# 1.7: Exercises

- Page ID
- 41256

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- What is the wavelength in free space of a signal at \(4.5\text{ GHz}\)?
- Consider a monopole antenna that is a quarter of a wavelength long. How long is the antenna if it operates at \(3\text{ kHz}\)?
- Consider a monopole antenna that is a quarter of a wavelength long. How long is the antenna if it operates at \(500\text{ MHz}\)?
- Consider a monopole antenna that is a quarter of a wavelength long. How long is the antenna if it operates at \(2\text{ GHz}\)?
- A dipole antenna is half of a wavelength long. How long is the antenna at \(2\text{ GHz}\)?
- A dipole antenna is half of a wavelength long. How long is the antenna at \(1\text{ THz}\)?
- 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]
- 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]
- The logarithm to base \(2\) of a number \(x\) is \(0.38\) (i.e., \(\log_{2}(x)=0.38\)). What is \(x\)?
- The natural logarithm of a number \(x\) is \(2.5\) (i.e., \(\ln (x)=2.5\)). What is \(x\)?
- The logarithm to base \(2\) of a number \(x\) is \(3\) (i.e., \(\log_{2}(x)=3\)). What is \(\log_{2}(\sqrt[2]{x})\)?
- What is \(\log_{3}(10)\)?
- What is \(\log_{4.5}(2)\)?
- Without using a calculator evaluate log \(\{[\log_{3} (3x) − \log_{3} (x)]\}\).
- 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.
- What is the power dissipated in the resistor in watts?
- What is the power dissipated in the resistor in \(\text{dBm}\)?

- The power of an RF signal is \(10\text{ mW}\). What is the power of the signal in \(\text{dBm}\)?
- The power of an RF signal is \(40\text{ dBm}\). What is the power of the signal in watts?
- An amplifier has a power gain of \(2100\).
- What is the power gain in decibels?
- If the input power is \(−5\text{ dBm}\), what is the output power in \(\text{dBm}\)? [Parallels Example 1.3.2]

- An amplifier has a power gain of \(6\). What is the power gain in decibels? [Parallels Example 1.3.2]
- A filter has a loss factor of \(100\). [Parallels Example 1.3.2]
- What is the loss in decibels?
- What is the gain in decibels?

- An amplifier has a power gain of \(1000\). What is the power gain in \(\text{dB}\)? [Parallels Example 1.3.2]
- 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 \(\text{dBm}\)?
- 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]
- 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]
- If the power input to the amplifier is \(10\text{ mW}\), what is the total power radiated by the antenna in \(\text{dBm}\)?
- What is the radiated power in watts?

- 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.
- What is the output power of the filter in watts?
- What is the output power of the filter in \(\text{dBW}\)?

- 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 \(\text{dB}\)?
- 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?
- 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?
- 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]
- 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]
- An amplifier has a power gain of \(1900\).
- What is the power gain in decibels?
- If the input power is \(−8\text{ dBm}\), what is the output power in \(\text{dBm}\)? [Parallels Example 1.3.2]

- An amplifier has a power gain of \(20\).
- What is the power gain in decibels?
- If the input power is \(−23\text{ dBm}\), what is the output power in \(\text{dBm}\)? [Parallels Example 1.3.2]

- An amplifier has a voltage gain of \(10\) and a current gain of \(100\).
- What is the power gain as an absolute number?
- What is the power gain in decibels?
- If the input power is \(−30\text{ dBm}\), what is the output power in \(\text{dBm}\)?
- What is the output power in \(\text{mW}\)?

- 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?
- 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

- \(3.25\text{ cm}\)

- \(2.096\)

- \(10\text{ dBm}\)
- \(10\text{ W}\)

- \(7.782\text{ dB}\)

- \(1.301\)
- \(50.12\text{ mW}\)
- (b) \(3.162\text{ W}\)