Electromagnetic Waves, Antennas, and RF: Thermal Noise, Noise Temperature, Noise Figure, G/T, and C/N0

Electromagnetic Waves, Antennas, and RF: Thermal Noise, Noise Temperature, Noise Figure, G∕T, and C∕N₀

EM25 established how transmitted electromagnetic power becomes received carrier power. A useful receiver, however, does not measure carrier power in isolation. It must distinguish the carrier from random fluctuations generated by the antenna environment and by the receiver electronics themselves.

The central chain in this article is

(1)

The objective is to derive the engineering quantities from thermal physics and power spectral density rather than introduce them as isolated link-budget rules [2, 1, 3, 4, 5].

1 From received carrier power to detectability

Suppose EM25 predicts a received carrier power C at the receiver input. Whether that carrier is useful depends on the competing noise power. If the receiver processes bandwidth B, then the thermal-noise power is ultimately of the form

(2)

where k is Boltzmann’s constant and T is an appropriate noise temperature.

The appearance of B means that noise power depends on receiver bandwidth. Carrier power does not acquire the same proportional dependence. This motivates a bandwidth-independent comparison between carrier power and noise density:

(3)

Here N₀ has units of W/Hz, so C∕N₀ has units of hertz and is conventionally reported in dB-Hz.

2 Thermal agitation and Johnson–Nyquist noise

A resistor at nonzero temperature contains charge carriers in continual thermal motion. Even when the average current is zero, the instantaneous voltage fluctuates randomly. In the classical low-frequency limit, the open-circuit voltage-noise power spectral density of a resistor R at absolute temperature T is

(4)

Over an ideal bandwidth B in which R and T may be regarded as constant,

(5)

Thus the RMS open-circuit noise voltage is

(6)

The noise voltage is random; the expression above specifies its mean-square strength, not a deterministic sinusoidal amplitude.

3 Why a matched resistor delivers kTB

Model the noisy resistor as an ideal noiseless resistance R in series with a random source having

(7)

Connect a matched load R_L = R. The load receives half of the open-circuit voltage, so

(8)

The average noise power delivered to the load is

N	=	(9)
	=	(10)
	= .	(11)

Therefore

(12)

The resistance has disappeared. A matched resistor at temperature T makes the same available thermal-noise power kTB regardless of its resistance value, within the assumptions of the classical model.

4 Noise spectral density N₀

If the available noise power in bandwidth B is

(13)

then the corresponding available noise-power spectral density is

(14)

The SI units are

(15)

Since one hertz is one inverse second, W/Hz is dimensionally equivalent to joules. In communications engineering, however, W/Hz keeps the spectral-density interpretation visible.

At the standard reference temperature

(16)

we have

N0	= kT0	(17)
	= (1.380649 × 10−23)(290)	(18)
	= 4.0039 × 10−21 W/Hz.	(19)

In logarithmic units,

(20)

and therefore

(21)

The familiar “−174 dBm/Hz thermal-noise floor” is therefore a noise-density value near room temperature, not the total noise power of a receiver.

For example, over B = 1 MHz,

(22)

so

(23)

5 Equivalent noise temperature

Noise temperature converts the noise produced by a component into the temperature of an equivalent thermal source. If a noisy two-port contributes an equivalent input noise density N_0,e, define

(24)

The parameter T_e does not necessarily equal the component’s physical temperature. It is an equivalent input quantity that reproduces the component’s noise degradation.

If the source at the input has noise temperature T_s, an ideal noiseless gain G would produce output noise

(25)

A real amplifier adds its own noise. Using equivalent input temperature T_e,

(26)

6 Noise factor and noise figure

Noise factor compares the input and output signal-to-noise ratios:

(27)

For the standard reference source temperature T₀ = 290 K,

(28)

Hence

(29)

Noise figure is the logarithmic form:

(30)

For example, if

(31)

then

(32)

and

(33)

7 Cascaded receiver noise: why the first stage matters

Consider cascaded stages with power gains G₁,G₂,… and noise factors F₁,F₂,…. Referred to the input, the total noise factor is

(34)

Equivalently, in noise-temperature form,

(35)

These are often called the Friis cascade formulas for noise. They are distinct from the Friis transmission equation of EM25, although both are historically associated with Harald Friis.

The key physical lesson is that large first-stage gain suppresses the input-referred contribution of later stages.

8 Passive loss before the LNA

A passive component with linear power loss L ≥ 1 has gain

(36)

At physical temperature T_p, its equivalent input noise temperature is

(37)

At the standard temperature T₀, its noise factor is simply

(38)

This result explains a major receiver-design rule: loss before the first low-noise amplifier is especially damaging. It attenuates the desired carrier before amplification and simultaneously contributes thermal noise.

For a 1.0 dB cable loss,

(39)

At 290 K,

(40)

9 Antenna noise temperature

An antenna receives thermal electromagnetic radiation from the directions in its pattern. The resulting available noise can be represented by an antenna noise temperature T_A such that

(41)

Conceptually, T_A is a pattern-weighted measure of environmental brightness temperature. It can depend on antenna pointing, frequency, ground pickup, atmosphere, sky background, and radome or feed losses.

The antenna temperature need not equal the antenna’s physical temperature. A receiving antenna pointed toward cold sky and the same antenna pointed toward warm ground can have very different T_A even though the metal structure itself is at the same physical temperature.

10 System noise temperature

At a chosen receiver input reference plane, define the total system noise temperature by

(42)

For a simple model in which antenna noise and receiver equivalent input noise are referred to the same plane,

(43)

If feed loss lies between the antenna and LNA, its effect must be transformed to the same reference plane rather than added blindly. The exact bookkeeping depends on where T_A, gain, and received carrier power are defined.

11 The receiver figure of merit G∕T

Received carrier power improves with receive antenna gain G_r, while thermal-noise density increases with system temperature T_sys. Their natural combined figure of merit is

(44)

In logarithmic units,

(45)

For example, if

(46)

then

(47)

and

(48)

A higher G∕T means a stronger carrier relative to the receiver’s thermal-noise density.

12 Carrier-to-noise-density ratio C∕N₀

Let the received carrier power at the chosen receiver reference plane be C. Since

(49)

we define

(50)

In decibel form,

(51)

Since

(52)

we may also write

(53)

This equation is one of the most important bridges between an RF link budget and receiver performance.

13 Direct C∕N₀ link-budget form

From EM25, the received carrier in dB form may be written schematically as

(54)

Substituting into the C∕N₀ equation and grouping G_r with T_sys gives

(55)

This form makes the transmit and receive roles visually distinct:

(56)

14 Relationship between C∕N₀ and C∕N

In a noise-equivalent bandwidth B,

(57)

Therefore

(58)

Hence

(59)

For example, if

(60)

and

(61)

then

(62)

This is not contradictory. A signal can have a useful spectral concentration or correlation structure even when its total carrier power is below the integrated wideband thermal noise. Later signal-processing articles can connect this idea to processing gain and E_b∕N₀.

15 GNSS-style numerical example

EM25 used an illustrative L1-like link at

(63)

with received carrier power

(64)

Take an illustrative system noise temperature

(65)

Then

N0	= 10 log 10(kTsys)	(66)
	= −228.60 + 10 log 10(300)	(67)
	≈−203.83 dBW/Hz.	(68)

Therefore

	= −157.50 − (−203.83)	(69)
	≈ 46.33 dB-Hz.	(70)

Thus

(71)

The purpose of this example is not to specify a particular GNSS satellite or receiver. It demonstrates how an EM25 received-power result becomes a receiver-quality metric once T_sys is known.

16 Classical approximation and quantum correction

The result N₀ = kT is the classical limit. At sufficiently high frequency or sufficiently low temperature, the available thermal-noise energy per mode is better represented by the Planck factor

(72)

When

(73)

use

(74)

so

(75)

At ordinary RF and microwave frequencies near room temperature, the classical approximation is usually excellent. The quantum correction becomes increasingly important for millimeter-wave, submillimeter-wave, cryogenic, and quantum receiver systems.

17 Reference-plane discipline

Noise calculations are extremely sensitive to reference-plane ambiguity. Before combining quantities, state explicitly:

• where carrier power C is defined;
• where antenna temperature T_A is defined;
• whether feed loss is before or after that plane;
• whether antenna gain is realized gain or gain excluding mismatch;
• whether a quoted receiver noise temperature already includes preceding losses;
• whether G∕T refers to the antenna terminals, receiver input, or another calibrated plane.

A physically correct formula can still produce the wrong answer if powers, gains, and noise temperatures are referred to different planes.

18 What EM26 adds to the series

EM25 answered

(76)

EM26 answers

(77)

The complete chain is now

(78)

The next article can add non-thermal interference and jamming. Once an interfering received power J is propagated through its own link, quantities such as J∕S, J∕N₀, and carrier degradation in the simultaneous presence of noise and interference can be derived without changing the underlying physics.

References