Stability, Noise, Nonlinearity and Distortion

Introduction

An RF circuit must do more than provide the required gain at its nominal operating point. It must remain stable for all expected source and load impedances, add acceptably little noise, and preserve signal integrity when driven by wanted signals and blockers. These requirements are interdependent: a source impedance chosen for minimum noise may reduce stability margin, while feedback used to improve linearity may alter gain and potentially create oscillation.

Stability and noise are principally small-signal concerns, whereas compression and distortion arise from large-signal nonlinear behavior. A reliable RF design therefore combines S-parameter analysis, noise-wave or noise-parameter models, nonlinear characterization, harmonic-balance simulation, and laboratory validation.

Learning Objectives

After completing this chapter, the reader should be able to:

• distinguish unconditional, conditional, and large-signal stability;
• apply the Rollett factor, determinant and stability measures correctly;
• construct and interpret source and load stability circles;
• calculate cascaded noise figure using available power gain;
• use four noise parameters to evaluate source-impedance choices;
• explain compression, harmonics, intermodulation, IIP2 and IIP3;
• interpret AM–AM and AM–PM conversion;
• evaluate stability, noise and linearity jointly in an RF receiver;
• select practical mitigation and verification techniques.

1. Small-Signal Stability of RF Two-Ports

1.1 Feedback and potential instability

For a linear two-port referenced to a real impedance, commonly 50 Ω,

\[

\begin{bmatrix}b_1\\b_2\end{bmatrix}=

\begin{bmatrix}S_{11}&S_{12}\\S_{21}&S_{22}\end{bmatrix}

\begin{bmatrix}a_1\\a_2\end{bmatrix}.

\]

A transistor may possess reverse transmission, represented by \(S_{12}\), through internal capacitances, package coupling or external feedback. The input reflection coefficient depends on the load termination:

\[

\Gamma_{in}=S_{11}+\frac{S_{12}S_{21}\Gamma_L}{1-S_{22}\Gamma_L},

\]

and the output reflection coefficient depends on the source termination:

\[

\Gamma_{out}=S_{22}+\frac{S_{12}S_{21}\Gamma_S}{1-S_{11}\Gamma_S}.

\]

A terminated small-signal two-port is stable when \(|\Gamma_{in}|<1\) and \(|\Gamma_{out}|<1\). These conditions imply positive real input and output resistances for the chosen passive terminations.

1.2 Conditional and unconditional stability

A network is unconditionally stable at a specified frequency and bias point if it is stable for every passive source and load termination:

\[

|\Gamma_S|\leq 1,\qquad |\Gamma_L|\leq 1.

\]

It is conditionally stable if only part of the passive termination region is stable. A conditionally stable circuit may operate correctly with nominal matching networks but oscillate because of antenna mismatch, component tolerance, switching transients or out-of-band impedances.

These definitions apply to a linearized two-port at one frequency. They do not by themselves establish stability across frequency, during startup, under compression, or in circuits with additional feedback paths.

1.3 Rollett stability criterion

Define

\[

\Delta=S_{11}S_{22}-S_{12}S_{21}

\]

and

\[

K=\frac{1-|S_{11}|^2-|S_{22}|^2+|\Delta|^2}

{2|S_{12}S_{21}|}.

\]

For a nondegenerate two-port with \(S_{12}S_{21}\neq0\), the standard Rollett conditions

\[

K>1,\qquad |\Delta|<1

\]

are necessary and sufficient for unconditional small-signal stability with passive terminations. Equality indicates a stability boundary and should not be treated as a robust design condition.

For a unilateral network, \(S_{12}=0\), the expression for \(K\) is singular. Stability must then be assessed directly; a strictly unilateral two-port is unconditionally stable when \(|S_{11}|<1\) and \(|S_{22}|<1\).

Alternative stability measures are

\[

\mu=\frac{1-|S_{11}|^2}

{|S_{22}-\Delta S_{11}^{*}|+|S_{12}S_{21}|},

\]

\[

\mu'=\frac{1-|S_{22}|^2}

{|S_{11}-\Delta S_{22}^{*}|+|S_{12}S_{21}|}.

\]

Values greater than unity indicate unconditional stability; values appreciably above unity provide useful design margin. Both input- and output-oriented measures should be inspected across the full frequency range of concern.

1.4 Stability circles

A source stability circle is the locus of \(\Gamma_S\) satisfying \(|\Gamma_{out}|=1\). Its center and radius are

\[

C_S=\frac{S_{11}^{}-\Delta^{}S_{22}}

{|S_{11}|^2-|\Delta|^2},

\qquad

r_S=\frac{|S_{12}S_{21}|}

{\left||S_{11}|^2-|\Delta|^2\right|}.

\]

A load stability circle is the locus of \(\Gamma_L\) satisfying \(|\Gamma_{in}|=1\):

\[

C_L=\frac{S_{22}^{}-\Delta^{}S_{11}}

{|S_{22}|^2-|\Delta|^2},

\qquad

r_L=\frac{|S_{12}S_{21}|}

{\left||S_{22}|^2-|\Delta|^2\right|}.

\]

These formulas require nonzero denominators. Degenerate cases should be evaluated from the defining boundary equations rather than by numerical division near zero.

The circle divides the plane into stable and unstable regions. The stable side is not determined by the circle alone. A test point must be evaluated. For example, at \(\Gamma_S=0\), \(\Gamma_{out}=S_{22}\); hence the origin is on the stable side when \(|S_{22}|<1\). Similarly, the load-plane origin is stable when \(|S_{11}|<1\).

#### Worked Example: Stability assessment

At one frequency, suppose

\[

S_{11}=0.5\angle-60^\circ,\quad S_{22}=0.4\angle-30^\circ,

\]

\[

S_{12}=0.05\angle20^\circ,\quad S_{21}=2\angle70^\circ.

\]

Then

\[

S_{11}S_{22}=0.2\angle-90^\circ=-j0.2,

\]

\[

S_{12}S_{21}=0.1\angle90^\circ=j0.1,

\]

so \(\Delta=-j0.3\) and \(|\Delta|=0.3\). Therefore,

\[

K=\frac{1-0.25-0.16+0.09}{2(0.1)}=3.4.

\]

Because \(K>1\) and \(|\Delta|<1\), the device is unconditionally stable at this frequency.

For the source circle,

\[

C_S=2.375\angle60^\circ,\qquad r_S=0.625.

\]

Its nearest point to the Smith-chart origin has magnitude \(2.375-0.625=1.75\), so it does not intersect the passive Smith chart. For the load circle,

\[

C_L=3.571\angle30^\circ,\qquad r_L=1.429,

\]

whose nearest magnitude is approximately 2.142. This also lies outside the passive chart, consistently confirming unconditional stability.

1.5 Practical stability verification

A responsible stability assessment includes:

• a frequency sweep extending well below and above the operating band;
• bias, temperature, process and component-tolerance corners;
• realistic package, grounding, via and enclosure parasitics;
• source and load mismatch sweeps;
• even-mode and odd-mode analysis for balanced or differential circuits;
• loop-gain, pole-zero or normalized determinant analysis for multi-loop circuits;
• large-signal stability checks under compression and modulated excitation.

Series gate/base resistors, lossy matching, feedback networks, source/emitter degeneration, ferrite loading and improved isolation can increase stability. Their penalties may include lower gain, higher noise figure or reduced output power.

2. Noise in RF Networks

2.1 Noise factor and equivalent noise temperature

Noise factor is defined under specified source-temperature and bandwidth conditions as

\[

F=\frac{(S/N)_{in}}{(S/N)_{out}},

\]

and noise figure is

\[

NF=10\log_{10}F\quad\text{dB}.

\]

With the standard reference temperature \(T_0=290\) K, equivalent input noise temperature is related to noise factor by

\[

T_e=T_0(F-1),\qquad F=1+\frac{T_e}{T_0}.

\]

Available thermal noise power from a matched source is approximately \(kTB\) when \(hf\ll kT\). At sufficiently high frequencies or cryogenic temperatures, the quantum expression must replace this classical approximation.

2.2 Passive loss

A passive attenuator at \(T_0\), with linear loss \(L\geq1\), has

\[

F=L,\qquad G_A=\frac{1}{L},

\]

where \(G_A\) is available power gain. Thus a 1 dB filter has \(L=10^{0.1}=1.259\), noise figure 1 dB and available gain \(1/1.259=0.794\), or −1 dB. Treating loss as gain produces an incorrect cascade result.

At physical temperature \(T_p\), the attenuator equivalent input noise temperature is

\[

T_e=(L-1)T_p,

\]

which reduces to \(F=L\) only when \(T_p=T_0\).

2.3 Friis cascade equation

For matched stages characterized by available power gains, the cascade noise factor is

\[

F_{tot}=F_1+\frac{F_2-1}{G_1}

+\frac{F_3-1}{G_1G_2}+\cdots.

\]

All factors and gains must be linear, not decibel quantities. For mismatched networks, transducer gain and noise-wave correlation may be needed; blindly applying the scalar Friis equation can be inaccurate.

#### Worked Example: Receiver noise figure

Consider a 1 dB input filter, an LNA with \(F=1.36\) and \(G=31.62\) (15 dB), and a mixer with \(F=8\). The filter has

\[

F_1=1.259,\qquad G_1=0.794.

\]

Hence

\[

F_{tot}=1.259+\frac{1.36-1}{0.794}

+\frac{8-1}{0.794(31.62)}.

\]

Therefore,

\[

F_{tot}=1.259+0.453+0.279=1.991,

\]

and

\[

NF_{tot}=10\log_{10}(1.991)\approx2.99\text{ dB}.

\]

The example illustrates why front-end loss is especially costly: it adds noise before the LNA and simultaneously reduces the gain suppressing later-stage noise.

2.4 Four noise parameters

At a specified frequency, bias, temperature and reference impedance, a linear two-port can be represented by four noise parameters:

• minimum noise factor \(F_{min}\);
• optimum source reflection coefficient \(\Gamma_{opt}\);
• equivalent noise resistance \(R_n\);
• reference impedance \(Z_0\).

For a passive source termination with \(|\Gamma_S|<1\),

\[

F=F_{min}+\frac{4R_n}{Z_0}

\frac{|\Gamma_S-\Gamma_{opt}|^2}

{(1-|\Gamma_S|^2)|1+\Gamma_{opt}|^2}.

\]

The minimum occurs at \(\Gamma_S=\Gamma_{opt}\). Constant-noise-factor circles can be plotted on the Smith chart. In practice, the optimum noise match often differs from the simultaneous conjugate match for maximum available gain. The selected \(\Gamma_S\) must therefore balance noise figure, gain, input return loss, bandwidth and stability.

Noise parameters are measurement- and model-sensitive. Connector de-embedding, tuner loss, calibration drift and device temperature can materially affect extracted values.

3. Nonlinear RF Behavior

3.1 Polynomial model

A weakly nonlinear, memoryless circuit may be approximated by

\[

y(t)=a_0+a_1x(t)+a_2x^2(t)+a_3x^3(t)+\cdots.

\]

The model is useful for interpreting distortion near the small-signal regime. It is not a universal large-signal transistor model and does not adequately describe strong compression, hysteresis, bias-network memory, trapping or thermal effects.

For \(x(t)=A\cos\omega t\), the important components up to third order are

\[

y_{DC}=a_0+\frac{a_2A^2}{2},

\]

\[

y_{\omega}=a_1A+\frac{3a_3A^3}{4},

\]

\[

y_{2\omega}=\frac{a_2A^2}{2},\qquad

y_{3\omega}=\frac{a_3A^3}{4}.

\]

Even-order terms generate DC and even harmonics, while odd-order terms affect the fundamental and generate odd harmonics.

3.2 Gain compression

As input power rises, output power ceases to follow the linear extrapolation. The input 1 dB compression point, \(P_{1dB,in}\), is the input level at which actual gain is 1 dB below small-signal gain. The corresponding output level is \(P_{1dB,out}\). Saturated output power is a different quantity and usually occurs at a higher drive level.

Compression points depend on frequency, impedance, bias, waveform statistics and thermal conditions. A modulated signal with a high peak-to-average power ratio may require substantial back-off even when its average power is below \(P_{1dB}\).

3.3 Harmonic distortion

Harmonic distortion appears at integer multiples of a single-tone frequency. Harmonics can violate emission masks, mix back into the wanted band or excite package and antenna resonances. Harmonic-balance simulation and spectrum-analyzer measurements are commonly used to quantify them.

Filters can suppress harmonics after they have been generated, but they do not remove in-band intermodulation products. Device bias, waveform symmetry, balanced architectures and feedback can reduce harmonic generation at its source.

4. Intermodulation and Intercept Points

4.1 Two-tone products

For two input tones at \(f_1\) and \(f_2\), second-order products include

\[

f_1+f_2,\qquad |f_1-f_2|,

\]

while important third-order products include

\[

2f_1-f_2,\qquad 2f_2-f_1.

\]

The third-order products are particularly troublesome because they lie close to the original tones when the tone spacing is small and may therefore fall inside the receiver or transmitter channel.

In the weakly nonlinear region, fundamental output power rises approximately 1 dB for each 1 dB increase in input power. Second- and third-order intermodulation powers rise with slopes of approximately 2 and 3 dB/dB, respectively.

4.2 IIP2, IIP3, OIP2 and OIP3

Intercept points are extrapolated figures rather than normal operating powers. If a two-tone test produces a fundamental-to-IM3 separation \(\Delta_3\) dB at per-tone input power \(P_{in}\), then

\[

IIP3=P_{in}+\frac{\Delta_3}{2}.

\]

For second-order distortion with separation \(\Delta_2\),

\[

IIP2=P_{in}+\Delta_2.

\]

For small-signal gain \(G_{dB}\),

\[

OIP3=IIP3+G_{dB},\qquad OIP2=IIP2+G_{dB}.

\]

These equations assume that measurements are made in the asymptotic weakly nonlinear region, before compression changes the slopes.

For equal two-tone inputs, the approximate input-referred third-order product is

\[

P_{IM3,in}=3P_{in}-2IIP3,

\]

with all quantities in dBm. Tone spacing, modulation bandwidth, termination impedance and measurement bandwidth must be reported with an intercept specification.

4.3 Cascaded IIP3

For matched stages with linear power gains \(G_i\) and input intercept powers expressed in the same linear power unit, a commonly used cascade approximation is

\[

\frac{1}{IIP3_{tot}}\approx

\frac{1}{IIP3_1}+\frac{G_1}{IIP3_2}

+\frac{G_1G_2}{IIP3_3}+\cdots.

\]

The formula assumes weak memoryless nonlinearities and additive third-order contributions without beneficial phase cancellation. If nonlinear contributions possess important phase, frequency dependence or memory, a complex behavioral or circuit-level analysis is required. Similar scalar cascade rules for IIP2 are less robust because even-order cancellation and balance errors can dominate.

5. AM–AM and AM–PM Distortion

For a narrowband input represented by complex envelope \(u(t)=r(t)e^{j\phi(t)}\), a nonlinear amplifier may be described as

\[

v(t)=A(r)e^{j[\phi(t)+\Phi(r)]}.

\]

The function \(A(r)\) describes AM–AM conversion, or amplitude-dependent gain. The function \(\Phi(r)\) describes AM–PM conversion, or amplitude-dependent phase shift.

AM–AM compression alters the signal envelope. AM–PM conversion transforms envelope fluctuations into phase fluctuations. Both effects create spectral regrowth and degrade error vector magnitude in QAM, OFDM and other variable-envelope signals. A single-tone compression measurement cannot fully predict modulated-signal performance because electrical and thermal memory can make the response depend on signal history.

Useful modulated-signal metrics include error vector magnitude, adjacent-channel leakage ratio, noise power ratio and adjacent-channel power ratio. Digital predistortion uses an approximate inverse model of the power amplifier. Memory-polynomial, Volterra-inspired and neural-network predistorters are increasingly used, although model complexity, training data quality, adaptation stability and power consumption must be controlled.

6. Integrated Receiver Case Study

Background

The following is an illustrative case. A receiver front end contains:

1. a 1 dB preselector filter;

2. an LNA with 15 dB gain, noise factor 1.36 and IIP3 of −5 dBm;

3. a mixer with noise factor 8 and IIP3 of +10 dBm.

The mixer has 5 dB conversion gain. The active devices have been verified as unconditionally stable across the specified termination and frequency range.

Problem

The design team must determine the receiver noise figure, cascaded IIP3 and the effect of two equal −35 dBm blockers. It must also identify whether the LNA or mixer dominates third-order distortion.

Analysis

The filter has linear gain

\[

G_F=10^{-1/10}=0.794

\]

and noise factor 1.259. As calculated earlier,

\[

F_{tot}=1.991,

\qquad NF_{tot}\approx2.99\text{ dB}.

\]

For IIP3, first refer the LNA–mixer cascade to the LNA input. Convert intercepts to milliwatts:

\[

IIP3_{LNA}=10^{-5/10}=0.3162\text{ mW},

\]

\[

IIP3_{mixer}=10^{10/10}=10\text{ mW}.

\]

The LNA gain is \(G_{LNA}=31.62\). Thus

\[

\frac{1}{IIP3_{LM}}=

\frac{1}{0.3162}+\frac{31.62}{10}

=3.162+3.162=6.324\text{ mW}^{-1}.

\]

Therefore,

\[

IIP3_{LM}=0.1581\text{ mW}\approx-8.01\text{ dBm}.

\]

The two reciprocal contributions are equal, so the LNA and mixer contributions are comparable; the distortion is not LNA-dominated.

The preceding ideal linear filter attenuates each input tone before it reaches the nonlinear stages. Referred to the receiver input,

\[

IIP3_{receiver}=\frac{0.1581}{0.794}=0.199\text{ mW}

\approx-7.01\text{ dBm}.

\]

For two equal blockers at −35 dBm per tone, the input-referred third-order product is approximately

\[

P_{IM3,in}=3(-35)-2(-7.01)\approx-90.98\text{ dBm}.

\]

The total conversion gain is approximately

\[

-1+15+5=19\text{ dB},

\]

so the corresponding output product is approximately −72 dBm, provided the circuit remains in the weakly nonlinear region.

Approach and Solution

The team should not improve sensitivity by simply increasing LNA gain. Greater gain reduces the mixer’s noise contribution but increases the mixer’s input drive and its input-referred cascade distortion contribution. Suitable actions include:

• using a lower-loss preselector where blocker rejection permits;
• selecting an LNA with a better noise-linearity trade-off;
• increasing mixer IIP3 or inserting controlled interstage attenuation;
• optimizing impedances for noise, gain, stability and linearity jointly;
• validating with realistic modulated blockers and mismatch conditions.

Outcome and Learning Points

The receiver achieves approximately 2.99 dB noise figure and −7.0 dBm input-referred IIP3. Front-end filter loss materially affects sensitivity, while LNA gain simultaneously improves cascaded noise performance and worsens the input-referred contribution of downstream nonlinearity. System optimization therefore requires a multi-objective rather than stage-by-stage approach.

7. Mitigation, Measurement and Design Practice

7.1 Mitigation techniques

Stability can be improved through resistive loading, feedback, degeneration, isolation and careful electromagnetic layout. Noise can be reduced by minimizing pre-LNA loss, using low-noise biasing, cooling selected systems and matching near \(\Gamma_{opt}\). Linearity can be improved through bias optimization, device sizing, balanced or differential architectures, feedback, feedforward correction, source degeneration, harmonic terminations and digital predistortion.

No technique is free of trade-offs. Resistive stabilization may raise noise figure; deep output-power back-off lowers efficiency; balanced circuits consume area and require accurate phase and amplitude balance.

7.2 Measurement limitations

Measured results can be corrupted by cable loss, fixture mismatch, analyzer noise floor, receiver compression or source harmonics. Two-tone sources should be isolated so that they do not intermodulate with each other. Output attenuation may be required to protect the analyzer, and its loss must be included in calibration.

Noise-figure measurements require accurate knowledge of source excess-noise ratio or calibrated cold-source parameters. Stability validation should include impedance variation rather than only a nominal 50 Ω test.

7.3 Governance and regulatory implications

Nonlinear emissions can interfere with licensed and safety-critical services. Designers must verify applicable spectral masks, spurious-emission limits, electromagnetic-compatibility requirements and exposure constraints. Reported NF, compression and intercept values should state test frequency, bias, temperature, bandwidth, tone spacing, calibration plane and uncertainty. Selectively reporting favorable conditions is technically misleading and may produce unsafe system margins.

Chapter Summary

RF design requires simultaneous control of stability, noise and nonlinearity. Unconditional small-signal stability can be assessed using \(K\), \(\Delta\), stability measures and correctly constructed stability circles. These tests must be supplemented by broadband, mismatch and large-signal verification.

Noise factor quantifies SNR degradation, while four noise parameters describe its dependence on source impedance. Friis’ equation shows why passive front-end loss and first-stage performance dominate receiver sensitivity. Nonlinear circuits exhibit compression, harmonics, intermodulation, AM–AM conversion and AM–PM conversion. IIP2 and IIP3 provide extrapolated weakly nonlinear metrics but do not replace realistic modulated-signal testing. Effective engineering balances noise, gain, stability, linearity, efficiency and regulatory compliance at system level.

Key Terms

AM–AM conversion: Dependence of output amplitude or gain on input-envelope amplitude.

AM–PM conversion: Dependence of output phase shift on input-envelope amplitude.

Available power gain: Ratio based on power available from the source and power available from the network output.

Compression point: Input or output power where gain has fallen by a specified amount from its small-signal value.

Conditional stability: Stability for only a subset of passive source and load terminations.

Error vector magnitude: Measure of the difference between measured and ideal modulation symbols.

Four noise parameters: \(F_{min}\), \(\Gamma_{opt}\), \(R_n\) and \(Z_0\), used to characterize two-port noise behavior.

Harmonic distortion: Spectral components at integer multiples of an excitation frequency.

IIP2: Input-referred extrapolated intersection of fundamental and second-order responses.

IIP3: Input-referred extrapolated intersection of fundamental and third-order responses.

Intermodulation: Generation of sum-and-difference frequencies from multiple input tones.

Noise factor: Linear ratio of input SNR to output SNR.

Noise figure: Noise factor expressed in decibels.

Noise temperature: Equivalent input temperature representing added network noise.

Rollett factor: Two-port stability parameter \(K\) used with \(|\Delta|\).

Spectral regrowth: Expansion of a modulated signal’s spectrum due to nonlinearity.

Stability circle: Reflection-coefficient boundary separating stable and potentially unstable termination regions.

Unconditional stability: Stability for every passive source and load termination at the stated operating condition.

Review Questions

Short Answer Questions

1. Distinguish between conditional and unconditional stability.

2. Why must \(K>1\) be used together with \(|\Delta|<1\)?

3. Why is a passive filter placed before an LNA particularly harmful to noise figure?

4. Distinguish the 1 dB compression point from IIP3.

5. Why are third-order intermodulation products especially troublesome in narrowband systems?

Long Answer Questions

1. Derive the source stability-circle equation from the condition \(|\Gamma_{out}|=1\), and explain how its stable side is identified.

2. Explain the four noise parameters and discuss noise matching versus gain matching.

3. Develop the polynomial interpretation of harmonic and intermodulation distortion up to third order.

4. Compare AM–AM and AM–PM distortion and explain their effects on OFDM and QAM signals.

5. Critically evaluate practical methods for improving RF stability and linearity, including their noise, gain and efficiency penalties.

Analytical / Application Questions

1. Given measured S-parameters at several frequencies, calculate \(K\), \(|\Delta|\), \(\mu\) and \(\mu'\), construct stability circles where required, and propose a stabilization network.

2. Analyze a receiver containing a lossy filter, LNA, variable-gain amplifier and mixer. Determine cascaded NF and IIP3 for each gain setting and recommend the best blocker-dependent operating mode.

3. Design a two-tone measurement plan for an RF amplifier, including tone spacing, calibration plane, power sweep, analyzer protection and checks for source-generated intermodulation.

Multiple Choice Questions with Answers

Question 1

For a nondegenerate two-port, which conditions establish unconditional small-signal stability for all passive terminations?

A. K > 1 and |Δ| < 1

B. K < 1 and |Δ| < 1

C. K > 1 and |Δ| > 1

D. K = 1 regardless of Δ

Correct Answer: A. K > 1 and |Δ| < 1

Question 2

For the worked stability example, what is the Rollett stability factor?

A. 0.30

B. 1.70

C. 3.40

D. 6.80

Correct Answer: C. 3.40

Question 3

A passive filter has 1 dB loss and is at the standard reference temperature. What are its approximate linear available gain and noise factor?

A. Gain 1.259 and noise factor 0.794

B. Gain 0.794 and noise factor 1.259

C. Gain 1.000 and noise factor 1.259

D. Gain 0.794 and noise factor 0.794

Correct Answer: B. Gain 0.794 and noise factor 1.259

Question 4

A cascade contains a 1 dB filter, an LNA with F = 1.36 and G = 31.62, and a mixer with F = 8. What is the approximate total noise figure?

A. 1.36 dB

B. 1.99 dB

C. 2.99 dB

D. 8.00 dB

Correct Answer: C. 2.99 dB

Question 5

At what source reflection coefficient does the four-noise-parameter equation attain its minimum noise factor?

A. ΓS = 0

B. ΓS = Γopt

C. ΓS = S11

D. ΓS = Δ

Correct Answer: B. ΓS = Γopt

Question 6

Which products are normally the most troublesome close-in third-order products for two tones f1 and f2?

A. 2f1 and 2f2

B. f1 + f2 and |f1 − f2|

C. 2f1 − f2 and 2f2 − f1

D. 3f1 and 3f2 only

Correct Answer: C. 2f1 − f2 and 2f2 − f1

Question 7

At a per-tone input power of −20 dBm, the measured fundamental-to-IM3 separation is 40 dB. What is IIP3?

A. −10 dBm

B. 0 dBm

C. +20 dBm

D. +40 dBm

Correct Answer: B. 0 dBm

Question 8

The LNA–mixer cascade in the case study has an IIP3 of 0.158 mW at the LNA input. After including the preceding 1 dB filter, what is the approximate receiver-input IIP3?

A. −9.0 dBm

B. −8.0 dBm

C. −7.0 dBm

D. +7.0 dBm

Correct Answer: C. −7.0 dBm

Question 9

What does AM–PM conversion describe?

A. Amplitude-dependent output phase shift

B. Frequency-dependent noise factor only

C. Conversion of RF power into DC power

D. Amplitude-independent gain compression

Correct Answer: A. Amplitude-dependent output phase shift

Question 10

Why is an intercept point not normally an achievable operating power?

A. It is defined only for passive networks

B. It is an extrapolated intersection of response slopes, often beyond compression

C. It equals the thermal noise floor

D. It is always lower than the 1 dB compression point

Correct Answer: B. It is an extrapolated intersection of response slopes, often beyond compression