Equivalent Noise Temperature

In communication and radar receiver design, quantifying noise added by each stage of a system is essential for predicting overall performance. The concept of equivalent noise temperature provides a unified, physically meaningful way to characterize the noise contribution of any two-port network, independent of the actual thermal environment it operates in. This framework is especially important in satellite links, radio telescopes, and microwave receivers where noise levels are extremely small and classical noise figure descriptions become inconvenient.

Core Concept: What Is Equivalent Noise Temperature

Any physical or electronic component, even when cooled below room temperature, adds some noise to the signal passing through it. The equivalent noise temperature (T_e) of a device is defined as the temperature at which a matched resistor, placed at the input of an otherwise noiseless version of that device, would produce the same output noise power as the actual device. It is not the physical temperature of the device. It is a mathematical construct that allows direct comparison of noise contributions using temperature units (Kelvin).

The noise power available from a resistor at temperature T over bandwidth B is given by the Nyquist formula: N = k T B, where k is Boltzmann's constant (1.38 x 10^-23 J/K). Using this relation, any noise power can be expressed as an equivalent temperature. A higher T_e means the device adds more noise. A cryogenically cooled low-noise amplifier might have T_e as low as 20 K, whereas a room-temperature amplifier might have T_e of 300–600 K.

The concept also applies to passive lossy components. A lossy transmission line with physical temperature T_phys and loss factor L (where L greater than 1 means loss) has an equivalent noise temperature given by T_e = (L - 1) T_phys. This means even a feeder cable connecting an antenna to a receiver contributes noise quantified in Kelvin.

Noise Figure and Its Relation to Equivalent Noise Temperature

The noise figure (F) of a device is the ratio of the signal-to-noise ratio at the input to the signal-to-noise ratio at the output, measured under a standard reference temperature T_0 = 290 K. Noise figure is dimensionless but commonly expressed in decibels. The relationship between noise figure and equivalent noise temperature is direct:

T_e = (F - 1) x T_0, and inversely, F = 1 + T_e / T_0. At T_0 = 290 K, a noise figure of F = 2 (3 dB) gives T_e = 290 K. A very low noise amplifier with F = 1.05 (0.21 dB) gives T_e = 14.5 K. For satellite and deep space applications, equivalent noise temperature is preferred because it gives finer resolution at low noise levels where decibel figures would be small fractions.

Friis Formula for Cascaded Stages

A receiver chain typically consists of multiple stages: a low-noise amplifier, a bandpass filter, a mixer, an IF amplifier, and a detector. Each stage adds its own noise. The Friis formula for noise temperature gives the total equivalent input-referred noise temperature of the cascade as: T_total = T_e1 + T_e2/G1 + T_e3/(G1 G2) + ..., where T_en is the equivalent noise temperature of the nth stage and Gn is the available power gain of the nth stage.

This formula reveals a critically important design principle: the first stage dominates. If G1 is large, the contributions of subsequent stages are divided by large gain values and become negligible. This is why a low-noise amplifier (LNA) is always placed first in the receive chain, immediately after the antenna, before any cable losses or mixing stages. Even a small cable loss before the LNA significantly raises the system noise temperature.

System Noise Temperature

The system noise temperature (T_sys) accounts for all noise sources seen by the receiver: the antenna temperature T_ant (which includes sky noise, ground radiation, and atmospheric effects picked up by the antenna beam) plus the receiver equivalent noise temperature T_rec. Thus T_sys = T_ant + T_rec. The signal-to-noise ratio at the receiver output is then SNR = P_signal / (k T_sys B), which directly governs link budget calculations in satellite communication systems.

The antenna temperature is not the physical temperature of the antenna metal. It is a weighted average of the brightness temperatures of all sources in the antenna beam pattern, including galactic background radiation (approximately 10 K at microwave frequencies), atmospheric noise, and ground thermal emission. For a dish antenna pointed at cold sky at 12 GHz, T_ant can be as low as 20 to 30 K.

Numerical Example

Consider a satellite receiver front end with three cascaded stages. The available gains and equivalent noise temperatures are known. We apply the Friis formula to find the total system equivalent noise temperature and then use it to compute the effective system noise temperature including antenna temperature.

Given:
LNA: G1 = 20 dB = 100 (linear), T_e1 = 50 K
Band-pass filter (passive, lossy): G2 = -1 dB ≈ 0.794, T_e2 = 75 K
IF Amplifier: G3 = 30 dB (gain not needed beyond stage 2), T_e3 = 200 K
Antenna temperature: T_ant = 30 K

Why this formula applies:
Friis cascaded noise temperature formula referred to input of LNA.

Formula:
T_rec = T_e1 + T_e2/G1 + T_e3/(G1 × G2)

Substitution:
T_rec = 50 + 75/100 + 200/(100 × 0.794)
T_rec = 50 + 0.75 + 200/79.4

Calculation:
T_rec = 50 + 0.75 + 2.52
T_rec = 53.27 K

System noise temperature:
T_sys = T_ant + T_rec = 30 + 53.27 = 83.27 K

Final Answer: T_sys ≈ 83.3 K

Exam Tip: In GATE problems, always convert dB gains to linear before applying Friis formula. The first stage T_e1 is never divided by any gain because it is already referred to the system input. A common trap is to divide T_e1 by G1, which is incorrect.

Mechanism: Noise Power Flow Through a Receiver Chain

• T_e is not physical temperature. It is the input-equivalent noise temperature of any two-port network.
• Friis formula: T_total = T_e1 + T_e2/G1 + T_e3/(G1 G2). First stage contribution is never divided.
• High gain in stage 1 (LNA) suppresses noise from all downstream stages, making LNA placement critical.
• System noise temperature: T_sys = T_ant + T_rec, directly used in link budget SNR calculations.
• Antenna temperature reflects sky and ground brightness temperatures weighted by beam pattern, not metal temperature.
• Relation between noise figure and noise temperature: T_e = (F - 1) x 290 K at standard reference.

Quick Revision

• Equivalent noise temperature T_e = (F - 1) T_0, where T_0 = 290 K is the standard reference temperature.
• Friis formula (cascaded): T_rec = T_e1 + T_e2/G1 + T_e3/(G1 G2) + ... All gains are linear, not dB.
• System noise temperature: T_sys = T_ant + T_rec. SNR = P_signal / (k T_sys B).
• First stage dominates. Large G1 makes all subsequent T_en/(...) terms negligible.
• T_ant is sky noise picked up by antenna beam. At microwave frequencies pointing at cold sky it can be 20-50 K.
• Lossy passive component at physical temperature T with loss L has T_e = (L - 1) T. Even cables add noise.
• Trap: Never divide T_e1 by G1 in Friis formula. T_e1 is directly the input-referred noise of stage 1.