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## Footnotes

### Chapter 1

1. The transit frequency is defined as the frequency at which the small-signal current gain of a device falls to unity.

2. In some cases, the modulator and the upconverter are one and the same. In some other cases, the modulation is performed in the digital domain before upconversion. Most receivers demodulate and detect the signal digitally, requiring only a downconverter in the analog domain.

### Chapter 2

1. It is helpful to remember that, for n = 1, each impulse in the above summation has an area of 1 and the corresponding sinusoid, a peak amplitude of 2.

2. Note that this expression should be considered as a fit across the signal swings of interest rather than as a Taylor expansion in the vicinity of x = 0. These two views may yield slightly different values for αj.

3. The factor k is necessary to ensure a proper dimension for y(t).

4. This effect is akin to the fact that nonlinearity can also be viewed as variation of the slope of the input/output characteristic with the input level.

5. Since a tone contains no randomness, it generally does not corrupt a signal. But a tone appearing in the spectrum of a signal may make the detection difficult.

6. It is assumed that no compression occurs so that the output fundamental tones also rise by 6 dB.

7. Note that this relationship holds for a third-order system and not necessarily if higher-order terms manifest themselves.

8. As seen in the next section, second-order nonlinearity also affects the IP3 in cascaded systems.

9. The spectrum of A cos ωt consists of two impulses, each with a weight of A/2. We drop the factor of 1/2 in the figures for simplicity.

10. As explained later, this is true even with a zero average current.

11. In practice, we make a guess for T, calculate Pn, increase T, recalculate Pn, and repeat until consecutive values of Pn become nearly equal.

12. This is also the conceptual operation of spectrum analyzers.

13. In the theory of signals and systems, the PSD is defined as the Fourier transform of the autocorrelation of a signal. These two views are equivalent.

14. Also called “spot noise.”

15. Recall that ideal inductors and capacitors store energy but do not dissipate it.

16. Strictly speaking, this is not correct because the noise of a receiving antenna is in fact given by the “background” noise (e.g., cosmic radiation). However, in RF design, the antenna noise is commonly assumed to be 4kTRrad.

17. More accurately, , where gd0 is the drain-source conductance in the triode region (even though the noise is measured in saturation) [3].

18. Because the input signal and the input noise are attenuated by the same factor.

19. We assume for simplicity that the reactive components of the input and output impedances are nulled but the final result is valid even if they are not.

20. Recall from Example 2.19 that the output noise of a circuit may depend on the source impedance driving it, but the source impedance noise is excluded from .

21. For simplicity, we assume the reactive parts of the impedances are cancelled but the final result is valid even if they are not.

22. Note that in conversion to dB or dBm, we take 10 log because these are power quantities.

23. Note that the integrated noise is a single value (e.g., 100 μVrms), not a density.

24. The term “L” is used because the capacitor and the inductor form the letter L in the circuit diagram.

25. This section can be skipped in a first reading.

26. From another point of view, in V0 exp(1t) = V0 cos ω1t + jV0 sin ω1t, the first term generates its own response, as does the second term; the two responses remain distinguishable by virtue of the factor j.

27. Other terms are excluded because they do not lead to a component at ω1 + ω2 + ω3.

### Chapter 3

1. Note that m has a dimension of 1/volt if xBB(t) is a voltage quantity.

2. In this case, m has a dimension of radian frequency/volt if xBB(t) is a voltage quantity.

3. We call these symmetric because omission of sideband signs would make them symmetric.

4. Basis functions must be orthogonal, i.e., have zero correlation.

5. More precisely, the two consecutive bits that are demultiplexed and appear at A and B together form a symbol.

6. Also known as QAM16.

7. This situation arises in our party analogy if two people speak much more loudly than others. Even with different languages, communication becomes difficult.

8. If a vehicle moves at a high speed or in an area with tall buildings, the power received by the base station from it can vary rapidly, requiring continuous feedback.

9. The sensitivity in GSM1800 is −101 dBm.

10. This mask and others described in this section symmetrically extend to the left.

11. In GSM1800 and GSM1900, 12 in-band exceptions are allowed.

12. The PA may need to deliver about +27 dBm to account for the loss of the duplexer.

13. However, if the TX leakage is large, the RX linearity must be quite higher.

14. This corresponds to the most common case of “power class 3.” Other power classes with higher output levels have also been specified.

15. Up to three exceptions are allowed for the third and higher adjacent channel powers, with a relaxed specification of −20 dBm.

16. As a rule of thumb, a receiver analog baseband output should be around 0 dBm.

17. CCK is a variant of QPSK.

18. Note that, unlike the 11a/g specification, this leakage is not with respect to the overall TX output power.

### Chapter 4

1. Shannon’s theorem states that the achievable data rate of a communication channel is equal to B log2(1 + SNR), where B denotes the bandwidth and SNR the signal-to-noise ratio.

2. The Q of a band-pass filter may be roughly defined as the center frequency divided by the −3-dB bandwidth.

3. As mentioned in Chapter 3, a duplexer consists of two band-pass filters, one for the TX band and another for the RX band.

4. In this book, we do not distinguish between heterodyne and “super heterodyne” architectures. The term heterodyne derives from hetero (different) and dyne (to mix).

5. These have also been called “superdyne” and “infradyne,” respectively.

6. As mentioned earlier, passive filters suffer from a trade-off between the in-band loss and the out-of-band attenuation.

7. Also called “limiting.”

8. Or only the odd harmonics of the LO if the LO and the mixer are perfectly symmetric (Chapter 6).

9. The spectrum of a real signal is symmetric with respect to the origin.

10. In fact, it is possible to remove one side without losing information.

11. Frequency division can be performed by a counter: for M input cycles, the counter produces one output cycle.

12. Fractional bandwidth is defined as the bandwidth of interest divided by the center frequency of the band.

13. The term homodyne originates from homo (same) and dyne (mixing) and has been historically used for only “coherent” reception.

14. Also because the LO in direct-conversion receivers employs inductors, which couple the LO waveform into the substrate, whereas the second LO in heterodyne architectures is produced by an inductor-less divider.

15. We assume that the mixer generates an output current.

16. As an extreme example, a noise component with a period of one day varies so slowly that it has negligible effect on a 20-minute phone conversation.

17. We use the terms “amplitude mismatch” and “gain mismatch” interchangeably.

18. This sum is called the “analytic signal” of I(t).

19. We can also consider this a quadrature down converter if ωIF < ωc. In Problem 4.14, we study the case ωIF > ωc.

20. Note that the ratio of the output image power and the input image power is not meaningful because it depends on the gain.

21. To calculate in dB, we write 20 log(1 + 10%) = 0.83 dB.

22. This occurs because the entire signal band must see a flat frequency response in the antenna/LNA/mixer chain.

23. The channel-select filters must, however, provide a bandwidth equal to the RF signal bandwidth rather than half of it [Fig. 4.72(a)].

24. For clarity, the plots are allowed to be negative even though Eq. (4.96) contains absolute values.

25. Also known as a “quadrature modulator” or a “vector modulator.”

26. In this conceptual model, we omit the frequency translation inherent in the upconverter.

27. The DACs may be embedded within the mixers themselves (Chapter 6).

28. Pulling can also occur if the injected signal frequency is close to a harmonic of the oscillator frequency, e.g., in the vicinity of 2ωLO. We call this effect “superharmonic pulling.”

29. This is true only if the differential PA incorporates “single-ended” inductors rather than one symmetric inductor (Chapter 7).

30. This is not strictly correct because the second harmonic of the PA output is also the third harmonic of the LO, potentially causing “superharmonic” pulling.

31. The higher harmonics are neglected here.

### Chapter 5

1. The IM3 components arising from second-order terms are neglected.

2. Note that Γ is sometimes defined as (ZinRS)/(Zin + RS), in which case it is expressed in decibels by computing 20 log Γ (rather than 10 log Γ).

3. In the presence of a front-end band-select filter, the LNA sees smaller changes in the source impedance.

4. That is, consider RP as one stage and the CS amplifier as another.

5. For example, if RS simply represents the low-frequency resistance of the wire, its value remains constant and Q = /RS rises linearly with frequency. For a parallel resistance, RP, to allow such a behavior for Q = RP/(), the resistance must rise in proprotion to ω2 rather than remain constant.

6. The input may also see additional capacitance due to electrostatic discharge (ESD) protection devices that are tied to VDD and ground.

7. For proper matching between the two transistors, M1 incorporates five unit transistors (e.g., gate fingers) and MB one unit transistor.

8. The body effect lowers the input resistance, but the feedback from the drain to the gate raises it. We therefore neglect both.

9. We also neglect channel-length modulation and body effect.

10. This is a rare case in which the transistor is too fast!

11. If CGS1/gm is constant and L1 increases, the input cannot remain matched and Eq. (5.95) is invalid.

12. The output impedance of the cascode is assumed much higher than R1.

13. This technique was originally devised for bipolar stages.

14. The input impedance of the feedback circuit is absorbed in ZL.

15. Where does the other half go?

16. Alternatively, capacitive coupling can be used in the feedback path. But the large value necessary for the capacitor would introduce additional parasitics.

17. To ensure stability in the presence of package parasitics, a capacitor of 10-20 pF must be placed between VDD and GND.

18. And since they employ large devices and hence have small mismatches.

19. To halve the input resistance, the transistor width and bias current must be doubled.

20. But the parasitic capacitance of ISS must be nulled.

21. Assuming that gm1 and ωT remain unchanged.

22. In reality, the outputs of two generators are summed for a two-tone test.

23. Note that one transistor turns off if the differential input reaches .

### Chapter 6

1. Due to nonlinearities, a component at 2ωLO still leaks to the input (Problem 6.3).

2. One exception is when an LO drives only a frequency divider to avoid injection pulling (Chapter 4).

3. Recall from basic analog circuits that the integral of this output noise from 0 to ∞ is equal to kT/C1.

4. It is helpful to remember that the peak amplitude of the first harmonic of a square wave is greater than the peak amplitude of the square wave.

5. Because the difference between VLO and must reach zero in ΔT seconds.

6. We neglect channel-length modulation here.

7. Since RP varies periodically, with a frequency equal to 2ωLO, we can express its value by a Fourier series and consider the first term as the average resistance.

8. The output resistance of M4 and M5 can be absorbed in RD for this calculation.

9. In this case, VOS2 represents the difference between the offsets of M3M4 and M5M6.

10. Note that MH1 and MH2 do not help the switching of the differential pairs because the 2ωLO waveforms at P and Q are identical (rather than differential).

11. In reality, each DAC is followed by a low-pass filter to suppress the DAC’s high-frequency output components.

12. The ac ground at the source nodes reduces third order nonlinearity (Chapter 5).

13. The threshold mismatch in fact leads to charge injection mismatch between the switches and a slight disturbance at the output at the LO frequency. But this disturbance carries litter energy because it appears only during LO transitions.

14. As explained in Chapter 4, the noise produced by a GSM transmitter in the receive band must be very small.

### Chapter 7

1. One may use the inner opening dimension, Din, rather than Dout or N.

2. The spirals are shorted to one another by vias, although the vias are not necessary.

3. But the number of turns must be at least 2 to create mutual coupling.

4. The outer dimension varies from 260 μmto110 μm in this experiment.

5. In fact, Eqs. (7.19) and (7.20) have been developed based on ASITIC simulations.

6. Note that the actual Q may be even lower due to other losses.

7. The equivalent (lumped) capacitance of the inductor is less than this value (Section 7.2.4).

8. Faraday’s law states that the voltage induced in a conducting circuit is proportional to the time derivative of the magnetic field.

9. A more accurate model would include mutual coupling such that Ltot = L1 + ... + Ln + nM.

10. But, in some technologies long lines require a wider spacing than short lines, in which case the minimum S may be 1 to 1.5 μm.

11. One can also view the single spiral as a loop antenna.

12. If the free terminal of L1 is grounded, the equivalent capacitance is quite larger.

13. We have neglected the fringe components for simplicity.

14. Doubling the width does not reduce Lu by a factor of 2 because placing two coupled wires in parallel does not halve the inductance.

15. Of course, semiconductor device simulators can be used here if the doping levels and the junction depths are known.

16. We assume that the gate resistance is minimized by proper layout.

### Chapter 8

1. If L1 has no series resistance, then its average voltage drop must be zero; thus, VX and VY must go above VDD and below VDD.

2. The voltage at node P falls at the crossings of VX and VY if M1 and M2 do not enter the triode region at any point. On the other hand, if each transistor enters the deep triode region in a half cycle, then VP is low most of the time and rises at the crossings at VX and VY.

3. This topology is also called a “negative-Gm oscillator.” This is not quite correct because it does not contain a negative transconductance but a negative conductance.

4. With large-signal oscillation, the nonlinearity of M1 and M2 shifts the output CM level slightly, but we neglect this effect here.

5. The series resistance, RS, decreases only slightly with ω because it is equal to the sum of the low-frequency component and the skin effect component, and because the latter varies with .

6. The center tap of L1 is tied to VDD but not shown.

7. The differential resistance, Rp, can be viewed as two resistors of value Rp/2 tied to VDD. The peak single-ended swing is therefore equal to (2/π)(Rp/2)ISS.

8. Note that I1 in this equation is in fact a charge quantity because it denotes the area under the impulses.

9. The initial condition in the tank can also be created by a current impulse and hence does not make the system nonlinear.

10. In this circuit, we typically scale the transistor widths in proportion to their bias currents; thus, gm3,4/gm1,2 = I1/ISS. For small-signal analysis, the coupling factor is equal to gm3,4/gm1,2.

11. In [26], the transistors are NMOS devices with the assumption that their bulks can be separated.

### Chapter 9

1. Of course, the transfer function represents the behavior for nonsinusoidal inputs as well.

2. Traditional PLLs are characterized by “acquisition range,” “pull-in range,” “capture range,” “lock range,” “tracking range,” etc. We will soon see that modern PLLs need not deal with these distinctions.

3. In Problem 9.10, we estimate IpR1.

4. Note that ζ is still the damping factor of the original second-order loop.

5. The skew is not completely cancelled because the capacitance seen by QB may be different from that seen by QA.

### Chapter 10

1. Note that Eqs. (10.3)(10.5) are written for frequency quantities, but they apply to phase quantities as well.

2. One may incorporate a simple analog filter even for FSK to improve the bandwidth efficiency.

3. The effect of the filter in the XBB path is neglected for simplicity.

4. The LPF removes the sum component at the output of MX1.

5. It is unfortunate that the overall circuit is called the “pulse swallow divider” and this block, the “swallow counter.”

6. In this book, we denote a latch by a single box and an FF by a double box.

7. It can be proved that, in fact, 2/3 of the distributed capacitance is absorbed by the ac ground (Chapter 7).

8. Except for the subthreshold leakage of the transistors.

9. But this terminology must not be confused with dynamic logic.

10. Capacitive coupling ensures that M1 and M2 can operate in saturation.

11. In a manner similar to the coupling mechanism in quadrature oscillators (Chapter 8).

12. We assume M1 and M2 experience abrupt switching.

### Chapter 11

1. This is only a simplistic view. Since a type-II PLL forces the average phase error to zero, the phase difference in fact fluctuates between positive and negative values.

2. The in-band noise can also be reduced by raising fCK, but in a synthesizer environment fCK = fREF.

3. As explained in conjunction with Eq. (11.25), Sq(f) is relatively flat in the frequency range of interest.

4. Since the magnitude of the error is halved, the PSD drops by 6 dB rather than 3 dB.

5. This choice of clock phases for the latches and the MUX allows master-slave operation between each latch and the MUX.

6. The double-edge operation is denoted by two hats at the clock input.

### Chapter 12

1. A lossless transformer with a turns ratio of 1:n transforms the load resistance down by a factor of n2 (why?).

2. At very high frequencies, the gate and channel resistances also contribute a real part.

3. With a large voltage swing, the transistor may also introduce significant nonlinearity.

4. This issue can be alleviated through the use of a low-threshold transistor for M2.

5. From another perspective, if power is not transferred, it is not necessarily dissipated.

6. Nonetheless, the PA output impedance as seen by the antenna must be somewhat close to 50 Ω to absorb reflections from the antenna. That is, PAs must typically achieve a reasonable |S22|.

7. It is assumed that AM/PM conversion in the output stage is negligible or can be corrected.

8. We define the delay as that between the times at which the input and the output reach 50% of their full swings.

9. This is equivalent to approximating φ0(VX) by the first two terms of its Taylor expansion.

10. The coupling of the output to the gate of each transistor through CGD does create some interaction.

11. In this case, the transformer in fact subtracts V2 from V1. Thus, V2 must be negated before reaching PA2.

12. If the input waveforms are represented by cosines, the imaginary part is given by (− tan θ)RL/2.

13. If the input waveforms are represented by cosines, then this real part is given by 2 cos2 θ/RL.

14. And forfeiting the benefits of cascode operation.

15. The operation frequency and the supply voltage are not mentioned. It is unclear which components are external.

16. The drain efficiency is 48% [30].

17. A zero must be added to this loop to ensure stability [33].

18. Of course, the drain signal contains stronger harmonics than the output signal does.

### Chapter 13

1. As explained in Section 13.3, this 9-dB “back-off” is quite conservative, leaving several decibels of margin for the TX nonlinearity.

2. Recall from Chapter 3 that the desired input is at 3 dB above the reference sensitivity in this test.

3. A 1- Vpp differential swing translates to a peak single-ended swing of 0.25 V, a reasonable value for a 1.2-V supply.

4. That is, for every dB of gain reduction, the NF must rise by no more than 1 dB.

5. The simulations in [1] suggest a P1dB of 20.5 dBm. The discrepancy possibly arises from different PA models.

6. With both I and Q inputs, the output voltage swing is higher by a factor of .

7. In this chapter, we represent the phase noise profile by either α/f2 (assuming a center frequency of zero) or α/(ffc)2 (assuming a center frequency of fc).

8. Since gmrO ≈ 10 and gm ≈ (40Ω)−1, we have rO ≈ 400 Ω.

9. This voltage gain is from the LNA input node to the output.

10. Note that ac and transient simulations yield slightly different voltage gains.

11. We assume that the input impedance of the mixer is much higher than 50 Ω.

12. The linearity is relaxed for the intermodulation of blockers but not for the compression of the desired signal.

13. Also called a “programmable-gain amplifier” (PGA).

14. We neglect the loss of the balun here. In practice, about 0.5 to 1 dB of margin must be allowed for the balun’s loss.

15. Note that the balun provides another 6 dB of voltage gain.

16. The phase noise of the frequency dividers following the VCOs is negligible.

17. For example, the crystal oscillator frequency may be dictated by the cell phone manufacturer or the baseband processor clock, etc.

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