RF Journey

01 / 10

Failure: three “-174 dBm/Hz” answers disagree

Three engineers quote the same noise-density reference. Why are their answers 12 dB apart?

The fictional telemetry gateway from 05.2 has a selected 2.450 GHz channel. It carries 20 kbit/s with uncoded, no-overhead QPSK mapping at 10 ksymbol/s. Ideal raised-cosine support is 13.5 kHz; D3 remains 80 kcomplex-sample/s. This module adds a separate 20.000 kHz receiver noise-equivalent bandwidth at the detector reference. It is a local allocation, not a standard filter or a replacement for signal support.

Opening workbook · available sensitivity at R2; 8.000 dB required R3 SNR + 2.000 dB RX-IMPL once; 20.000 kHz ENBW, Ts = 290 K unless marked erroneous
Worksheet	Noise input / criterion error	Claimed sensitivity dBm	Decision
A · correct	−128.165007875 dBm input-equivalent noise; +8 +2 dB	−118.165007875	Conditional receiver threshold under the stated model.
B · bandwidth error	200 kHz substituted for 20 kHz: +10 dB integrated noise	−108.165007875	10 dB too pessimistic; restore the detector ENBW.
C · bandwidth + duplicate loss	B plus another 2 dB RX-IMPL charge	−106.165007875	12 dB error; remove duplication after repairing bandwidth.

Before comparing any numbers, align input plane, source temperature, receiver NF convention, integrated bandwidth and output criterion. A connector number referred after a lossy front end is not an R2 antenna-feed number. An 8 dB detection allocation is not a BER, PER or EVM specification.

Common misconception−174 dBm/Hz is the universal total receiver noise floor.

It approximates available thermal power density near 290 K. Integrating a declared bandwidth and referring receiver-added noise to one plane are separate steps. At exact k and 290 K, the density is −173.975187194 dBm/Hz, before receiver noise.

The useful deliverable is a conditional statement whose plane, mode and criterion survive the handoff.

02 / 10

Define sensitivity at a plane and quality threshold

Where is the minimum available power defined, and what must the detector do with it?

Define conditional sensitivity as the smallest available wanted power at R2 that supplies the required R3 detector SNR plus a separately allocated implementation loss. This matched, small-signal screen excludes blockers. Gain and NF are synthetic, flat over the selected channel, at nominal bias and active-device temperature 25 °C. T0 = 290 K is the noise normalization; RX-FE is physically 290 K in this fixture.

Frozen plane ledger · p05-m03-receiver-ledger-v1 · matched available-power quantities
Stage / semantic plane mapping	Gain / NF at T0	Boundary condition
RX-FE · R2 → R1-LNA-in	−1.500 / 1.500 dB	Protection/filter/switch; matched passive loss at physical 290 K.
RX-LNA · R1-LNA-in → R1-LNA-out	+20.000 / 1.000 dB	Fixed active gain/NF, independent of source-temperature edits.
RX-MIX · R1-LNA-out → A1-IF	−6.000 / 6.000 dB	Effective single-signal-channel NF; image-noise contribution already accounted for.
RX-DRV · A1-IF → A1-ADC-in	+30.000 / 4.000 dB	Matched power-transfer teaching boundary, 100 Ω differential.
A1 → A0 → R3	Analog input → codes → detector	R3 is the decision boundary, never the antenna port. D0/D1/D2/D3 retain payload/bit/symbol/pulse-shaping meanings.

RF reference impedance is 50 Ω at R2/R1/R0. A1’s 100 Ω differential load is only a teaching conversion from volts to power; a real ADC switched input is not asserted to be a resistor. Available, delivered and accepted power coincide only under the declared match. Moving the plane requires its actual gain/loss and noise transformation, not a renamed column.

\begin{aligned}P_{\mathrm{sens,R2}} &= 10 \log_{10}(\frac{k T_{0} B}{1 \mathrm{mW}}) \\ &+ \mathrm{NF} + \mathrm{SNR}_{\mathrm{required}} + L_{\mathrm{impl}}\end{aligned}

Definition under matched Ts = T0 = 290 K; k = 1.380649×10⁻²³ J/K, B in Hz; SNR required and implementation loss in dB.

Quality contract · illustrative requirements only
Criterion	Evidence needed	What 8 dB here means
SNR	Signal/noise definition, filtering and R3 decision plane	An explicit teaching allocation.
BER / PER	Waveform, coding, receiver, bit/packet population and error target	No named-technology threshold inferred.
EVM	Reference normalization, synchronization/equalization and sample population	Not interchangeable with SNR when non-noise impairments dominate.

Go deeperA system allocation differs from the preceding LNA model

04.4 studies a bilateral two-port with source/load-dependent gain and noise. The +20 dB/1 dB LNA here is a new synthetic matched system allocation, not a substitute for that device’s S/noise-parameter result. Carry actual terminations, image convention and bandwidth before replacing a row with hardware data.

The criterion is now explicit. Next, assign the bandwidth that actually integrates its noise.

03 / 10

Noise bandwidth is not a generic channel width

Which bandwidth belongs beside kT?

Bandwidth ledger · equal numerical values need not share a definition
Quantity / plane	Nominal value	Meaning / consequence
Receiver ENBW · R3 reference	20.000 kHz	Equal-area noise integration for the selected response. Used once in this scalar cascade.
Ideal RC total support · D3 / translated RF	13.5 kHz	Null-to-null width = (1 + 0.35) × 10 ksymbol/s. It is not measured occupied bandwidth.
Digital complex sample rate · D3	80 ksample/s	Representation/processing rate; not integrated receiver noise bandwidth.
Real converter sample rate · A0 benchmark	80 MS/s	40 MHz Nyquist interval for the ideal quantization comparison, before channel selection.
Channel / occupied / analyzer RBW	Not supplied as ENBW	Channel allocation, power-containment width and instrument resolution each need their own definition.

Think about itIf only ENBW changes from 20 to 200 kHz, do NF and integrated noise both rise by 10 dB?

Answer

NF is unchanged in the flat model. Noise power grows by ten, so integrated noise and conditional sensitivity worsen by 10log10(10) = 10 dB. Reusing the same required SNR and implementation loss is a conditional arithmetic exercise; a real changed filter requires a new receiver-quality argument.

\begin{aligned}B_{\mathrm{ENBW}}&=\frac{\int_0^\infty|H(f)|^2\,\mathrm df}{|H|_{\max}^2}\\N_{\mathrm{in}}&=k(T_s+T_{\mathrm{e,total}})B_{\mathrm{ENBW}}\end{aligned}

Definition for a flat one-sided input noise density and normalized response. H is the amplitude transfer function; B is its noise-equivalent width, not necessarily its 3 dB width.

Using the exact SI Boltzmann constant, k × 290 × 20000 gives −130.964887238 dBm available source noise. Adding the matched receiver’s 2.799879362 dB NF gives −128.165007875 dBm input-equivalent noise at R2. The receiver allocation uses that same bandwidth all the way through; it does not integrate unrelated per-stage bandwidths repeatedly.

Common misconceptionReceiver bandwidth equals data rate.

20 kHz and 20 kbit/s happen to share a numeral here. One is a filter integral; the other counts a precisely defined population of bits per second. Modulation, coding, pulse shaping and detection connect them through a model.

Go deeperWhen the scalar model must stop

If edited filters impose different frequency-dependent shapes, each stage’s added-noise spectrum must be weighted by the downstream response before integration. The ledger returns “inspect—scalar ENBW model insufficient.” It does not manufacture a spectral integral from a collection of widths. The required response/noise spectra remain an evidence request.

Bandwidth selects how much noise is admitted. Source temperature and the first loss determine what noise arrives at the LNA.

04 / 10

Source temperature and loss before the LNA

Does cooling the source change the receiver’s quoted NF?

Think about itHold the active stages fixed and change only source temperature from 290 to 400 K. Does receiver Te rise?

Answer

No. Te,total stays 262.568258826 K and NF stays 2.799879362 dB. The input-equivalent integrated noise rises from −128.165007875 to −127.376560950 dBm because Ts + Te increases. This is a hotter source, not a hot-device prediction.

\begin{aligned}G &= \frac{1}{L} \qquad T_{\mathrm{e}} = (L - 1)T_{\mathrm{p}} \\ F &= 1 + \frac{(L - 1)T_{\mathrm{p}}}{T_{0}} \\ \frac{N_{\mathrm{out}}}{k B} &= \frac{T_{\mathrm{s}}}{L} + (1 - \frac{1}{L})T_{\mathrm{p}}\end{aligned}

Matched passive loss L ≥ 1, physical Tp in K; G is a linear power gain. The physical output noise relation also provides a limiting-case check.

Independent temperature / loss checks · 20 kHz; active parameters fixed
Case	Derived result	Decision consequence
Ts = 200 K	Nin,R2 = −128.937108932 dBm	Use Ts + Te, not a non-290 K floor plus fixed NF in dB.
Ts = 400 K	Nin,R2 = −127.376560950 dBm	Antenna brightness/source noise is not device physical temperature.
RX-FE loss 1.5 → 0.5 dB at 290 K	NF 2.799879362 → 1.799879362 dB; gain 42.5 → 43.5 dB	Exactly 1 dB better NF, but 1 dB less converter headroom.
Loss = 0 dB, any permitted Tp	G = 1, F = 1, Te = 0	A lossless stage adds no thermal noise.
Tp = Ts with positive loss	Physical source noise remains kTsB at passive output	The signal attenuates; unchanged physical output noise does not imply unchanged SNR.

Common misconceptionFront-end loss is harmless if later gain is high.

Loss at the input reduces wanted power before low-noise gain and adds thermal noise. Later gain amplifies that degraded signal/noise pair. It cannot restore the lost SNR.

Rohde & Schwarz’s cold-source guidance distinguishes passive-interconnect physical temperature from DUT temperature. Antenna/source temperature is an equivalent scene-dependent input temperature. Pattern, ground, atmosphere and antenna/feed losses belong to Path 06; neither air temperature nor antenna-metal temperature alone supplies Ts. The next step is to allocate receiver-added noise stage by stage.

05 / 10

Use Friis as an allocation

Which stage is worth improving, and which number shows why?

Apply the Friis model from RF Fundamentals with compatible available-power gains and factors. Convert dB to linear units first. RX-MIX’s negative gain does not make it a passive attenuator; its effective noise convention is part of the allocation.

\begin{aligned}F_{\mathrm{total}}&=F_1+\frac{F_2-1}{G_1}+\frac{F_3-1}{G_1G_2}+\frac{F_4-1}{G_1G_2G_3}\\T_{\mathrm{e,total}}&=\sum_i\frac{T_{\mathrm{e},i}}{\prod_{j<i}G_j}\\F_{\mathrm{total}}&=1+\frac{T_{\mathrm{e,total}}}{T_0}\end{aligned}

Definition · Gi = 10^(gain dB/10), Fi = 10^(NF dB/10). Equivalent temperatures are referred to R2; products contain only preceding stages.

Independent nominal Friis terms · full precision retained by the solver
Stage	Linear G	Linear F	Input-referred term
RX-FE	0.707945784384	1.412537544623	1.412537544623
RX-LNA	100.000000000000	1.258925411794	0.365741865416
RX-MIX	0.251188643151	3.981071705535	0.042108757073
RX-DRV	1000.000000000000	2.511886431510	0.085019621943

Noise factors add under this Friis definition; NF values in dB do not. The first rectangle includes the reference factor 1. A smaller later contribution prevents additional SNR loss; it does not undo earlier loss.

Ftotal = 1.905407789054971; NF = 2.799879362 dB; Te = 262.568258826 K; total gain = 42.500000 dB. The first term’s 1.412537544623 includes the reference 1. Its excess 0.412537544623 is still the largest added-noise contribution; the LNA contributes 0.365741865416, mixer 0.042108757073 and driver 0.085019621943. Improving early loss is effective in this fixture.

Think about itMove the same 1.5 dB passive loss after the LNA. Is total gain unchanged, and is NF unchanged?

Answer

Total gain remains 42.5 dB. NF falls to 1.430708 dB because LNA gain now discounts the loss contribution. This hypothetical matched rearrangement leaves the LNA exposed to whatever the front end had rejected. Noise improvement is not permission to remove protection or blocker filtering.

Independent nominal physical power propagation · 20 kHz integrated noise
Stage output	Wanted dBm	Physical noise dBm	Cumulative NF dB
RX-FE → R1-LNA-in	-106.500000	-130.964887	1.500000
RX-LNA → R1-LNA-out	-86.500000	-109.964887	2.500000
RX-MIX → A1-IF	-92.500000	-115.863247	2.601640
RX-DRV → A1-ADC-in	-62.500000	-85.665008	2.799879

Common misconceptionCascaded NF values add in dB, and gain recovers SNR already lost.

The waterfall adds linear Friis terms, not NF heights in dB. Preceding gain makes later input-referred contributions smaller; it does not reverse the degradation already introduced by an earlier stage.

Go deeperReconcile an actual mixer before replacing RX-MIX

Analog Devices’ mixer-noise analysis distinguishes SSB, DSB and effective image-terminated cases. Our 6 dB is a declared effective single-signal-channel teaching NF with image noise already accounted for. Real substitution needs source/image terminations, their temperatures, conversion gains and selection bandwidths. Do not apply an automatic 3 dB correction or cascade a raw DSB figure.

The noise ledger is now traceable. The next conversion relates that noise to the rate and energy population used by the detector requirement.

06 / 10

Connect SNR, Eb/N0, bit rate, symbol rate, and coding

When do 8 dB SNR and 8 dB Eb/N0 describe the same requirement?

\begin{aligned}\mathrm{SNR}&=\frac CN\\C/N_0&=\mathrm{SNR}_{\mathrm{dB}}+10\log_{10}\left(\frac{B_{\mathrm{ENBW}}}{1\,\mathrm{Hz}}\right)\\E_b/N_0&=C/N_0-10\log_{10}\left(\frac{R_b}{1\,\mathrm{bit/s}}\right)\\&=\mathrm{SNR}_{\mathrm{dB}}+10\log_{10}\left(\frac{B_{\mathrm{ENBW}}}{R_b}\right)\end{aligned}

Definition · C and N at the same R3 reference and ENBW; C/N0 in dB-Hz, Rb is the explicitly selected bit population per second.

Independent rate exercise · required R3 SNR held at 8 dB
ENBW / bit population	C/N0 dB-Hz	Eb/N0 dB	Physical interpretation
20 kHz / 20 kbit/s information = mapped	51.010300	8.000000	Uncoded QPSK, no overhead: 2 bits/symbol, 10 ksymbol/s.
10 kHz / same 20 kbit/s population	48.000000	4.989700	Arithmetic variant only; do not assume the unchanged 13.5 kHz-support waveform and detector still meet quality.
Same symbol rate, coding/overhead introduced	Requires exact rate	Requires energy definition	Mapped/coded bits, information bits and payload bits no longer share one Rb.

The 10 kHz variant needs a specified receiver/filter transfer function, waveform/equalization choice and a re-evaluated required SNR. ENBW smaller than the support is a question about weighted filtering, not a universal brick-wall rejection test. A data rate alone cannot answer it.

With full QPSK mapping, Rmapped = 2Rs. If coding and framing exist, write the actual relation, for example Rinformation = Rc × Rcoded and Rpayload = ηframe × Rinformation under those explicitly defined efficiencies. Count coding gain only from a stated code, receiver, channel and error criterion. Revisit Signals & Modulation for the waveform model; Path 07 owns exact technology thresholds.

Common misconceptionSNR, Eb/N0 and required BER are interchangeable labels.

SNR is a power ratio within bandwidth. Eb/N0 is tied to an energy/bit population. BER and PER are outcomes of a specified receiver and signal/channel model. The conversions above establish no standard sensitivity or packet-error guarantee.

Once the quality criterion is fixed, distinguish penalties needed by that receiver from reserves needed by the link plan.

07 / 10

Keep every margin in its own row

Is an extra 3 dB a worse receiver, a production allowance, or a design reserve?

Default margin ledger · sensitivity criterion: 8.000 dB required R3 SNR + 2.000 dB RX-IMPL once; 20.000 kHz ENBW, Ts = 290 K
Allocation / owner	Value	Where charged / evidence
RX-IMPL / Receiver systems	2 dB	Once in sensitivity; distinct from required R3 SNR.
RX-PROD / Production	2 dB	Once in a separate receive-power planning target. Synthetic reserve, not measured spread.
RX-DESIGN / Receiver systems	1 dB	Once in the planning target; design discretion, not NF.
LINK-FADE / Propagation → Path 06	Unknown	Unresolved fading allowance; never silently zero.
MEAS-U / Validation → Path 08	Unknown	Measurement method not declared; separate from reserves.
ADC-LOSS / Receiver systems → 05.6	0.156451259 dB comparison	Optional independently added output-noise row; excluded by default and never hidden in RX-IMPL.

\begin{aligned}P_{\mathrm{sens}} &= -128.165007875 + 8 + 2 \\ &= -118.165007875 \mathrm{dBm} \\ P_{\mathrm{planning}} &= P_{\mathrm{sens}} + 2 + 1 = -115.165007875 \mathrm{dBm}\end{aligned}

Derived · R2 available powers, 8.000 dB required R3 SNR + 2.000 dB RX-IMPL once; 20.000 kHz ENBW, Ts = 290 K. The planning target is before an unknown fading allowance.

Common misconceptionFading margin belongs inside NF.

NF describes receiver noise under reference conditions. Fading is a propagation/availability question, production reserve is an allocation, and measurement uncertainty describes knowledge of a result. Folding them into NF hides owners and invites duplicate charges.

Go deeperUncertainty needs coefficients and covariance

NIST TN 1297, Appendix A, Eq. A-3 propagates standard uncertainties with sensitivity coefficients and covariance. In compact form u²y = cᵀΣc. For two additive-dB contributions with c1 = c2 = +1, uy = √(u1² + u2² + 2ρu1u2). This local exercise is not the derivative model of a complete receiver.

Independent uncertainty exercise · not added to the margin ledger
Basis	Calculation	Reported result
Independent standard uncertainties	√(0.3² + 0.4²)	0.5 dB standard uncertainty; no coverage factor assigned.
Perfect positive correlation	√(0.3² + 0.4² + 2×0.3×0.4)	0.7 dB standard uncertainty.
Only ±0.3 / ±0.4 dB bounded tolerances	Conservative extreme interval ±(0.3 + 0.4)	±0.7 dB interval; no Gaussian confidence/yield percentage.

Do not quadrature-add unknown phase errors, reserves, bounded fading or uncharacterized correlations. Record “unknown” until the uncertainty basis exists.

The same discipline applies at the converter: distinguish its noise contribution from the maximum permitted signal level.

08 / 10

Distribute gain without sacrificing dynamic range

How much gain hides converter noise without clipping the largest in-band signal?

Use a bounded 05.6 preview: an ideal 12-bit, real 80 MS/s converter with white, independent quantization noise. The full-scale reference is a 2.000 Vpp differential sine across a teaching 100 Ω load. ADI MT-001 supplies the ideal full-Nyquist SNR convention and the warning that quantization error need not be white or independent in real records.

\begin{aligned}P_{\mathrm{FS}} &= \frac{(\frac{2}{2\sqrt{2}})^{2}}{100} = 5 \mathrm{mW} \\ P_{\mathrm{FS}} &= 6.989700043 \mathrm{dBm} \\ N_{\mathrm{adc}} &= P_{\mathrm{FS}} - 74 + 10\log_{10}(\frac{20000}{40000000}) \\ &= -100.020599913 \mathrm{dBm} \text{at A1}\end{aligned}

Derived ideal benchmark. The rounded 6.02N + 1.76 convention is explicitly frozen at 74.00 dB; full-scale sine RMS is distinct from its rail peak.

Add independent converter and analog output noise in linear power. With analog output noise −85.665007875 dBm, the extra degradation is 10log10[1 + 10^((Nadc − Nanalog,out)/10)] = 0.156451259 dB. The optional ADC-LOSS row includes this once; the ordinary 2 dB implementation allocation remains a separately defined residual loss.

\begin{aligned}G_{\mathrm{min}} &= N_{\mathrm{adc}} - N_{\mathrm{in}} - 10\log_{10}(10^{\frac{D}{10}} - 1) \\ &= 37.280152770 \mathrm{dB}\end{aligned}

Derived from Nadc/Nanalog,out ≤ 10^(D/10)−1, with D = 0.5 dB. Nin is integrated analog input-equivalent noise at R2; Nadc is integrated independent noise at A1.

\begin{aligned}G_{\mathrm{max}} &= P_{\mathrm{FS}} + 3.010299957 - P_{\mathrm{max,in}} - \mathrm{CF} - \mathrm{reserve} \\ &= 41.000000 \mathrm{dB}\end{aligned}

Local maximum-level screen: largest in-band R2 average input −40 dBm, waveform crest factor CF = 6 dB, peak reserve 3 dB. The +3.0103 correction changes sine RMS full scale to rail peak.

Think about itDoes the default 42.5 dB gain clear both bounds? What changes if only driver gain becomes 28 dB?

Answer

The allowed total-gain interval is 37.280153–41.000000 dB. Default gain exceeds the upper bound by 1.5 dB despite clearing converter noise. Reducing driver gain from 30 to 28 dB gives 40.5 dB total, 0.5 dB remaining peak headroom and 0.245413 dB added noise. Both local screens clear. With its input-referred NF held fixed and no later analog stage, changing only this final gain leaves analog cascade NF unchanged; the solver still recomputes it.

Gain choice · local no-blocker screen; conditional sensitivity retains the stated R3 criterion
Choice	Analog output noise dBm	Converter added loss dB	Consequence
42.5 dB nominal	−85.665008	0.156451	Noise clears, maximum level fails by 1.5 dB.
40.5 dB; driver 28 dB	−87.665008	0.245413	Noise clears, maximum level clears with 0.5 dB.
40.5 dB plus ADC-LOSS once	Same analog noise	0.245413	Conditional sensitivity R2 −117.919595 dBm; 8.000 dB required R3 SNR + 2.000 dB RX-IMPL once; 20.000 kHz ENBW, Ts = 290 K; ADC-LOSS included separately.

Common misconceptionMaximize first-stage gain; a low calculated noise floor proves headroom.

Early gain suppresses later noise but also raises large-signal levels. The −105 dBm weak wanted signal cannot bound overload; a separate largest-input and crest/headroom contract is essential. When minimum gain exceeds maximum gain, the interval is empty and the allocation needs a different noise, level, filter or converter boundary.

This is not an AGC design. Real gain changes may alter NF, bandwidth, linearity and settling; recompute every supported state. Blockers, peak statistics and time-dependent control belong to 05.4. Driver input impedance, jitter, spurs and clipping probability belong to 05.6.

09 / 10

Check temperature, frequency, spread, and mode corners

Which corner is physically credible, and which row limits it?

Independent corners · R2 conditional sensitivity, 8 dB R3 SNR + 2 dB RX-IMPL; 20 kHz ENBW; ADC-LOSS excluded
Corner / changed input	Gain / NF dB	Nin dBm	Sensitivity dBm	Binding consequence
Nominal · Ts 290 K	42.500 / 2.799879	-128.165008	-118.165008	RX-FE largest excess term; converter level fails.
Hot source · only Ts 400 K	42.500 / 2.799879	-127.376561	-117.376561	Higher source noise; active-device NF unchanged.
Lower preloss · only 0.5 dB at 290 K	43.500 / 1.799879	-129.165008	-119.165008	1 dB better NF; 2.5 dB above converter gain maximum.
Reduced driver · only gain 28 dB	40.500 / 2.799879	-128.165008	-118.165008	0.5 dB headroom; 0.245413 dB added converter noise.

These are one-at-a-time synthetic cases. No device temperature coefficient or manufacturing distribution is invented. In particular, “hot source” is not “all devices hot,” and “lower preloss” changes only a matched 290 K attenuator. A correlated hot-bias/process mode needs a joint parameter set, not the independently worst number from every unrelated data-sheet plot.

Unresolved operating-corner evidence
Dimension	Record required	Owner / decision
Frequency / acquisition	Loss, gain, NF and filter/noise response over tuning and wider acquisition modes	Receiver systems; the narrow nominal filter may not support 05.2 clock-error acquisition.
Device temperature / bias / spread	Paired gain/NF curves, conditions, guaranteed limits or measured population	Components / Production; choose credible correlated combinations.
Source / antenna scene	Equivalent Ts at R2 with loss and scene assumptions	Path 06; preserve receiver Te.
Gain / blocker state	Real stage gain, NF, compression and settling in each mode	05.4; compare noise and large-signal constraints together.
Measurement	Reference planes, method, calibration and uncertainty	Path 08; no production yield or PER guarantee from the scalar table.

Do not select a single headline sensitivity for all modes. Carry a row for each supported mode and ask which specific evidence could overturn its decision.

10 / 10

Reconcile the gateway sensitivity budget

Can you repair the workbook and defend one conditional gain distribution?

Inspect nominal: distinguish input-equivalent noise from physical output noise, and locate the converter-level failure.
Load Flawed-budget repair. Explain the dB-in-Friis and 200 kHz errors before applying each repair.
Expose RX-FE, charge RX-IMPL once, and move unresolved fading into LINK-FADE.
Load lower preloss. Predict the exact 1 dB NF improvement and inspect the new maximum-level conflict.
Load source 200 K and 400 K. Confirm that source changes leave device NF and receiver Te fixed.
Return to nominal, compare 42.5 dB with both converter bounds, then load driver 28 dB. Recompute and copy the complete conditional record.

Class 1 · local teaching model

Receiver Budget Ledger

Repair the bookkeeping, then choose a gain distribution. Apply commits a complete draft; each preset restores all its inputs and evidence. Static worked tables below remain the reviewed nominal reference.

Complete scenario preset

Signal, noise and criterion at named planes

Source temperature at R2 (K)200–400 K; step 1.

Common detector ENBW (kHz)1–1000 kHz; step 0.1.

Required detector SNR at R3 (dB)0–25 dB; step 0.1.

RX-IMPL · implementation loss (dB)0–10 dB; step 0.1.

Uncoded mapped / information bit rate (kbit/s)1–1000 kbit/s; step 0.1.

Wanted average input at R2 (dBm)-150–-20 dBm; step 0.1.

Largest expected in-band average input at R2 (dBm)-100–0 dBm; step 0.1.

Fixture metadata: 2.450 GHz, nominal bias and 25 °C active-device characterization; fixed T0 = 290 K. RX-FE is physically 290 K by default. Common ENBW is selected at R3; the mixer/IF filter marks selection, and other stages are flat over that band. A changed response shape needs a spectral model.

Edit stage chain · 4 of 12 rows

Default lineage: R2 → R1-LNA-in → R1-LNA-out → A1-IF → A1-ADC-in. A reordered/expanded chain uses local-after-ID planes and needs compatible interfaces. Stage data remain attached to stable IDs.

1. RX-FE

Stage labelUp to 80 characters.

Stage class

Matched passive loss (dB)0–30 dB; step 0.1.

Passive physical temperature (K)200–400 K; step 1.

Response across common ENBW

Derived: G = 0.707945784; F = 1.412537545; NF = 1.500000 dB; Te = 119.635888 K. NF follows physical temperature; F = loss only at 290 K.

RX-FE input P1dB metadata · optional

Compression evidence

Waveform, frequency, temperature and input plane conditionsUp to 240 characters. Blank conditions keep compression evidence unknown. A CW threshold is not modulated blocker proof.

2. RX-LNA

Stage labelUp to 80 characters.

Stage class

Signed matched available-power gain (dB)-30–60 dB; step 0.1.

Noise parameter evidenceSelecting supplied initializes 0 visibly; enter the actual allocation before applying.

Controlling noise representationThe alternate representation is derived; conversion preserves full precision, including off-step values.

Noise figure at 290 K (dB)0–30 dB; step 0.1.

Response across common ENBW

Derived: G = 100.000000000; F = 1.258925412; NF = 1.000000 dB; Te = 75.088369 K. Changing source temperature does not change this stage’s NF.

RX-LNA input P1dB metadata · optional

Compression evidence

Waveform, frequency, temperature and input plane conditionsUp to 240 characters. Blank conditions keep compression evidence unknown. A CW threshold is not modulated blocker proof.

3. RX-MIX

Stage labelUp to 80 characters.

Stage class

Signed matched available-power gain (dB)-30–60 dB; step 0.1.

Noise parameter evidenceSelecting supplied initializes 0 visibly; enter the actual allocation before applying.

Controlling noise representationThe alternate representation is derived; conversion preserves full precision, including off-step values.

Noise figure at 290 K (dB)0–30 dB; step 0.1.

Response across common ENBW

Mixer image-noise convention

Derived: G = 0.251188643; F = 3.981071706; NF = 6.000000 dB; Te = 864.510795 K. Changing source temperature does not change this stage’s NF.

RX-MIX input P1dB metadata · optional

Compression evidence

Waveform, frequency, temperature and input plane conditionsUp to 240 characters. Blank conditions keep compression evidence unknown. A CW threshold is not modulated blocker proof.

4. RX-DRV

Stage labelUp to 80 characters.

Stage class

Signed matched available-power gain (dB)-30–60 dB; step 0.1.

Noise parameter evidenceSelecting supplied initializes 0 visibly; enter the actual allocation before applying.

Controlling noise representationThe alternate representation is derived; conversion preserves full precision, including off-step values.

Noise figure at 290 K (dB)0–30 dB; step 0.1.

Response across common ENBW

Derived: G = 1000.000000000; F = 2.511886432; NF = 4.000000 dB; Te = 438.447065 K. Changing source temperature does not change this stage’s NF.

RX-DRV input P1dB metadata · optional

Compression evidence

Waveform, frequency, temperature and input plane conditionsUp to 240 characters. Blank conditions keep compression evidence unknown. A CW threshold is not modulated blocker proof.

Margin provenance and uncertainty

Implementation is fixed to RX-IMPL / Receiver systems. Add only distinct reserves here. Fading remains outside the displayed planning target, even when a value is supplied.

Reserve row 1

Allocation IDUp to 40 characters.

Nominal · four stages · committed budget

Illustrative matched small-signal allocation; 2.450 GHz, nominal bias, active devices at 25 °C; passive temperature separately declared. RF R2/R1/R0: 50 Ω; A1 teaching driver: 100 Ω differential matched power transfer. Common R3 detector ENBW. No hardware or standards performance claim.

Criterion: 8.000 dB required R3 SNR, 20.000 kHz ENBW, 290 K source, RX-IMPL 2.000 dB once. Uncoded QPSK rate exercise 20.000 kbit/s; support metadata 13.5 kHz; D3 80 ksample/s. A0 benchmark 80 MS/s is separate.

Conditional sensitivity · available input at R2 · criterion above: -118.165008 dBm
Planning target · R2 · same criterion + known production/design reserves, before fading: -115.165008 dBm
Input-equivalent integrated analog noise · R2: -128.165008 dBm
Physical integrated analog output noise · A1-ADC-in: -85.665008 dBm
Receiver NF at fixed T0 = 290 K / total gain: 2.799879 / 42.500000 dB
Receiver equivalent temperature · R2: 262.568259 K

Independent decision axes · a scalar clearance is conditional
Axis	State	Reason / evidence request
Conditional wanted signal	screen clears	R2 wanted -105.000000 dBm versus -118.165008 dBm; 8.000 dB required R3 SNR, 20.000 kHz ENBW, 290 K source, RX-IMPL 2.000 dB once. This is a scalar allocation.
Converter added noise	screen clears	0.156451 dB added versus 0.500000 dB allowed; G ≥ 37.280153 dB. Ideal independent white quantization only.
Converter maximum level	fails supplied limit	G ≤ 41.000000 dB; -1.500000 dB remaining below the supplied peak/reserve bound. Largest input -40.000000 dBm at R2.
Gain interval	fails supplied limit	Permissible scalar interval 37.280153–41.000000 dB. Recompute Friis after redistributing real stage gain.
Filter / rate evidence	screen clears	Declared nominal 20 kHz ENBW, 13.5 kHz ideal RC support, 20 kbit/s uncoded mapping. Flat scalar allocation only.
Stage compression / blockers	inspect—missing/out-of-domain evidence	Input P1dB is a conditioned single-tone comparison only. Missing rows, modulated blockers, peak statistics, reciprocal mixing and AGC require 05.4 evidence.
Driver / actual converter	inspect—missing/out-of-domain evidence	100 Ω is a teaching power-transfer boundary. Switched input, settling, common mode, jitter, SFDR and clipping probability belong to 05.6.
Link / production / measurement	inspect—missing/out-of-domain evidence	Reserve-loaded target excludes fading. Known fading rows are recorded separately; a propagation model and measurement uncertainty method remain required.

Largest receiver-added contribution: RX-FE, excess factor 0.412537545 referred to R2. Improving this term offers the largest direct noise opportunity at the committed gain distribution. The supplied converter peak bound is violated; better NF alone does not repair that level constraint.

Linear Friis terms · R2 input referral · common detector ENBW
Stage	G [1]	F [1]	Term [1]	Σ gain dB	Σ F / NF dB	Σ Te K
RX-FE	0.707945784384	1.412537544623	1.412537544623	-1.500000	1.412537544623 / 1.500000	119.635888
RX-LNA	100.000000000000	1.258925411794	0.365741865416	18.500000	1.778279410039 / 2.500000	225.701029
RX-MIX	0.251188643151	3.981071705535	0.042108757073	12.500000	1.820388167112 / 2.601640	237.912568
RX-DRV	1000.000000000000	2.511886431510	0.085019621943	42.500000	1.905407789055 / 2.799879	262.568259

Physical plane ledger · powers in dBm integrated over common ENBW
Stage / local plane lineage	Signal in	Signal out	Noise in	Noise out	Stage gain / NF / Te	P1dB evidence
RX-FE · Protection / filter / switch · R2 → R1-LNA-in	-105.000000	-106.500000	-130.964887	-130.964887	-1.500 dB / 1.500 dB / 119.636 K	inspect—input P1dB or conditions unknown
RX-LNA · LNA · R1-LNA-in → R1-LNA-out	-106.500000	-86.500000	-130.964887	-109.964887	20.000 dB / 1.000 dB / 75.088 K	inspect—input P1dB or conditions unknown
RX-MIX · Mixer / IF filter · R1-LNA-out → A1-IF	-86.500000	-92.500000	-109.964887	-115.863247	-6.000 dB / 6.000 dB / 864.511 K	inspect—input P1dB or conditions unknown
RX-DRV · IF / ADC driver · A1-IF → A1-ADC-in	-92.500000	-62.500000	-115.863247	-85.665008	30.000 dB / 4.000 dB / 438.447 K	inspect—input P1dB or conditions unknown

Reordering creates explicit local-after-ID interstage planes; these are a hypothetical matched chain requiring new interface evidence. R2 remains the antenna feed; R3 remains the detector. Input-referred cumulative noise differs from physical noise at each output. RX-FE sets the largest excess-factor contribution in the nominal chain; early gain discounts later noise.

Converter comparison · 05.6 teaching benchmark, independently added at A1
Quantity	Value / unit	Condition
Full-scale sine RMS reference	6.989700 dBm	2 Vpp differential across the teaching 100 Ω; rail peak is +3.010299957 dB above sine RMS.
Independent in-band converter noise	-100.020600 dBm	Ideal 12-bit real 80 MS/s; 74.00 dB full-Nyquist SNR; flat white-noise allocation to the selected ENBW.
Added degradation	0.156451 dB	ADC-LOSS · comparison only; excluded from RX-IMPL and sensitivity.
Minimum / maximum total gain	37.280153 / 41.000000 dB	Noise allocation versus largest input + crest + peak reserve; not a complete AGC design.

Margin provenance · R2 receive-power planning; never hidden inside NF
Allocation ID	Category / owner	Value dB	Destination
RX-IMPL	Implementation / Receiver systems	2.000	Included once in sensitivity; required R3 SNR stays separate.
RX-PROD	production / Production	2.000	Added once to before-fading planning target.
RX-DESIGN	design / Receiver systems	1.000	Added once to before-fading planning target.
LINK-FADE	fading / Propagation	unknown	Separate Path 06 allowance; excluded from the displayed before-fading target.
ADC-LOSS	Converter / Receiver systems → 05.6	comparison only	Independent output noise; never duplicated in RX-IMPL.
MEAS-U	Measurement / Validation → Path 08	unknown	No method declared; not a design reserve.

Uncertainty layer: Disabled; actual measurement uncertainty remains unknown.

Rate exercise: C/N0 = 51.010300 dB-Hz; Eb/N0 = 8.000000 dB at the required SNR. Exact bit population: uncoded information bits = mapped bits, overhead excluded. Edited rates do not update the frozen waveform or validate a new receiver.

Model / fixture p05-m03-receiver-ledger-v1. Full precision is retained internally; dB/K displays use 6 decimals, linear tables 12, compact conditions 3; tiny/large values use scientific notation. Calculation precision is not device evidence.

Printable synthesis · reviewed allocation

Use the 40.5 dB gain state for this local screen

Retain RX-FE −1.5 dB/1.5 dB NF, RX-LNA +20/1, RX-MIX −6/6 and RX-DRV +28/4 at the stated matched planes, nominal bias and active-device 25 °C conditions; RX-FE is physically 290 K. Keep the original four-row 42.5 dB fixture frozen for the 05.4 handoff, and label the 28 dB driver state as a variant.

Analog F = 1.905407789054971, Te = 262.568258826 K, NF = 2.799879362 dB. Source kTB = −130.964887238 dBm, R2 input-equivalent noise = −128.165007875 dBm, A1 analog output noise = −87.665007875 dBm. Conditional R2 sensitivity is −118.165007875 dBm for 8.000 dB required R3 SNR + 2.000 dB RX-IMPL once; 20.000 kHz ENBW, Ts = 290 K. Planning target is −115.165007875 dBm before unknown fading, with RX-PROD 2 dB and RX-DESIGN 1 dB added once.

The 40.5 dB total gain lies between 37.280152770 and 41.000000 dB, giving 0.5 dB headroom under the local −40 dBm / 6 dB crest / 3 dB reserve contract. Independent converter added noise is 0.245412512 dB. If ADC-LOSS is explicitly included once, conditional R2 sensitivity becomes −117.919595363 dBm for the same criterion plus ADC-LOSS; before-fading planning target becomes −114.919595363 dBm. Measurement uncertainty stays unknown.

RX-FE remains the largest nominal receiver-added noise contribution. The default upper-gain conflict is set by the supplied largest input, crest and rail boundary. This variant clears those two scalar converter screens only; it has no verified blocker, driver, PER, production or stability guarantee.

Repair workbook · explain the error before recalculating
Flaw	Repair	Corrected consequence
dB values entered directly into Friis	Convert gain and NF with 10^(dB/10); add the resulting linear terms	F = 1.905407789054971, NF = 2.799879362 dB.
200 kHz used instead of 20 kHz	Restore declared R3 ENBW; no change to 13.5 kHz support or D3 rate	Noise/sensitivity improve exactly 10 dB in this flat model.
Pre-LNA loss hidden	Restore RX-FE at R2 before the LNA	R2 retains the antenna-feed plane; RX-FE excess factor 0.412537544623 is visible.
Implementation loss duplicated	RX-IMPL owns 2 dB exactly once	Remove the extra 2 dB error; nominal conditional sensitivity −118.165007875 dBm for the stated R3 criterion.
Fading disguised as NF	Return it to LINK-FADE / Propagation	Unknown allowance remains unknown; planning and receiver sensitivity stay distinct.

Sensitivity / gain handoff · evidence requests with owners
Owner	Preserve	Request before hardware acceptance
05.4 · Receiver / coexistence	Frozen p05-m03-receiver-ledger-v1 stage IDs, planes, nominal 42.5 dB gain and separate driver-28 variant	Actual blocker levels/spectra, stage P1dB/IIP3, filtering, reciprocal mixing, gain-state NF, detector thresholds and settling.
05.6 · Converter / driver	A1 100 Ω teaching boundary, 2 Vpp sine, 80 MS/s vs D3 80 ksample/s, ADC-LOSS ID	Actual differential input model, drive/common mode, quantization/thermal noise, in-band spectrum, jitter, SFDR and peak/clipping contract.
Path 06 / Path 07	R2 available-power threshold and exact QPSK/rate/ENBW/SNR population	Antenna/source temperature and fading model; technology/receiver-specific BER/PER/EVM criterion.
Production / Path 08	RX-PROD reserve distinct from MEAS-U	Characterized joint gain/NF spread, method, calibration, traceability and uncertainty.

Your synthesis should name the limiting row, retain every margin ID/owner and explain one condition that could reverse the selected gain state. Use 05.1’s requirement grammar to turn each unknown into a discriminating evidence request.

Ungraded review

Check your understanding

Answer each question in your own words, then reveal the model answer.

01Repair the flawed workbook in the correct order.
Model answer
Convert dB to linear G/F; restore 20 kHz R3 ENBW; expose 1.5 dB matched RX-FE loss at R2; charge RX-IMPL 2 dB once; move fading to unresolved LINK-FADE. Nominal F=1.905407789054971 and NF=2.799879362 dB. Conditional R2 sensitivity = −118.165007875 dBm for required R3 SNR 8 dB, 20 kHz ENBW, Ts 290 K and 2 dB implementation once.
02Reproduce the mixer’s linear Friis term.
Model answer
GFE=10^(−1.5/10), GLNA=100, FMIX=10^(6/10). (FMIX−1)/(GFE×GLNA)=0.042108757072807. This is a dimensionless contribution to F, not a 0.042 dB NF segment.
03Why does the 10× bandwidth error produce 10 dB?
Model answer
For fixed flat density, N=k(Ts+Te)B is proportional to B. 10log10(200/20)=10 dB. NF is unchanged; with the same stated SNR/implementation criterion the conditional sensitivity shifts by the same 10 dB.
04Compute the independent and correlated uncertainty exercises.
Model answer
For two additive-dB standard uncertainties 0.3 and 0.4 with c1=c2=1, covariance zero gives √(0.09+0.16)=0.5 dB. Perfect positive correlation adds 2×0.3×0.4 inside the root, giving 0.7 dB. Bounded tolerances instead give a conservative ±0.7 dB interval without a probability claim.
05Reject “maximize first-stage gain” using both converter screens.
Model answer
Noise requires total gain ≥37.280152770 dB; the supplied largest-input/crest/reserve screen requires ≤41 dB. Default 42.5 fails maximum level. Driver 28 dB produces 40.5 total and 0.245412512 dB added converter noise, clearing both scalar screens. More early gain can overload later stages and does not undo prior SNR loss.
06Which temperatures and quality claims remain separate?
Model answer
T0=290 K normalizes NF. Ts=200/290/400 K is source noise at R2. RX-FE physical Tp sets passive noise. Active-device 25 °C is fixed fixture metadata with no supplied temperature law. R3 SNR 8 dB is an illustrative allocation; no named BER/PER/EVM or production guarantee follows.

Sources and further study

Access checked 7 September 2026. Synthetic stage values and converter/load assumptions belong to p05-m03-receiver-ledger-v1, not to a manufacturer or standard. Model p05-m03-receiver-ledger-v1; rules p05-m03-budget-rules-v1; serialization p05-m03-ledger-json-v1; display p05-m03-display-v1. The independent Decimal oracle uses 50-digit arithmetic and forward physical noise propagation. Tolerances: relative 10⁻¹⁰ for G/F/terms, absolute 10⁻⁶ dB and 10⁻⁶ K. No measured or normative receiver threshold is claimed.

NIST SP 330, 2019 edition, §2: exact SI k = 1.380649×10⁻²³ J/K. The requested constants CGI endpoint returned an ill-formed-request page; SP 330 and the NIST CODATA table supplied the primary cross-check.
D. Staelin, MIT 6.661, Lecture 2: Thermal Noise, Spring 2003 course, slide 16 (the supplied deck carries Fall 2001 footers): kTB / Rayleigh–Jeans thermal available-power foundation carried forward from RF Fundamentals Noise.
Leffel and Stumpf, Rohde & Schwarz 1SL378, The Cold Source Technique for Noise Figure Measurements, version 0e, December 2021, §§3.2–3.4 and §4.5.3: available thermal noise, equivalent temperature and the distinction between passive-interconnect temperature and DUT temperature. The local passive-loss equation is independently checked by thermal equilibrium; measurement procedure remains deferred.
Analog Devices, System Noise-Figure Analysis for Modern Radio Receivers, 14 June 2013, “Noise-Figure Definitions,” “Single-Sideband Noise Factor,” “Double-Sideband Noise Factor” and cascade/equivalent-temperature sections: gain, image termination and NF-convention checks. No data-sheet mixer NF is transplanted.
Walt Kester, Analog Devices MT-001, served seven-page ©2009 edition, pp. 2–5, Eqs. 9–10: ideal full-scale sine SNR, Nyquist bandwidth, process gain and quantization-correlation limits. The 74.00 dB teaching convention and 100 Ω boundary are explicit local assumptions.
NIST TN 1297, 1994 edition, §5 and Appendix A, Eq. A-3: standard uncertainty, sensitivity coefficients and covariance. Our two-term c1=c2=1 examples are local derived arithmetic, not an uncertainty analysis of a characterized receiver.
W. F. Egan, Practical RF System Design, Wiley–IEEE, 2003, Chapter 2 “Gain” (pp. 7–45) and Chapter 3 “Noise Figure” (pp. 47–90). Publisher contents consulted for the curriculum reading path; full chapter text was unavailable. Numerical anchors are independently derived, not attributed to unread pages.

The passive formula is also checked by thermal equilibrium: output noise temperature = Ts/L + (1−1/L)Tp. Repository RF Fundamentals Noise and the pure shared noise helpers use the same matched conventions. Tools’ fixed-reference thermal helper is not used for variable source temperature.

Sensitivity, Gain& Noise Budget

Failure: three “-174 dBm/Hz” answers disagree

Define sensitivity at a plane and quality threshold

Noise bandwidth is not a generic channel width

Source temperature and loss before the LNA

Use Friis as an allocation

Connect SNR, Eb/N0, bit rate, symbol rate, and coding

Keep every margin in its own row

Distribute gain without sacrificing dynamic range

Check temperature, frequency, spread, and mode corners

Reconcile the gateway sensitivity budget

1. RX-FE

2. RX-LNA

3. RX-MIX

4. RX-DRV

Nominal · four stages · committed budget

Use the 40.5 dB gain state for this local screen

Check your understanding

Sources and further study

Sensitivity, Gain
& Noise Budget