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2.7: Exercises

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    1. An antenna only radiates \(45\%\) of the power input to it. The rest is lost as heat. What input power (in \(\text{dBm}\)) is required to radiate \(30\text{ dBm}\)?
    2. The output stage of an RF front end consists of an amplifier followed by a filter and then an antenna. The amplifier has a gain of \(27\text{ dB}\), the filter has a loss of \(1.9\text{ dB}\), and of the power input to the antenna, \(35\%\) is lost as heat due to resistive losses. If the power input to the amplifier is \(30\text{ dBm}\), calculate the following:
      1. What is the power input to the amplifier in watts?
      2. Express the loss of the antenna in \(\text{dB}\).
      3. What is the total gain of the RF front end (amplifier + filter)?
      4. What is the total power radiated by the antenna in \(\text{dBm}\)?
      5. What is the total power radiated by the antenna in \(\text{mW}\)?
    3. The output stage of an RF front end consists of an amplifier followed by a filter and then an antenna. The amplifier has a gain of \(27\text{ dB}\), the filter has a loss of \(1.9\text{ dB}\), and of the power input to the antenna, \(45\%\) is lost as heat due to resistive losses. If the power input to the amplifier is \(30\text{ dBm}\), calculate the following:
      1. What is the power input to the amplifier in watts?
      2. Express the loss of the antenna in decibels.
      3. What is the total gain of the RF front end (amplifier + filter)?
      4. What is the total power radiated by the antenna in \(\text{dBm}\)?
      5. What is the total power radiated by the antenna in milliwatts?
    4. Thirty five percent of the power input to an antenna is lost as heat, what is the loss of the antenna in \(\text{dB}\).
    5. Only \(65\%\) of the power input to an antenna is radiated with the rest lost to dissipation in the antenna, what is the gain of the antenna in \(\text{dB}\)? (This is not the antenna gain.)
    6. The efficiency of an antenna is \(66\%\). If the power input to the antenna is \(10\text{ W}\) what is the power radiated by the antenna in \(\text{dBm}\)?
    7. An antenna with an input of \(1\text{ W}\) operates in free space and has an antenna gain of \(12\text{ dBi}\). What is the maximum power density at \(100\text{ m}\) from the antenna?
    8. A transmitter has an antenna with an antenna gain of \(10\text{ dBi}\), the resistive losses of the antenna are \(50\%\), and the power input to the antenna is \(1\text{ W}\). What is the \(\text{EIRP}\) in watts?
    9. A transmitter has an antenna with an antenna gain of \(20\text{ dBi}\), the resistive losses of the antenna are \(50\%\), and the power input to the antenna is \(100\text{ mW}\). What is the \(\text{EIRP}\) in watts?
    10. An antenna with an antenna gain of \(8\text{ dBi}\) radiates \(6.67\text{ W}\). What is the \(\text{EIRP}\) in watts? Assume that the antenna is \(100\%\) efficient.
    11. An antenna has an antenna gain of \(10\text{ dBi}\) and a \(40\text{ W}\) input signal. What is the \(\text{EIRP}\) in watts?
    12. An antenna with \(5\text{ W}\) of input power has an antenna gain of \(20\text{ dBi}\) and an antenna efficiency of \(25\%\) and all of the loss is due to resistive losses in the antenna. [Parallels Example 2.5.2]
      1. How much power in \(\text{dBm}\) is lost as heat in the antenna?
      2. How much power in \(\text{dBm}\) is radiated by the antenna?
      3. What is the \(\text{EIRP}\) in \(\text{dBW}\)?
    13. An antenna with an efficiency of \(50\%\) has an antenna gain of \(12\text{ dBi}\) and radiates \(100\text{ W}\). What is the \(\text{EIRP}\) in watts?
    14. An antenna with an efficiency of \(75\%\) and an antenna gain of \(10\text{ dBi}\). If the power input to the antenna is \(100\text{ W}\),
      1. What is the total power in \(\text{dBm}\) radiated by the antenna? (b) What is the \(\text{EIRP}\) in \(\text{dBm}\)?
    15. The output stage of an RF front end consists of an amplifier followed by a filter and then an antenna. The amplifier has a gain of \(27\text{ dB}\), the filter has a loss of \(1.9\text{ dB}\), and of the power input to the antenna, \(45\%\) is lost as heat due to resistive losses. If the power input to the amplifier is \(30\text{ dBm}\), calculate the following:
      1. What is the power input to the amplifier in watts?
      2. Express the loss of the antenna in decibels.
      3. What is the total gain of the RF front end (amplifier + filter)?
      4. What is the total power radiated by the antenna in \(\text{dBm}\)?
      5. What is the total power radiated by the antenna in milliwatts?
    16. A communication system operating at \(2.5\text{ GHz}\) includes a transmit antenna with an antenna gain of \(12\text{ dBi}\) and a receive antenna with an effective aperture area of \(20\text{ cm}^{2}\). The distance between the two antennas is \(100\text{ m}\).
      1. What is the antenna gain of the receive antenna?
      2. If the input to the transmit antenna is \(1\text{ W}\), what is the power density at the receive antenna if the power falls off as \(1/d^{2}\), where \(d\) is the distance from the transmit antenna?
      3. Thus what is the power delivered at the output of the receive antenna?
    17. Consider a point-to-point communication system. Parabolic antennas are mounted high on a mast so that ground effects do not exist, thus power falls off as \(1/d^{2}\). The gain of the transmit antenna is \(20\text{ dBi}\) and the gain of the receive antenna is \(15\text{ dBi}\). The distance between the antennas is \(10\text{ km}\). The effective area of the receive antenna is \(3\text{ cm}^{2}\). If the power input to the transmit antenna is \(600\text{ mW}\), what is the power delivered at the output of the receive antenna?
    18. Consider a \(28\text{ GHz}\) point-to-point communication system. Parabolic antennas are mounted high on a mast so that ground effects do not exist, thus power falls off as \(1/d^{2}\). The gain of the transmit antenna is \(20\text{ dBi}\) and the gain of the receive antenna is \(15\text{ dBi}\). The distance between the antennas is \(10\text{ km}\). If the power output from the receive antenna is \(10\text{ pW}\), what is the power input to the transmit antenna?
    19. An antenna has an effective aperture area of \(20\text{ cm}^{2}\). What is the antenna gain of the antenna at \(2.5\text{ GHz}\)?
    20. An antenna operating at \(28\text{ GHz}\) has an antenna gain of \(50\text{ dBi}\). What is the effective aperture area of the antenna?
    21. A \(15\text{ GHz}\) receive antenna has an antenna gain of \(20\text{ dBi}\). If the power density at the receive antenna is \(1\text{ nW/cm}^{2}\), what is the power at the output of the antenna? [Parallels Example 2.6.2]
    22. Two identical antennas are used in a pointto-point communication system, each having a gain of \(50\text{ dBi}\). The system has an operating frequency of \(28\text{ GHz}\) and the antennas are at the top of masts \(100\text{ m}\) tall. The RF link between the antennas consists only of the direct line-of-sight path.
      1. What is the effective aperture area of each antenna?
      2. How does the power density of the propagating signal rolloff with distance.
      3. If the separation of the transmit and receive antennas is \(10\text{ km}\), what is the path loss in decibels?
      4. If the separation of the transmit and receive antennas is \(10\text{ km}\), what is the link loss in decibels?
    23. A transmitter and receiver operating at \(2\text{ GHz}\) are at the same level, but the direct path between them is blocked by a building and the signal must diffract over the building for a communication link to be established. This is a classic knife-edge diffraction situation. The transmit and receive antennas are each separated from the building by \(4\text{ km}\) and the building is \(20\text{ m}\) higher than the antennas (which are at the same height). Consider that the building is very thin. It has been found that the path loss can be determined by considering loss due to free-space propagation and loss due to diffraction over the knife edge.
      1. What is the additional attenuation (in decibels) due to diffraction?
      2. If the operating frequency is \(100\text{ MHz}\), what is the attenuation (in decibels) due to diffraction?
      3. If the operating frequency is \(10\text{ GHz}\), what is the attenuation (in decibels) due to diffraction?
    24. A hill is \(1\text{ km}\) from a transmit antenna and \(2\text{ km}\) from a receive antenna. The receive and transmit antennas are at the same height and the hill is \(20\text{ m}\) above the height of the antennas. What is the additional loss caused by diffraction over the top of the hill? Treat the hill as causing knifeedge diffraction and the operating frequency is \(1\text{ GHz}\).
    25. Two identical antennas are used in a pointto-point communication system, each having a gain of \(30\text{ dBi}\). The system has an operating frequency of \(14\text{ GHz}\) and the antennas are at the top of masts \(100\text{ m}\) tall. The RF link between the antennas consists only of the direct LOS path.
      1. What is the effective aperture area of each antenna?
      2. How does the power density of the propagating signal rolloff with distance?
      3. If the separation of the transmit and receive antennas is \(10\text{ km}\), what is the path loss? Ignore atmospheric loss.
    26. The three main cellular communication bands are centered around \(450\text{ MHz},\: 900\text{ MHz},\) and \(2\text{ GHz}\). Compare these three bands in terms of multipath effects, diffraction around buildings, object (such as a wall) penetration, scattering from trees and parts of trees, and ability to follow the curvature of hills. Complete the table below with the relative attributes: high, medium, and low.
      Characteristic \(450\text{ MHz}\) \(900\text{ MHz}\) \(2\text{ GHz}\)
      Multipath
      Scattering
      Penetration
      Following curvature
      Range
      Antenna size
      Atmospheric loss

      Table \(\PageIndex{1}\)

    27. Describe the difference in multipath effects in a central city area compared to multipath effects in a desert. Your description should be approximately \(4\) lines long and not use a diagram
    28. Wireless LAN systems can operate at \(2.4\text{ GHz},\: 5.6\text{ GHz},\: 40\text{ GHz}\) and \(60\text{ GHz}\). Contrast with explanation the performance of these schemes inside a building in terms of range.
    29. At \(60\text{ GHz}\) the atmosphere strongly attenuates a signal. Discuss the origin of this and indicate an advantage and a disadvantage.
    30. Short answer questions. Each part requires a short paragraph of about five lines and a figure, where appropriate, to illustrate your understanding.
      1. Cellular communications systems use two frequency bands to communicate between the basestation and the mobile unit. The bands are generally separated by \(50\text{ MHz}\) or so. Which band (higher or lower) is used for the downlink from the basestation to the mobile unit and what are the reasons behind this choice?
      2. Describe at least two types of interference in a cellular system from the perspective of a mobile handset.
    31. The three main cellular communication bands are centered around \(450\text{ MHz},\: 900\text{ MHz},\) and \(2\text{ GHz}\). Compare these three bands in terms of multipath effects, diffraction around buildings, object (such as a wall) penetration, scattering from trees and parts of trees, and the ability to follow the curvature of hills. Use a table and indicate the relative attributes: high, medium, and low.
    32. Describe Rayleigh fading in approximately \(4\) lines and without using a diagram.
    33. In several sentences and using a diagram describe Rayleigh fading and the impact it has on radio communications.
    34. A transmitter and receiver operate at \(100\text{ MHz}\), are at the same level, and are separated by \(4\text{ km}\). The signal must diffract over a building half way between the antennas that is \(20\text{ m}\) higher than the direct path between the antennas. What is the attenuation (in decibels) due to diffraction?
    35. A transmitter and receiver operate at \(10\text{ GHz}\), are at the same level, and are \(4\text{ km}\) apart. The signal must diffract over a building that is half way between the antennas and is \(20\text{ m}\) higher than the line between the antennas. What is the attenuation (in \(\text{dB}\)) due to diffraction?

    2.10.1 Exercises By Section

    \(†\)challenging

    \(§2.3 1†\)

    \(§2.5 2†, 3†, 4, 5, 6†, 7†, 8†, 19, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21\)

    \(§2.6 22†, 23†, 24†, 25†, 27†, 28, 29, 30†, 31, 32, 33, 34, 35\)

    2.10.2 Answers to Selected Exercises

    1. (e)\(53.23\text{ dBm}\)
    1. \(0.251\:\mu\text{W}\)
    2. \(14.3\text{ pW}\)

    This page titled 2.7: Exercises is shared under a CC BY-NC license and was authored, remixed, and/or curated by Michael Steer.

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