# 6.14: Exercises

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1. A mixer has an LO at $$28.2\text{ GHz}$$. The mixer is used to convert a signal at $$28.1\text{ GHz}$$ to an IF at $$100\text{ MHz}$$, and has a conversion loss of $$13\text{ dB}$$ and an image rejection of $$40\text{ dB}$$. Two signals are presented to the mixer, one at $$28.1\text{ GHz}$$ with a power of $$1\text{ pW}$$ and the other at $$28.3\text{ GHz}$$ with a power of $$10\:\mu\text{W}$$. [Parallels Example 6.2.1]
1. What is the power of the (intended) signal at the IF in $$\text{dBm}$$?
2. What is the signal-to-interference ratio at the IF (ignoring noise)?
2. Consider the single-ended diode mixer in Figure 6.2.5(a).
1. Develop a symbolic expression for the voltage at the test point. The diode is modeled by $$i_{D} = a_{1}v_{D} + a_{2}v^{2}_{D} + a_{3}v^{3}_{D}$$.
2. What are the frequencies and amplitudes of the components of the spectrum at the test point?
3. A mixer in a receiver has a conversion loss of $$16\text{ dB}$$. If the applied RF signal has an available power of $$100\:\mu\text{W}$$, what is the available power of the IF at the output of the mixer?
4. The RF signal applied to the input of a mixer has a power of $$1\text{ nW}$$ and the output of the mixer at the IF has a power level of $$100\text{ pW}$$. What is the conversion loss of the mixer in decibels?
5. A mixer in a receiver has a conversion gain of $$10\text{ dB}$$. If the applied RF signal has a power of $$100\:\mu\text{W}$$, what is the available power of the IF at the output of the mixer?
6. A mixer in a receiver has a conversion loss of $$6\text{ dB}$$. If the applied RF signal has a power of $$1\:\mu\text{W}$$, what is the available power of the IF at the output of the mixer?
7. A mixer has an LO at $$18\text{ GHz}$$. The mixer is used to convert a signal at $$18.5\text{ GHz}$$ to an IF at $$500\text{ MHz}$$. Two signals are presented to the mixer, one at $$18.5\text{ GHz}$$ with a power of $$100\text{ nW}$$ and an interfering signal at $$17.5\text{ GHz}$$ with a power of $$10\text{ nW}$$. If the image rejection is $$20\text{ dB}$$ and the conversion loss is $$10\text{ dB}$$.
1. What is the signal power at the IF?
2. What is the interference power at the IF?
3. What is the signal-to-interference ratio (in decibels) at the IF?
8. A mixer has an LO at $$100\text{ GHz}$$. The mixer is used to convert a signal at $$110\text{ GHz}$$ to an IF at $$10\text{ GHz}$$. Two signals are presented to the mixer, one at $$110\text{ GHz}$$ with a power of $$10\text{ nW}$$ and an interfering signal at $$90\text{ GHz}$$ with a power of $$5\text{ nW}$$. If the image rejection is $$40\text{ dB}$$ and the conversion loss is $$20\text{ dB}$$, what is the signal-to-interference ratio (in decibels) at the IF?
9. A mixer has an LO at $$18\text{ GHz}$$. The mixer is used to convert a signal at $$18.5\text{ GHz}$$ to an IF at $$500\text{ MHz}$$. The RF signal at $$18.5\text{ GHz}$$ has a power of $$100\text{ pW}$$. In addition, noise with a power of $$1\text{ pW}$$ is applied to the mixer at $$18.5\text{ GHz}$$ and $$17.5\text{ GHz}$$ ($$1\text{ pW}$$ at $$18.5\text{ GHz}$$ and $$1\text{ pW}$$ at $$17.5\text{ GHz}$$). If the image rejection is $$6\text{ dB}$$ and the conversion loss is $$10\text{ dB}$$. Ignore noise contributions from the mixer. What is the SNR (in decibels) at the IF?
10. A mixer in a communication system has an LO at $$5.5\text{ GHz}$$. The mixer is used to convert a $$10\text{ MHz}$$ bandwidth signal at $$5.6\text{ GHz}$$ to an IF at $$100\text{ MHz}$$. The RF signal at $$5.6\text{ GHz}$$ has a power of $$100\text{ pW}$$. The image rejection is ideal and the conversion loss is $$10\text{ dB}$$. The mixer has a single-sideband noise figure of $$6\text{ dB}$$.
1. What is the noise power at the input if the source is held at standard temperature ($$290\text{ K}$$)?
2. What is the input SNR (in decibels)?
3. What is the SNR (in decibels) at the IF?
11. The double-balanced ring diode mixer shown below has the special characteristic that the LO and RF tones are suppressed at the IF output port. Develop a symbolic expression for the voltage at the IF port. The diodes are matched and are modeled by $$i_{D} = a_{1}v_{D} + a_{2}v^{2}_{D} + a_{3}v^{3}_{D}$$. The LO voltage, at the LO terminal, is $$v_{\text{LO}} = A \cos(\omega_{\text{LO}}t)$$ and the RF voltage, at the RF terminal, is $$v_{\text{RF}} = B \cos(\omega_{\text{RF}}t)$$. Consider a $$1:1$$ winding ratio. That is, the number of windings on the secondary on each side of the center tap is equal to the number of windings on the primary.

1. A diode double-balanced mixer has an LO at $$100\text{ GHz}$$ and has an input RF signal of $$101\text{ GHz}$$. What will be the frequencies of the main signals at the IF?
2. The phase noise of an oscillator was measured as $$−120\text{ dBc/Hz}$$ at $$10\text{ kHz}$$ offset. What is the normalized phase noise at $$1\text{ MHz}$$ offset, assuming that the phase noise power varies as the inverse of frequency?
3. The phase noise of an oscillator was measured as $$−130\text{ dBc/Hz}$$ at $$10\text{ kHz}$$ offset. What is the normalized phase noise at $$1\text{ MHz}$$ offset, assuming that the phase noise power varies as the inverse of frequency?
4. The phase noise of an oscillator was measured as $$−125\text{ dBc/Hz}$$ at $$100\text{ kHz}$$ offset. What is the normalized phase noise at $$1\text{ MHz}$$ offset, assuming that the phase noise power varies as the square of the inverse of frequency?
5. The phase noise of an oscillator was measured as $$−125\text{ dBc/Hz}$$ at $$100\text{ kHz}$$ offset. What is the normalized phase noise at $$1\text{ MHz}$$ offset, assuming that the phase noise power varies inversely with frequency offset?
6. When $$0\text{ V}$$ is applied to a VCO, the output frequency is $$1\text{ GHz}$$. When the input to the VCO is $$10\text{ mV}$$, the sinusoidal output of the VCO has a frequency of $$1.01\text{ GHz}$$. What is the tuning gain of the VCO?
7. If a sinusoidal voltage is applied to the input of an analog VCO, describe the signal at the output of the VCO.
8. Describe the design of a times-two frequency divider using a frequency multiplier based on a diode and one or more bandpass filters. That is, sketch the circuit at the block diagram level.
9. Describe the design of a times-three frequency divider using a frequency multiplier based on a diode and one or more bandpass filters. That is, sketch the circuit at the block diagram level.

## 6.14.1 Exercises By Section

$$†$$challenging, $$‡$$very challenging

$$§6.2 1, 2†, 3, 4, 5, 6, 7†, 8†, 9†, 10†$$

$$§6.3 11‡, 12$$

$$§6.4 13†, 14†, 15†, 16$$

$$§6.5 17, 18$$

$$§6.9 19, 20$$

## 6.14.2 Answers to Selected Exercises

1. $$-26\text{ dBm}$$
1. $$0\text{ dBm}$$
1. (b) $$10\text{ pW}$$
1. $$19.0\text{ dB}$$
2. (b) $$28\text{ dB}$$
1. One signal at $$1\text{ GHz}$$
1. $$1\text{ GHz/V}$$

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