Noise in communication systems and signal-to-noise ratio

Noise sources (thermal, shot, flicker), SNR, noise factor and temperature, and Friis' formula for cascaded receiver stages.

Drafted with Aria, reviewed by the AiCanCode.org team. Spotted an error? Use Give Feedback at the bottom of the page.

Why it matters

Noise sets the floor on what any communication link or measuring instrument can detect. The weakest signal a receiver can use, the number of bits an ADC really resolves and the error rate of a digital link are all decided by the signal-to-noise ratio (SNR). Knowing where noise comes from and how it adds through a chain of amplifiers tells you which stage to spend money on.

Key ideas

External and internal noise. External noise arrives with the signal: atmospheric (lightning), galactic and solar noise, and man-made interference (ignition, switching supplies). Internal noise is generated inside the circuits and is what the receiver designer controls.

Thermal (Johnson–Nyquist) noise. Random thermal motion of charge carriers in any resistance at absolute temperature T produces an open-circuit noise voltage with mean-square value v_n² = 4kTRB. It is white (flat spectrum) up to very high frequencies and Gaussian. The maximum noise power a resistor can deliver to a matched load is P_n = kTB, independent of R. At T = 290 K, kT = 4.00 × 10⁻²¹ W/Hz, i.e. −174 dBm/Hz.

Shot noise. Current made of discrete charges crossing a junction fluctuates: i_n² = 2qIB. It dominates in diodes, photodiodes and BJTs at low current and is central to optical receivers.

Flicker (1/f) noise rises at low frequencies and matters below about 1 kHz in semiconductor devices and sensors; chopper and lock-in techniques move the signal away from it. Partition noise appears where current divides between electrodes. In optical systems, quantum noise (photon arrival statistics) and relative intensity noise of lasers also appear.

Signal-to-noise ratio. SNR = P_s/P_n, usually quoted in dB as 10·log₁₀(P_s/P_n). For voltages across the same resistance, SNR(dB) = 20·log₁₀(V_s/V_n). Only noise inside the system bandwidth counts, so the noise bandwidth B must be stated; narrowing B improves SNR for a narrowband signal.

Noise factor and noise figure. A real amplifier adds noise, so the SNR at its output is worse than at its input. The noise factor is F = SNR_in/SNR_out (≥ 1), measured with the source at T₀ = 290 K; the noise figure is NF = 10·log₁₀F dB. The equivalent noise temperature is T_e = (F − 1)T₀, a convenient measure for very low-noise stages (satellite LNAs, where NF is a fraction of a dB).

Cascades — Friis' formula. For stages with noise factors F₁, F₂, … and available power gains G₁, G₂, …, F = F₁ + (F₂ − 1)/G₁ + (F₃ − 1)/(G₁G₂) + …. The first stage dominates if its gain is high, which is why receivers put a low-noise amplifier (LNA) right after the antenna and why a lossy cable or mixer placed before the LNA is costly: a passive loss L at T₀ has F = L.

Noise in modulation systems. The figure of merit is the output SNR divided by the baseband SNR (the SNR with the same transmitted power sent without modulation). With coherent detection: DSB-SC and SSB give 1; conventional AM gives μ²/(2 + μ²) ≤ 1/3; FM above threshold gives 3β²/2 for a sinusoid-equivalent message power, so wideband FM trades bandwidth for SNR. Envelope and FM detectors show a threshold effect at low input SNR.

Formulas

v_n,rms = √(4kTRB) k = 1.381 × 10⁻²³ J/K Boltzmann's constant, T absolute temperature (K), R resistance (Ω), B noise bandwidth (Hz), v_n (V).

P_n = kTB Available thermal noise power (W) into a matched load.

i_n,rms = √(2qIB) q = 1.602 × 10⁻¹⁹ C, I DC current (A); shot noise current (A).

SNR = P_s/P_n, SNR(dB) = 10·log₁₀(P_s/P_n)

F = SNR_in/SNR_out, NF = 10·log₁₀F (dB), T_e = (F − 1)·T₀, T₀ = 290 K

F_total = F₁ + (F₂ − 1)/G₁ + (F₃ − 1)/(G₁G₂) + … F and G as linear ratios (not dB). Applies to cascaded, impedance-matched stages.

T_e,total = T_e1 + T_e2/G₁ + T_e3/(G₁G₂) + …

Worked examples

Example 1 (standard). Find the RMS thermal noise voltage of a 1 kΩ resistor at 290 K over a 1 MHz bandwidth, and the available noise power in dBm.

  1. v_n = √(4kTRB) = √(4 × 1.381 × 10⁻²³ × 290 × 10³ × 10⁶).
  2. = √(1.602 × 10⁻¹¹) = 4.00 × 10⁻⁶ V = 4.0 µV.
  3. P_n = kTB = 1.381 × 10⁻²³ × 290 × 10⁶ = 4.00 × 10⁻¹⁵ W.
  4. In dBm: 10·log₁₀(4.00 × 10⁻¹⁵ / 10⁻³) = −114 dBm. Answer: v_n ≈ 4.0 µV; P_n ≈ 4.0 fW ≈ −114 dBm.

Example 2 (GATE level). A receiver chain has an LNA (NF 2 dB, gain 20 dB), a mixer (NF 10 dB, conversion gain −6 dB) and an IF amplifier (NF 15 dB). Find the overall noise figure and equivalent noise temperature.

  1. Convert to ratios: F₁ = 10^0.2 = 1.585, G₁ = 100; F₂ = 10; G₂ = 10^−0.6 = 0.251; F₃ = 10^1.5 = 31.62.
  2. F = 1.585 + (10 − 1)/100 + (31.62 − 1)/(100 × 0.251).
  3. = 1.585 + 0.090 + 30.62/25.12 = 1.585 + 0.090 + 1.219 = 2.894.
  4. NF = 10·log₁₀(2.894) = 4.61 dB; T_e = (2.894 − 1) × 290 = 549 K. Answer: NF ≈ 4.6 dB, T_e ≈ 549 K. The noisy IF stage still matters because the mixer has loss; more LNA gain would reduce its contribution.

Example 3 (SNR through a stage). An amplifier with NF = 5 dB receives a signal with SNR 30 dB. What is the output SNR?

  1. In dB, SNR_out = SNR_in − NF (from F = SNR_in/SNR_out).
  2. SNR_out = 30 − 5 = 25 dB. Answer: 25 dB.

Common mistakes

  • Using dB values directly in Friis' formula. Convert NF and gain to linear ratios first.
  • Writing 4kTB for noise power. 4kTRB is mean-square open-circuit voltage; the available (matched) power is kTB.
  • Using 10·log for a voltage ratio. Voltages and currents use 20·log₁₀.
  • Calling F = 3.16 a "noise figure of 3.16". The noise factor is 3.16; the noise figure is 5 dB.
  • Forgetting that a lossy cable before the LNA adds its full loss (in dB) to the system noise figure.
  • Using the 3 dB bandwidth instead of the equivalent noise bandwidth for filters with gentle skirts (for a single RC pole, B_N = (π/2)·f_3dB).

For GATE IN

  • Thermal noise voltage and power numericals; parallel and series resistor combinations (add mean squares).
  • Noise figure, noise temperature and Friis' formula for two or three stages, including a lossy line.
  • SNR in dB, and output SNR after an amplifier of given NF.
  • Shot noise in diodes and photodiodes (links to the optical detector topic).
  • Figure-of-merit comparisons of AM, DSB-SC, SSB and FM.

Quick check

  1. What is kT at 290 K in dBm/Hz?
  2. An amplifier has F = 2. What is its noise temperature?
  3. Two stages: F₁ = 2, G₁ = 10, F₂ = 11. What is the overall noise factor?
  4. Which noise type has a spectral density proportional to DC current? Answers: 1. −174 dBm/Hz. 2. 290 K. 3. 2 + 10/10 = 3. 4. Shot noise.

See it move

All Instrumentation animations
Explore Signal-to-Noise Ratio (SNR)

Adjust the signal and noise power to see how the Signal-to-Noise Ratio (SNR) changes. Observe how a higher SNR results in a clearer signal.

Equations used
  • SNR = P_signal / P_noise — SNR is the signal-to-noise ratio, P_signal is the power of the signal (W), P_noise is the power of the noise (W)
  • SNR_dB = 10 * log10(SNR) — SNR_dB is the signal-to-noise ratio in decibels (dB)

Interview questions

All Communication and Optical Instrumentation interview questions

Try answering each one aloud before you open it.

  1. 1.What is noise in communication systems?Concept

    Noise in communication systems refers to any unwanted electrical signals that interfere with the transmission and reception of the desired signal. It can originate from various sources such as thermal noise, intermodulation noise, crosstalk, and impulse noise. Noise can degrade the quality of the communication signal, leading to errors in data transmission.

  2. 2.Explain the concept of signal-to-noise ratio (SNR).Concept

    Signal-to-noise ratio (SNR) is a measure used to quantify the level of a desired signal to the level of background noise. It is usually expressed in decibels (dB). A higher SNR indicates a clearer and more distinguishable signal from the noise, which generally results in better communication quality.

  3. 3.How does thermal noise affect communication systems?Concept

    Thermal noise, also known as Johnson-Nyquist noise, is generated by the random motion of electrons in a conductor due to thermal agitation. It affects communication systems by adding a continuous background noise that can interfere with the signal, especially in low-power or weak signal scenarios. This type of noise is unavoidable and sets a fundamental limit on the performance of communication systems.

  4. 4.Why is SNR important in communication systems?Application

    SNR is important because it determines the quality and reliability of a communication system. A higher SNR means that the signal is much stronger than the noise, leading to fewer errors and better performance. It is crucial for designing systems that need to operate efficiently in noisy environments, such as wireless communication networks.

  5. 5.What happens if the SNR is too low in a communication system?Application

    If the SNR is too low, the noise level is comparable to or greater than the signal level, making it difficult to distinguish the signal from the noise. This can lead to increased error rates, loss of data integrity, and poor communication quality. In extreme cases, the communication system may fail to function properly.

  6. 6.Explain how noise figure is related to SNR.Concept

    Noise figure is a measure of degradation of the SNR as a signal passes through a system or device. It is defined as the ratio of the input SNR to the output SNR. A lower noise figure indicates that the system adds less noise to the signal, preserving the SNR and thus maintaining better signal quality.

  7. 7.Why is it important to minimise noise in optical communication systems?Application

    The receiver's SNR sets the bit-error rate, so for a target BER noise decides the receiver sensitivity and hence the maximum span or bit rate. The main sources are shot noise of the signal and dark current, thermal noise of the preamplifier load resistor, laser relative intensity noise and, in amplified links, ASE noise from optical amplifiers. Designers reduce them with low-noise transimpedance amplifiers, APDs (internal gain to beat thermal noise), cooled detectors and optical filtering.

  8. 8.What is the impact of crosstalk on communication systems?Application

    Crosstalk occurs when a signal transmitted on one channel or circuit creates an undesired effect on another channel. It can lead to interference and degradation of the signal quality in communication systems. Crosstalk is particularly problematic in densely packed systems, such as data centers, where multiple signals are transmitted in close proximity.

  9. 9.Calculate the SNR in dB if the signal power is 10 mW and the noise power is 1 mW.Numerical

    To calculate the SNR in dB, use the formula: SNR (dB) = 10 * log10(Signal Power / Noise Power). Here, Signal Power = 10 mW and Noise Power = 1 mW. SNR (dB) = 10 * log10(10 / 1) = 10 * log10(10) = 10 * 1 = 10 dB.

  10. 10.A communication system has an input SNR of 30 dB and an output SNR of 25 dB. What is the noise figure of the system?Numerical

    The noise factor is F = SNR_in/SNR_out in linear terms: 10³/10^2.5 = 1000/316.2 = 3.16. The noise figure is NF = 10·log₁₀F = 5 dB, which is simply 30 dB − 25 dB. Note the distinction: 3.16 is the noise factor (a ratio), 5 dB is the noise figure.

Finished this topic? Mark it so your progress, study plan and readiness keep up.

Stuck on something here?

PreviousSuperheterodyne receiversNext Pulse code modulation and quantisation