# 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.