Noise in RF Systems — RF Theory

// 01 — Thermal Noise

Thermal Noise Fundamentals

All resistive elements generate noise due to random thermal motion of electrons. This Johnson-Nyquist noise is present in every resistor, transmission line, and lossy component — regardless of whether any signal is present. It sets the absolute floor below which no signal can be detected.

Thermal Noise — Key Formulas

P_n = k·T·B (watts) = −174 dBm/Hz at T₀ = 290 K
V_n = √(4·k·T·R·B) (V_rms, open circuit across R)
I_n = √(4·k·T·G·B) (A_rms, through conductance G)
k = 1.381×10⁻²³ J/K, T₀ = 290 K, B = noise bandwidth (Hz)

Why −174 dBm/Hz? P = kT₀ = 1.381×10⁻²³ × 290 = 4.004×10⁻²¹ W = 4.004×10⁻¹⁸ mW → 10·log₁₀(4.004×10⁻¹⁸) = −173.97 ≈ −174 dBm/Hz. The single most important number in RF system design.

Noise Floor vs Bandwidth

Example 1 — Noise power in 50 Ω at T=290 K

Bandwidth	Noise Power	dBm	V_n (50 Ω)
1 Hz	4.004×10⁻²¹ W	−174.0 dBm	0.895 nV
200 kHz (GSM)	8.008×10⁻¹⁶ W	−111.0 dBm	400 nV
20 MHz (WiFi)	8.008×10⁻¹⁴ W	−101.0 dBm	4.00 μV
100 MHz (5G NR)	4.004×10⁻¹³ W	−94.0 dBm	8.94 μV
1 GHz (wideband)	4.004×10⁻¹² W	−84.0 dBm	28.3 μV

Noise floor (dBm) = −174 + 10·log₁₀(B[Hz]). Verification 20 MHz: −174+73.0 = −101.0 dBm ✓ · V_n=√(4×kT₀×50×20M)=4.00 μV ✓

Bandwidth matters enormously: GSM 200 kHz → WiFi 20 MHz = 10·log₁₀(100) = 20 dB higher noise floor. Wideband systems require lower NF to compensate.

// 02 — Noise Figure

Noise Figure & Noise Temperature

The Noise Figure (NF) quantifies how much SNR degrades through a component. Ideal = 0 dB. Every real amplifier, mixer, filter and cable has NF > 0 dB.

Noise Factor F and Noise Figure NF

F = SNR_in/SNR_out (linear, ≥1) · NF = 10·log₁₀(F) dB
T_e = T₀·(F−1) = 290·(F−1) K · F = 1 + T_e/T₀

NF=3 dB → F=2 → T_e=290 K · NF=1 dB → F=1.259 → T_e=75 K (LNA) · NF=10 dB → T_e=2610 K

Lossy passives: A 3 dB attenuator/cable at T₀ has NF = 3 dB — adds same noise as it attenuates signal. This is why lossy feed cables before an LNA devastate system NF.

NF (dB)	F	T_e (K)	Typical Component
0.5	1.122	35 K	Best cryogenic LNA
1.0	1.259	75 K	Low-noise LNA (Qorvo QPL9065)
2.0	1.585	170 K	Typical LNA
3.0	2.000	290 K	Mixer, moderately lossy path
6.0	3.981	867 K	Typical mixer, driver amplifier
10.0	10.00	2610 K	Lossy passive, poor amplifier

Y-Factor Measurement

// Y-Factor Measurement Procedure

Equipment: Calibrated noise source (Keysight 346B, ENR≈15 dB) + Spectrum analyser or NFA

1. Noise source ON → record P_hot · Noise source OFF → record P_cold
2. Y = P_hot/P_cold (linear) · F = (10^(ENR/10)−Y)/(Y−1) · NF = 10·log₁₀(F)

Example: ENR=15 dB, P_hot=−62 dBm, P_cold=−66 dBm
Y = 10^(4/10) = 2.512 · F = (31.62−2.512)/(2.512−1) = 29.11/1.512 = 19.25 · NF = 12.84 dB
(This is system NF including analyser — apply 2nd-stage correction to get DUT NF alone)

// 03 — Friis Cascaded Noise Figure

Friis Cascaded Noise Figure

The Friis formula shows that the first stage dominates system NF — which is why an LNA always comes first in a receiver chain.

Friis Formula

F_total = F₁ + (F₂−1)/G₁ + (F₃−1)/(G₁G₂) + … (all linear, not dB)
If G₁ is large → later stages contribute negligibly · If G₁=1 (0 dB loss) → F_total=F₁·F₂

Worked Example — 2.4 GHz WiFi Receiver

Example 2 — Cable → BPF → LNA → Mixer → IF Amp

Cable

1.0 dB

→

BPF

1.5 dB

→

LNA

NF=1.5 dB
G=18 dB

→

Mixer

NF=8 dB
G=−6 dB

→

IF Amp

NF=5 dB
G=20 dB

1

Linear: F₁=1.259,G₁=0.794 · F₂=1.413,G₂=0.708 · F₃=1.413,G₃=63.10 · F₄=6.310,G₄=0.251 · F₅=3.162

2

G₁G₂=0.562 · G₁G₂G₃=35.46 · G₁G₂G₃G₄=8.901

3

F_total = 1.259 + 0.413/0.794 + 0.413/0.562 + 5.310/35.46 + 2.162/8.901
= 1.259 + 0.520 + 0.735 + 0.150 + 0.243 = 2.907

4

NF_total = 10·log₁₀(2.907) = 4.63 dB

✓ System NF = 4.63 dB. Cable+BPF before LNA contribute 2.5 dB of the total. Removing cable alone drops system NF to ~3.5 dB — biggest single improvement.

Design rule: Every 1 dB of loss before the LNA adds exactly 1 dB to system NF. Place the LNA as close to the antenna as physically possible.

Cascaded NF Calculator →

// 04 — Receiver Sensitivity

Receiver Sensitivity

S_min = −174 + NF + 10·log₁₀(B) + SNR_min (dBm)

Practical Sensitivity Examples

Example 3 — Sensitivity for common wireless standards

Standard	BW	10·log(B)	NF	SNR_min	Sensitivity
GSM	200 kHz	53 dB	8 dB	10 dB	−103 dBm
WiFi 802.11b	22 MHz	73.4 dB	8 dB	4 dB	−88.6 dBm
LTE 10 MHz	9 MHz	69.5 dB	7 dB	−1 dB	−98.5 dBm
GPS L1	2 MHz	63 dB	3 dB	−27 dB	−135 dBm
Bluetooth LE	2 MHz	63 dB	10 dB	−3 dB	−104 dBm

GSM: −174+8+53+10 = −103 dBm ✓ · LTE: −174+7+69.5−1 = −98.5 dBm ✓ · GPS achieves −135 dBm through spread-spectrum processing gain (~43 dB)

Example 4 — What does −103 dBm feel like physically?

−103 dBm = 50.1 fW = 50.1×10⁻¹⁵ W
Across 50 Ω: V = √(P·R) = √(2.505×10⁻¹²) = 1.58 μV peak
A GSM phone detects 1.58 μV while the LO runs at ~1 V in the same IC — 9 orders of magnitude of dynamic range in one chip.

// 05 — Linearity

Linearity — P1dB & IIP3

1 dB Compression Point

P1dB

P1dB_out = P1dB_in + G − 1 dB · P1dB_in ≈ IIP3 − 9.6 dB (theoretical)
LNA: P1dB_out ≈ 10–20 dBm · Driver: 20–30 dBm · PA (1W): ~30 dBm

Third-Order Intercept — IIP3

IIP3 and OIP3

IIP3 = P_in + ΔIM3/2 where ΔIM3 = P_fund−P_IM3 at output (dBm)
OIP3 = IIP3 + G · IM3 grows at 2:1 slope vs fundamental

Two-Tone Test — Worked Example

Example 5 — LNA IIP3 from two-tone measurement

G=20 dB, f₁=2.400 GHz, f₂=2.401 GHz, P_in=−30 dBm. Measure P_IM3 at 2f₁−f₂=2.399 GHz = −62.0 dBm

1

P_fund=−30+20=−10 dBm ✓ (no compression) · ΔIM3=−10−(−62)=52.0 dBc

2

IIP3 = P_in+ΔIM3/2 = −30+26 = −4.0 dBm

3

OIP3 = −4+20 = +16.0 dBm · P1dB_in ≈ −4−9.6 = −13.6 dBm

4

Verify: raise P_in 10 dB → P_fund rises 10 dB, P_IM3 rises 30 dB → ΔIM3 drops 20 dBc (2:1 slope) ✓

✓ IIP3 = −4.0 dBm, OIP3 = +16.0 dBm, P1dB_out ≈ +5.4 dBm. IM3 slope = 2:1 confirms 3rd-order behaviour.

// 06 — Cascaded Linearity

Cascaded IIP3

The last stage dominates cascaded IIP3 — opposite of Friis for NF. High gain helps noise but hurts linearity.

Cascaded IIP3

1/IIP3_total = 1/IIP3₁ + G₁/IIP3₂ + G₁G₂/IIP3₃ + … (all linear mW)

Example 6 — LNA (G=20 dB, IIP3=−4 dBm) → Mixer (IIP3=+10 dBm)

1

1/IIP3_total = 1/0.398 + 100/10.0 = 2.513 + 10.0 = 12.513 mW⁻¹

2

IIP3_total = 1/12.513 = 0.0799 mW = −11.0 dBm

✓ System IIP3 = −11.0 dBm vs LNA alone at −4.0 dBm. LNA gain of 20 dB delivers 100× more signal to the mixer — fundamental gain/linearity trade-off.

// 07 — Dynamic Range

Dynamic Range

SFDR and LDR

SFDR = (2/3)·(IIP3 − P_noise) (dB)
LDR = P1dB_in − S_min (dB)

WiFi 20 MHz — Full Dynamic Range Budget

Example 7 — NF=5 dB, IIP3=−10 dBm, P1dB_in=−19.6 dBm, B=20 MHz, SNR_min=10 dB

1

Noise floor = −174+5+73 = −96 dBm

2

Sensitivity = −96+10 = −86 dBm

3

SFDR = (2/3)·(−10−(−96)) = (2/3)·86 = 57.3 dB

4

LDR = −19.6−(−86) = 66.4 dB

✓ Handles −86 dBm to −19.6 dBm (LDR=66.4 dB). SFDR=57.3 dB — spurious-free operating window.

Trade-off: SFDR improves +1.5 dB per dB of IIP3 improvement vs +0.67 dB per dB of NF improvement. Better to improve linearity than NF for SFDR, but sensitivity requires low NF. Both matter.

// 08 — Practical Design

Practical Receiver Design

5G NR 3.5 GHz Front-End — NF Budget

Example 8 — 3.5 GHz, 100 MHz BW, 9 dB NF budget (3GPP UE)

Chain: Ant. Switch (1 dB loss) → BPF (1.5 dB loss) → LNA (NF=1.5 dB, G=16 dB) → Mixer (NF=10 dB)

1

G₁=0.794, G₁G₂=0.562, G₁G₂G₃=0.562×39.81=22.38

2

F_total=1.259+0.413/0.794+0.413/0.562+9.0/22.38=1.259+0.520+0.735+0.402=2.916

3

NF=10·log₁₀(2.916) = 4.65 dB — 4+ dB margin vs 9 dB budget ✓

✓ Standard front-end achieves NF=4.65 dB. Margin covers PCB trace losses, connectors and temperature drift.

Common NF Mistakes

Mistake	Effect	Fix
Long cable before LNA	Each dB adds 1 dB to system NF	Mount LNA at antenna (tower-top amp)
High-loss BPF before LNA	2 dB filter = 2 dB NF penalty	Use SAW/BAW <1 dB IL, or move filter after LNA
Too much LNA gain	Compresses mixer, degrades IIP3	Optimise gain for NF/IIP3 trade-off
Input mismatch at LNA	Mismatch loss adds to NF; LNA NF not achieved	Match to Γ_opt, not Γ_MS
NF without 2nd-stage correction	Measurement too high — includes analyser NF	Apply Y-factor correction or use dedicated NFA

Cascaded NF Calculator → Amplifier Stability → Plot LNA S-params →

// 09 — Try the Tools

Put This Theory Into Practice

Every noise and linearity calculation on this page has a corresponding RFLab tool. Use them to compute your own receiver chain budget and verify S-parameters of your LNA or amplifier.

Noise & Signal Chain Calculators

⚡

CALCULATOR

Cascaded Noise Figure

Enter NF and gain for up to 8 stages — Friis formula computed automatically. Replicate Example 2 (cable → BPF → LNA → mixer → IF amp) and see each stage's contribution to total NF = 4.63 dB. Instantly shows which stage dominates.

⚡

CALCULATOR

Attenuator Design

Design pi/T attenuator pads to pad down LNA gain after the LNA. A 3 dB pad after the LNA: NF up 3 dB in that stage, IIP3 up 3 dB — use to find the optimum gain/linearity trade-off point from Example 6.

⚡

CALCULATOR

Γ Reflection Calculator