Noise in Communication Systems

Noise sets the ultimate limit on receiver performance. GATE EC tests thermal noise, noise figure, and SNR through cascaded stages (1–2 marks), central to link-budget reasoning.

Key results

• Thermal (Johnson) noise power: N = kTB (available power); noise voltage v_n = √(4kTBR). k = 1.38×10⁻²³ J/K.
• Noise Figure F = SNR_in/SNR_out = 1 + T_e/T₀ (T_e = equivalent noise temperature, T₀ = 290 K). NF(dB) = 10 log F.
• Friis formula (cascade): F_total = F₁ + (F₂−1)/G₁ + (F₃−1)/(G₁G₂) + … → the FIRST stage dominates.
• Shot noise ∝ current; flicker (1/f) noise dominant at low frequency.
• SNR degrades along a chain; a low-noise first stage (high gain, low F) protects the whole system.

Noise type	Dependence
Thermal	kTB (temperature, bandwidth)
Shot	2qI·B (device current)
Flicker (1/f)	∝ 1/f (low frequency)
Friis total F	F₁ + (F₂−1)/G₁ + …

Exam Tricks & Tips

• 🎯 Thermal noise power = kTB — proportional to temperature and bandwidth (independent of resistance for available power).
• 🎯 Friis: F_total ≈ F₁ + (F₂−1)/G₁ — the first stage's noise figure and gain dominate; use a high-gain LNA first.
• 🎯 Noise figure F = 1 + T_e/290; convert to dB with 10 log F.
• 🎯 Cascaded noise: later stages matter less because their contribution is divided by preceding gains.
• 🎯 Doubling bandwidth doubles thermal noise power (3 dB more noise).
• ❌ Common mistake: adding noise figures in dB directly for a cascade — use Friis in linear (ratio) form, not dB.

Expected exam pattern

1–2 marks: compute thermal noise power/voltage, noise figure, equivalent noise temperature, or cascade noise figure via Friis.

Quick recap

Thermal N=kTB (k=1.38×10⁻²³). F=1+T_e/290, NF=10 log F. Friis: F=F₁+(F₂−1)/G₁+…, first stage dominates → low-noise high-gain front end. Bandwidth ↑ → noise ↑.

Cover the answer, recall, then check. 11 cards on noise in communication systems for GATE EC.

Q1. Write the available thermal noise power.
A1. N = kTB, where k = 1.38×10⁻²³ J/K, T = temperature (K), B = bandwidth (Hz).

Q2. Write the thermal noise voltage across a resistor R.
A2. v_n(rms) = √(4kTBR).

Q3. Define noise figure.
A3. F = SNR_in/SNR_out = 1 + T_e/T₀ (T₀ = 290 K); NF(dB) = 10·log F.

Q4. State the Friis cascade noise formula.
A4. F_total = F₁ + (F₂−1)/G₁ + (F₃−1)/(G₁G₂) + …

Q5. Which stage dominates the total noise figure and why?
A5. The first stage — later stages' noise contributions are divided by the preceding gains.

Q6. Relate equivalent noise temperature to noise figure.
A6. T_e = (F − 1)·T₀, with T₀ = 290 K.

Q7. How does bandwidth affect thermal noise power?
A7. Noise power is proportional to bandwidth (double B → double noise, +3 dB).

Q8. What causes shot noise?
A8. The discrete/random arrival of charge carriers; its power ∝ 2qI·B (device current).

Q9. Where does flicker (1/f) noise dominate?
A9. At low frequencies (its PSD rises as 1/f).

Q10. Why is a low-noise amplifier placed first in a receiver?
A10. Its low F and high gain set the system noise figure (Friis), protecting SNR for the whole chain.

Q11. Can noise figures in dB be added directly for a cascade?
A11. No — convert to linear ratios and apply Friis; only certain simplifications add in dB.

Noise sets the ultimate limit on how weak a signal a receiver can recover. Quantifying it — noise figure, noise temperature, and the SNR at each stage — lets engineers design receivers that squeeze out the last dB. GATE tests the Friis cascade formula, noise-figure conversions, and SNR calculations, all high-value and formula-driven.

Core concept: Thermal and device noise degrade the signal-to-noise ratio; noise figure F (or noise temperature Te) quantifies how much a stage worsens SNR, and Friis' formula gives the cascade total.

The theory

Beginner — noise sources and thermal noise

• Thermal (Johnson) noise: available noise power from a resistor = kTB, where k = 1.38×10⁻²³ J/K, T temperature, B bandwidth. Its RMS voltage across R is √(4kTRB).
• Shot noise: from discrete charge crossing a junction, i²_shot = 2qI_DC·B.
• Flicker (1/f) noise: dominant at low frequency.

Intermediate — noise figure and temperature

• Noise Figure F = (SNR_in)/(SNR_out) ≥ 1; in dB, NF = 10·log₁₀F. It measures how much a stage degrades SNR (F = 1, 0 dB = noiseless).
• Noise temperature Te = (F − 1)·T0, with reference T0 = 290 K. Conversely F = 1 + Te/T0. Te is convenient for low-noise (near-ideal) devices.
• An attenuator of loss L (as a ratio) has F = L.

Advanced — Friis cascade formula

For stages with gains G1, G2, … and noise figures F1, F2, …, the total noise figure:
F_total = F1 + (F2 − 1)/G1 + (F3 − 1)/(G1·G2) + …
Key insight: the first stage dominates — a high-gain, low-noise first amplifier (LNA) suppresses the noise contribution of all later stages (divided by the preceding gain). This is why receivers put an LNA right at the antenna.
Equivalent temperature form: Te = Te1 + Te2/G1 + Te3/(G1G2) + …

Worked example

A receiver: LNA with G1 = 20 dB (=100), F1 = 2 dB; second stage F2 = 10 dB (=10). Find total noise figure.
F1 = 10^(0.2) = 1.585, F2 = 10^(1.0) = 10, G1 = 100.
F_total = F1 + (F2 − 1)/G1 = 1.585 + (10 − 1)/100 = 1.585 + 0.09 = 1.675 → NF = 10·log(1.675) = 2.24 dB.
The noisy second stage added only 0.24 dB thanks to the LNA's 100× gain — the Friis lesson.

GATE relevance

1–2 mark NAT/MCQ: thermal noise power (kTB), noise figure ↔ noise temperature conversion, Friis cascade total F, SNR at input/output, and why the first stage dominates. Very common, formula-based, high-scoring.

Exam tricks and shortcuts

• Thermal noise power = kTB; F = 1 + Te/T0 (T0 = 290 K); attenuator F = its loss L.
• Friis: F_total = F1 + (F2−1)/G1 + … — later stages divided by preceding gain.
• Put a high-gain low-NF stage first to minimise total NF.
• Convert dB to ratio before using Friis (it needs linear F and G).
• Mnemonic: "First stage sets the noise floor."

Key formulas

• Thermal noise power: N = kTB (k = 1.38×10⁻²³, B = bandwidth).
• Noise figure: F = SNR_in/SNR_out; in dB F(dB) = 10 log F.
• Friis (cascade): F = F₁ + (F₂−1)/G₁ + (F₃−1)/(G₁G₂) + …
• Equivalent noise temperature: T_e = (F − 1)T₀.
• SNR: = signal power/noise power.

Noise figure quantifies SNR degradation; the first amplifier stage's noise dominates a cascade (Friis).

Worked Example

Problem: (a) Find the RMS thermal (Johnson) noise voltage of a 1 kΩ resistor at T = 300 K over a bandwidth B = 1 MHz (k = 1.38 × 10⁻²³ J/K). (b) A receiver has two cascaded stages: stage 1 with noise figure F₁ = 2 and gain G₁ = 10, followed by stage 2 with F₂ = 4. Find the overall noise figure.

Solution:
(a) Thermal noise RMS voltage:
V_n = √(4kTRB).
Inside the root: 4 × 1.38 × 10⁻²³ × 300 × 1000 × 10⁶
= 5.52 × 10⁻²³ × 300 × 10⁹
= 1.656 × 10⁻²⁰ × 10⁹ = 1.656 × 10⁻¹¹.
V_n = √(1.656 × 10⁻¹¹) ≈ 4.07 × 10⁻⁶ V = 4.07 µV.

(b) Friis formula for cascaded noise figure (linear ratios):
F = F₁ + (F₂ − 1)/G₁ = 2 + (4 − 1)/10 = 2 + 0.3 = 2.3.
In decibels: 10·log₁₀(2.3) ≈ 3.6 dB.

Answer: (a) V_n ≈ 4.07 µV RMS; (b) overall noise figure F = 2.3 (≈ 3.6 dB).