is \(100\text{ K}\). The antenna is connected to a bandpass filter with a bandwidth of \(20\text{ MHz}\) and an insertion loss of \(1\text{ dB}\). [Parallels Examples 4.3.1 and 4.3.2]
      1. What is the available noise power in a \(20\text{ MHz}\) bandwidth at the output of the antenna?
      2. What is the noise figure of the bandpass filter (consider only the passband)?
      3. What is the excess noise power at the output of the filter? (Consider only the passband).
      4. What is the total available noise power in the passband at the output of the filter?
    10. A receive amplifier with a gain of \(30\text{ dB}\), a noise figure of \(2\text{ dB}\), and bandwidth of \(5\text{ MHz}\) is connected to an antenna which has a noise temperature of \(20\text{ K}\). [Parallels Example 4.3.1]
      1. What is the available noise power presented to the input of the amplifier in the \(5\text{ MHz}\) bandwidth (recall that the antenna noise temperature is \(20\text{ K}\)?
      2. If instead the input of the amplifier is connected to a resistor held at standard temperature, what is the available noise power presented to the input of the amplifier in the \(5\text{ MHz}\) bandwidth?
      3. What is the noise factor of the amplifier?
      4. What is the excess noise power of the amplifier referred to the its output?
      5. What is the effective noise temperature of the amplifier when the amplifier is connected to the antenna with a noise temperature of \(20\text{ K}\). That is, what is the effective noise temperature of the resistor in the Thevenin equivalent circuit of the amplifier output?
    11. A receive amplifier has a bandwidth of \(5\text{ MHz}\), a \(1\text{ dB}\) noise figure, a linear gain of \(20\text{ dB}\). The minimum acceptable SNR is \(10\text{ dB}\).
      1. What is the output noise power in \(\text{dBm}\)?
      2. What is the minimum detectable output signal in \(\text{dBm}\)?
      3. What is the minimum detectable input signal in \(\text{dBm}\)?
    12. A \(75\:\Omega\) attenuator has a loss of \(16\text{ dB}\) and is between a source with a Thevenin impedance of \(75\:\Omega\) and a load of \(75\:\Omega\).
      1. What is the noise power, \(N_{i}\), available from the \(75\:\Omega\) source resistor at standard temperature \((290\text{ K})\) in a \(1\text{ MHz}\) bandwidth?
      2. Now consider that the source is connected to the attenuator which is also connected to the load. If the source generates a modulated signal that is \(1\text{ MHz}\) wide and has an available power, \(S_{i}\), of \(10\text{ fW}\), what is SNR\(_{i}\) at the input to the attenuator at standard temperature?
      3. With the attenuator connected to the source, what is the Thevenin equivalent impedance looking into the output of the attenuator?
      4. Calculate the noise power, \(N_{o}\), available from the attenuator with the source attached at standard temperature \((290\text{ K})\) in a \(1\text{ MHz}\) bandwidth?
      5. What is the signal power, \(S_{o}\), delivered to the load?
      6. What is the SNR at the load, SNR\(_{o}\)?
      7. What is the noise factor of the attenuator?
      8. What is the noise figure of the attenuator?
    13. The system shown below is a receiver with bandpass filters, amplifiers, and a mixer. [Parallels Example 4.3.3]

    clipboard_eee71648f3f142417bdfd4cf5ebcddc11.png

    Figure \(\PageIndex{1}\)

    1. What is the total gain of the system?
    2. What is the noise factor of the first filter?
    3. What is the system noise factor?
    4. What is the system noise figure?
    1. An amplifier consists of three cascaded stages with the following characteristics:
        Stage 1 Stage 2 Stage 3
      Gain \(10\text{ dB}\) \(15\text{ dB}\) \(30\text{ dB}\)
      NF \(0.8\text{ dB}\) \(2\text{ dB}\) \(2\text{ dB}\)
      Table \(\PageIndex{2}\)
      What is the noise figure (NF) and gain of the cascade amplifier?
    2. The front end of a receiver for a cellular phone has a bandpass filter with a \(25\text{ MHz}\) passband and a loss in the passband of \(2\text{ dB}\) and is followed by two amplifier stages. The first stage has a gain of \(20\text{ dB}\) and a noise figure of \(0.5\text{ dB}\) and the second stage has a gain of \(60\text{ dB}\) and a noise figure of \(2\text{ dB}\).
      1. Sketch the system block diagram.
      2. What is the gain of the system?
      3. What is the noise figure of the filter?
      4. What is the noise figure of the system?
      5. The system is now connected to an antenna with an effective noise temperature of \(30\text{ K}\) that delivers a signal of \(10\text{ pW}\) to the bandpass filter. Determine the noise temperature at the output of the system and hence the output noise power in the \(25\text{ MHz}\) bandwidth. First calculate the excess noise temperature added by the system to the output.
      6. Determine the signal-to-noise ratio at the output of the front-end system.
    3. The RF front end of a communications unit consists of an amplifier followed by a mixer. The amplifier has a gain of \(20\text{ dB}\) and a noise figure of \(4\text{ dB}\). The mixer has a conversion loss of \(6\text{ dB}\) and a double-sideband noise figure of \(8\text{ dB}\).
      1. Why is the double-sideband noise figure sometimes used with a mixer but not with an amplifier?
      2. If the amplifier has sufficient bandwidth to amplify both the RF and image frequency, what is the noise figure of the cascade? Note that the overall noise figure is a single-sideband noise figure.
    4. The first stage of a two-stage amplifier has a linear gain of \(40\text{ dB}\) and a noise figure if \(3\text{ dB}\). The second stage has a gain of \(10\text{ dB}\) and a noise figure of \(5\text{ dB}\).
      1. What is the overall gain of the amplifier?
      2. What is the overall noise figure of the amplifier?
    5. A subsystem consists of a matched filter with an insertion loss of \(2\text{ dB}\) then an amplifier with a gain of \(20\text{ dB}\) and a noise figure, NF, of \(3\text{ dB}\).
      1. What is the overall gain of the subsystem?
      2. What is NF of the filter?
      3. What is NF of the subsystem?
    6. A subsystem consists of a matched amplifier with a gain of \(20\text{ dB}\) and a noise figure of \(2\text{ dB}\), followed by a \(2\text{ dB}\) attenuator, and then another amplifier with a gain of \(10\text{ dB}\) and NF of \(3\text{ dB}\).
      1. What is the overall gain of the subsystem?
      2. What is NF of the attenuator?
      3. What is NF of the subsystem?
    7. Consider a digitally modulated signal and briefly describe the impact of a nonlinear amplifier on the signal. You must include several negative effects. Use one or more diagrams.
    8. An amplifier has a linear gain of \(30\text{ dB}\) and an output-referred \(1\text{ dB}\) gain compression point of \(13\text{ dBm}\). What is the input-referred \(1\text{ dB}\) gain compression point of the amplifier?
    9. An amplifier has a linear gain of \(30\text{ dB}\) and an input-referred \(1\text{ dB}\) gain compression point of \(−30\text{ dBm}\). What is the output-referred \(1\text{ dB}\) gain compression point of the amplifier?
    10. An amplifier has an output power of \(10\text{ dBm}\) when the gain of a single tone is compressed by \(1\text{ dB}\). What is the maximum output power of an undistorted 64-QAM signal? (A 64-QAM signal which has a PMEPR of \(7.8\text{ dB}\). [Parallels Example 4.5.1]
    11. The input-referred \(1\text{ dB}\) gain compression point of an amplifier with a linear gain of \(30\text{ dB}\) is \(0\text{ dBm}\).
      1. What is the gain of the amplifier at \(1\text{ dB}\) gain compression?
      2. What is the output power at \(1\text{ dB}\) gain compression?
      3. Consider amplifying an 8-PSK signal with a PMEPR of \(3.3\text{ dB}\). What is the maximum output power of the undistorted 8-PSK signal? Maximum acceptable distortion is when the envelope peak is compressed by \(1\text{ dB}\) gain.
    12. The gain of an amplifier at the \(1\text{-dB}\) gain compression point is \(40\text{ dB}\) and the input power is \(−7\text{ dBm}\).
      1. What is the power of the amplifier’s output signal?
      2. If the power input to the amplifier is reduced to \(−20\text{ dBm}\), what is the amplifier’s output power now?
    13. Briefly describe intermodulation distortion with a a two-tone signal. Use a diagram.
    14. Briefly describe what is meant by \(1\text{ dB}\) gain compression. Use a diagram.
    15. A single-stage amplifier has a linear gain of \(16\text{ dB}\), 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 noise floor at the output of the amplifier is \(−60\text{ dBm}\). The communication protocol has a minimum SNR, SNR\(_{\text{MIN}}\), of \(6\text{ dB}\).
      1. What is the dynamic range of the amplifier?
      2. What is the SFDR of the amplifier?
    16. A receiver system comprising a filter and two cascaded amplifiers has an overall linear gain of \(80\text{ dB}\), an output \(1\text{ dB}\) gain compression point of \(−10\text{ dBm}\), and an output-referred third-order intercept point, \(\text{OIP3} = 10\text{ dBm}\). The noise floor at the output of the amplifier is \(−80\text{ dBm}\). What is the spurious free dynamic range of the receiver?
    17. A power amplifier has a linear 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 \(\text{OIP3} = 60\text{ dBm}\). The noise floor at the output of the amplifier is \(−70\text{ dBm}\). What is the dynamic range of the amplifier if the required minimum SNR at the output is \(6\text{ dB}\)?
    18. A room-temperature two-stage amplifier in a receiver has a bandwidth of \(100\text{ MHz}\), a noise figure of \(3\text{ dB}\), a linear gain of \(32\text{ dB}\), and an output-referred third-order intercept point, \(\text{OIP3}\), of \(27\text{ dBm}\). The minimum SNR of the receiver system is \(16\text{ dB}\).
      1. What is the output noise power in \(\text{dBm}\)?
      2. What is the difference between the input-and output-referred spurious free dynamic ranges?
      3. What is the SFDR in \(\text{dB}\)?
      4. What is the difference between the input-and output-referred dynamic ranges?
      5. What is the minimum detectable output signal in \(\text{dBm}\)?
      6. What is the output-referred DR in \(\text{dB}\)?
    19. When determining the dynamic range of an amplifier the gain compression level is not used. Briefly discuss why.

    Exercises by Section

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

    \(§4.2\: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\)

    \(§4.3\: 12†, 13, 14, 15†, 16†, 17, 18, 19\)

    \(§4.5\: 20, 21, 22, 23, 24, 25, 26, 27\)

    \(§4.6\: 28, 29, 30, 31, 32\)

    Answers to Selected Exercises

    1. (g) \(39.8\)
    2. (d) \(5.17\text{ dB}\)
    1. (b) \(3\text{ dB}\)
    1. (b) \(60\text{ dB}\)

    4.11: Exercises is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

    1. Back to top
    • Was this article helpful?

    Recommended articles

    1. Article type
      Section or Page
      Autonumber Section Headings
      title with space delimiters
    2. Tags
      This page has no tags.