What is the Friis noise equation?

The Friis equation calculates the total noise factor of cascaded stages: F_total = F1 + (F2-1)/G1 + (F3-1)/(G1*G2) + ... Each stage's noise contribution is divided by the total gain preceding it. This means the first stage (usually the LNA) dominates: if G1 is large enough, the second and subsequent terms become negligible. Example: LNA with NF=1 dB (F=1.26), G=20 dB (G=100), followed by mixer with NF=8 dB (F=6.31): F_total = 1.26 + (6.31-1)/100 = 1.26 + 0.053 = 1.313. NF_total = 10*log10(1.313) = 1.18 dB. The mixer's 8 dB NF only adds 0.18 dB to the system because of the 20 dB LNA gain.

How does cascaded IP3 work?

The cascaded OIP3 (output third-order intercept) for series stages is: 1/OIP3_total = 1/OIP3_1 + G1/OIP3_2 + (G1*G2)/OIP3_3 + ... Unlike noise, gain makes things WORSE for linearity: each stage's distortion is amplified by all subsequent gain. The last stage (usually the ADC driver or mixer) dominates because it sees the highest signal levels. Example: LNA (OIP3=+20 dBm, G=20 dB), Mixer (OIP3=+15 dBm): 1/OIP3_total = 1/100 + 100/31.6 = 0.01 + 3.16 = 3.17. OIP3_total = 1/3.17 = 0.316 mW = -5 dBm. The high-gain LNA reduces the system OIP3 to -5 dBm, far below either stage individually.

What is the SFDR and how does cascade affect it?

Spurious-Free Dynamic Range (SFDR) is the range between the noise floor and the level where spurious products equal the noise: SFDR = (2/3)*(OIP3 - NF - 10*log10(kTB)). Higher OIP3 and lower NF both improve SFDR. But they fight each other in cascade: more LNA gain improves NF but degrades OIP3. The optimal LNA gain for maximum SFDR is a key design tradeoff. Typical results: SFDR = 60-70 dB for a wideband receiver (100 MHz BW). SFDR = 90-100 dB for a narrowband receiver (10 kHz BW). The instantaneous bandwidth directly affects SFDR through the noise floor term.

System Analysis

Cascade Analysis

/kas-kayd/ (cascaded noise & linearity)

Cascade analysis calculates the overall NF, gain, and linearity of multi-stage RF chains using the Friis equation. The first stage (LNA) dominates noise figure; the last stage dominates linearity. More LNA gain improves NF but degrades IP3, creating the fundamental sensitivity vs. dynamic range tradeoff in all receiver design.

Category: System Analysis

Key Formula: Friis: F = F₁ + (F₂-1)/G₁

Tradeoff: NF vs. IP3

Understanding Cascade Analysis

Every RF receiver is a cascade of stages: filter, LNA, mixer, IF amplifier, ADC. Each stage adds gain, noise, and distortion. The Friis equation tells us that the first stage dominates noise (because its noise is not reduced by any preceding gain). The cascade IP3 equation tells us the last stage dominates linearity (because all preceding gain amplifies the signals to their highest level). This creates the central tension in receiver design: you want high LNA gain for low NF but low gain for high IP3. Optimizing this tradeoff is what receiver design is all about.

Cascade Equations

Cascade Analysis:
Cascade analysis calculates the overall NF, gain, and linearity of multi-stage RF chains using the Friis equation. The first stage (LNA) dominates noise figure; the...

Key specifications:
1 dB | 20 dB | 8 dB | 1.18 dB | 0 dB | 1 mW

Power: P(dBm) = 10log(P_mW), 0dBm = 1mW

Typical Receiver Cascade

Stage	Gain	NF	OIP3	Running NF	Running OIP3
Preselector filter	-2 dB	2 dB	+50 dBm	2.0 dB	+50 dBm
LNA	+20 dB	1.0 dB	+22 dBm	3.1 dB	+22 dBm
Image filter	-1 dB	1 dB	+50 dBm	3.1 dB	+18 dBm
Mixer	-7 dB	8 dB	+15 dBm	3.2 dB	-2 dBm
IF amplifier	+30 dB	4 dB	+30 dBm	3.2 dB	-5 dBm

Key Equations

Decibel conversion:
Power: dB = 10log(P₂/P₁)
Voltage: dB = 20log(V₂/V₁)

dBm to watts:
P(W) = 10^{(dBm−30)/10}
0 dBm = 1 mW, +30 dBm = 1 W

Wavelength:
λ = c/f = 300/f(MHz) meters

Comparison

Aspect	Cascade Analysis Spec	Typical Range	Impact	Design Note
Primary function	Cascade analysis calculates the overall...	Application-dep.	Critical	Verify in sim
Operating range	The first stage (LNA) dominates noise fi...	Application-dep.	Critical	Verify in sim
Performance	More LNA gain improves NF but degrades I...	Application-dep.	Critical	Verify in sim
Integration	dynamic range tradeoff in all receiver d...	Application-dep.	Critical	Verify in sim
Trade-off	Understanding Cascade Analysis Every RF...	Application-dep.	Critical	Verify in sim

Common Questions

Frequently Asked Questions

Friis equation?

F_total = F1 + (F2-1)/G1 + ... Each stage's noise divided by preceding gain. First stage dominates when G1 is large. LNA NF=1dB, G=20dB + mixer NF=8dB: system NF = 1.18 dB. The mixer's 8 dB NF adds only 0.18 dB.

Cascaded IP3?

1/OIP3_total = 1/OIP3_1 + G1/OIP3_2 + ... Gain makes linearity WORSE (amplifies signals). Last stage dominates. LNA G=20dB, OIP3=+20dBm + mixer OIP3=+15dBm: system OIP3 = -5 dBm. High gain kills linearity.

SFDR optimization?

SFDR = (2/3)*(OIP3 - NF - 10*log(kTB)). More gain: better NF, worse IP3. Less gain: worse NF, better IP3. Optimal LNA gain balances both. Wideband: SFDR ~60-70 dB. Narrowband: 90-100 dB.

Related Terms

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