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1.7: Exercises

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    1. What is the wavelength in free space of a signal at \(4.5\text{ GHz}\)?
    2. Consider a monopole antenna that is a quarter of a wavelength long. How long is the antenna if it operates at \(3\text{ kHz}\)?
    3. Consider a monopole antenna that is a quarter of a wavelength long. How long is the antenna if it operates at \(500\text{ MHz}\)?
    4. Consider a monopole antenna that is a quarter of a wavelength long. How long is the antenna if it operates at \(2\text{ GHz}\)?
    5. A dipole antenna is half of a wavelength long. How long is the antenna at \(2\text{ GHz}\)?
    6. A dipole antenna is half of a wavelength long. How long is the antenna at \(1\text{ THz}\)?
    7. A transmitter transmits an FM signal with a bandwidth of \(100\text{ kHz}\) and the signal is received by a receiver at a distance \(r\) from the transmitter. When \(r = 1\text{ km}\) the signal power received by the receiver is \(100\text{ nW}\). When the receiver moves further away from the transmitter the power received drops off as \(1/r^{2}\). What is \(r\) in kilometers when the received power is \(100\text{ pW}\). [Parallels Example 1.3.1]
    8. A transmitter transmits an AM signal with a bandwidth of \(20\text{ kHz}\) and the signal is received by a receiver at a distance \(r\) from the transmitter. When \(r = 10\text{ km}\) the signal power received is \(10\text{ nW}\). When the receiver moves further away from the transmitter the power received drops off as \(1/r^{2}\). What is \(r\) in kilometers when the received power is equal to the received noise power of \(1\text{ pW}\)? [Parallels Example 1.3.1]
    9. The logarithm to base \(2\) of a number \(x\) is \(0.38\) (i.e., \(\log_{2}(x)=0.38\)). What is \(x\)?
    10. The natural logarithm of a number \(x\) is \(2.5\) (i.e., \(\ln (x)=2.5\)). What is \(x\)?
    11. The logarithm to base \(2\) of a number \(x\) is \(3\) (i.e., \(\log_{2}(x)=3\)). What is \(\log_{2}(\sqrt[2]{x})\)?
    12. What is \(\log_{3}(10)\)?
    13. What is \(\log_{4.5}(2)\)?
    14. Without using a calculator evaluate log \(\{[\log_{3} (3x) − \log_{3} (x)]\}\).
    15. A \(50\:\Omega\) resistor has a sinusoidal voltage across it with a peak voltage of \(0.1\text{ V}\). The RF voltage is \(0.1\cos (\omega t)\), where \(\omega\) is the radian frequency of the signal and \(t\) is time.
      1. What is the power dissipated in the resistor in watts?
      2. What is the power dissipated in the resistor in \(\text{dBm}\)?
    16. The power of an RF signal is \(10\text{ mW}\). What is the power of the signal in \(\text{dBm}\)?
    17. The power of an RF signal is \(40\text{ dBm}\). What is the power of the signal in watts?
    18. An amplifier has a power gain of \(2100\).
      1. What is the power gain in decibels?
      2. If the input power is \(−5\text{ dBm}\), what is the output power in \(\text{dBm}\)? [Parallels Example 1.3.2]
    19. An amplifier has a power gain of \(6\). What is the power gain in decibels? [Parallels Example 1.3.2]
    20. A filter has a loss factor of \(100\). [Parallels Example 1.3.2]
      1. What is the loss in decibels?
      2. What is the gain in decibels?
    21. An amplifier has a power gain of \(1000\). What is the power gain in \(\text{dB}\)? [Parallels Example 1.3.2]
    22. An amplifier has a gain of \(14\text{ dB}\). The input to the amplifier is a \(1\text{ mW}\) signal, what is the output power in \(\text{dBm}\)?
    23. An RF transmitter consists of an amplifier with a gain of \(20\text{ dB}\), a filter with a loss of \(3\text{ dB}\) and then that is then followed by a lossless transmit antenna. If the power input to the amplifier is \(1\text{ mW}\), what is the total power radiated by the antenna in \(\text{dBm}\)? [Parallels Example 1.3.4]
    24. The final stage of an RF transmitter consists of an amplifier with a gain of \(30\text{ dB}\) and a filter with a loss of \(2\text{ dB}\) that is then followed by a transmit antenna that looses half of the RF power as heat. [Parallels Example 1.3.4]
      1. If the power input to the amplifier is \(10\text{ mW}\), what is the total power radiated by the antenna in \(\text{dBm}\)?
      2. What is the radiated power in watts?
    25. A \(5\text{ mW}\)-RF signal is applied to an amplifier that increases the power of the RF signal by a factor of \(200\). The amplifier is followed by a filter that losses half of the power as heat.
      1. What is the output power of the filter in watts?
      2. What is the output power of the filter in \(\text{dBW}\)?
    26. The power of an RF signal at the output of a receive amplifier is \(1\:\mu\text{W}\) and the noise power at the output is \(1\text{ nW}\). What is the output signal-tonoise ratio in \(\text{dB}\)?
    27. The power of a received signal is \(1\text{ pW}\) and the received noise power is \(200\text{ fW}\). In addition the level of the interfering signal is \(100\text{ fW}\). What is the signal-to-noise ratio in \(\text{dB}\)? Treat interference as if it is an additional noise signal.age gain of \(1\) has an input impedance of \(100\:\Omega\), a zero output impedance, and drives a \(5\:\Omega\) load. What is the power gain of the amplifier?
    28. A transmitter transmits an FM signal with a bandwidth of \(100\text{ kHz}\) and the signal power received by a receiver is \(100\text{ nW}\). In the same bandwidth as that of the signal the receiver receives \(100\text{ pW}\) of noise power. In decibels, what is the ratio of the signal power to the noise power, i.e. the signal-to-noise ratio (SNR) received by the receiver?
    29. An amplifier with a voltage gain of \(20\) has an input resistance of \(100\:\Omega\) and an output resistance of \(50\:\Omega\). What is the power gain of the amplifier in decibels? [Parallels Example 1.3.1]
    30. An amplifier with a voltage gain of \(1\) has an input resistance of \(100\:\Omega\) and an output resistance of \(5\:\Omega\). What is the power gain of the amplifier in decibels? Explain why there is a power gain of more than \(1\) even though the voltage gain is \(1\). [Parallels Example 1.3.1]
    31. An amplifier has a power gain of \(1900\).
      1. What is the power gain in decibels?
      2. If the input power is \(−8\text{ dBm}\), what is the output power in \(\text{dBm}\)? [Parallels Example 1.3.2]
    32. An amplifier has a power gain of \(20\).
      1. What is the power gain in decibels?
      2. If the input power is \(−23\text{ dBm}\), what is the output power in \(\text{dBm}\)? [Parallels Example 1.3.2]
    33. An amplifier has a voltage gain of \(10\) and a current gain of \(100\).
      1. What is the power gain as an absolute number?
      2. What is the power gain in decibels?
      3. If the input power is \(−30\text{ dBm}\), what is the output power in \(\text{dBm}\)?
      4. What is the output power in \(\text{mW}\)?
    34. An amplifier with \(50\:\Omega\) input impedance and \(50\:\Omega\) load impedance has a voltage gain of \(100\). What is the (power) gain in decibels?
    35. An attenuator reduces the power level of a signal by \(75\%\). What is the (power) gain of the attenuator in decibels?

    1.7.1 Exercises by Section

    †challenging

    \(§1.2 1, 2, 3, 4, 5, 6, 7, 8 \)

    \(§1.3 9, 10, 11, 12, 13, 14, 15, 16, 17 18, 19, 20, 21, 22, 23† , 24†, 25† 26, 27, 28, 29, 30, 31, 32, 33, 34, 35\)

    1.7.2 Answers to Selected Exercises

    1. \(3.25\text{ cm}\)
    1. \(2.096\)
    1. \(10\text{ dBm}\)
    2. \(10\text{ W}\)
    1. \(7.782\text{ dB}\)
    1. \(1.301\)
    2. \(50.12\text{ mW}\)
    3. (b) \(3.162\text{ W}\)

    This page titled 1.7: Exercises is shared under a CC BY-NC license and was authored, remixed, and/or curated by Michael Steer.

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