Electromagnetic Waves, Antennas, and RF: RF Interference and Jamming

Electromagnetic Waves, Antennas, and RF: RF Interference and Jamming, J∕S, J∕N₀, C∕(N + J), and Receiver Degradation

EM25 determined how much desired RF Power reaches a receiver. EM26 then compared that desired carrier power with thermal-noise density through C∕N₀. Real receivers must also operate in the presence of other radio-frequency signals. These may be unintended cochannel emissions, adjacent-channel leakage, harmonics, spurious radiation, or deliberate jamming. The same received-power bookkeeping applies in each case; what changes is the physical source and, often, the spectral structure of the interference [1, 2, 4, 5].

This article develops receiver-side interference metrics for analysis and robustness studies. The central chain is

(1)

The most important conceptual point is that interference power, thermal-noise density, and signal power are different kinds of quantities. They can be combined only after they have been referred to a common receiver reference plane and, when necessary, integrated over the same effective bandwidth.

1 Notation and receiver reference plane

Let

• S denote the desired received signal power;
• C denote the desired received carrier or signal power when using the conventional C∕N₀ notation;
• J denote received interference power after the relevant propagation and antenna effects;
• N₀ = kT_sys denote thermal-noise power spectral density;
• N = N₀B_n denote thermal-noise power in receiver equivalent noise bandwidth B_n;
• J₀ denote an equivalent interference power spectral density when a density model is appropriate.

For the power-ratio derivations below, S and C may refer to the same desired received RF power. The letter C is retained where the conventional carrier-to-noise notation is useful.

The basic quantities have different dimensions:

This distinction prevents one of the most common errors in interference analysis: adding or subtracting a total power directly from a spectral density without first accounting for bandwidth.

2 Jam- or interference-to-signal ratio J∕S

At a specified receiver reference plane,

(2)

In decibels,

(3)

If S = C, then the same ratio may be written J∕C.

A negative J∕S means the interference power is below the desired-signal power. A positive J∕S means the interference power is above it. Neither case, by itself, predicts receiver failure. Receiver performance also depends on thermal noise, bandwidth, spectral overlap, modulation, coding, filtering, correlation processing, front-end linearity, and other implementation details.

Worked example: received-power ratio

Suppose

(4)

Then

dB	= −136 − (−130)	(5)
	= −6 dB.	(6)

Therefore

(7)

In linear form,

(8)

Thus the interference power is about 25.1% of the desired received power.

3 A second RF link produces the received interference power

EM25 showed that a desired transmitter produces received power through its own link budget. An interfering transmitter obeys the same propagation physics. For two independent links,

SdBW	= EIRPS,dBW − LS,path − LS,other + Gr(ΩS),	(9)
JdBW	= EIRPJ,dBW − LJ,path − LJ,other + Gr(ΩJ).	(10)

The receive-antenna gain can differ because the desired signal and the interfering signal may arrive from different directions. Their polarizations and frequencies may also differ, in which case the corresponding mismatch and propagation terms must be included separately.

Subtracting the two received-power equations gives

(11)

This formulation is particularly useful for receiver susceptibility studies because it separates transmitter, path, and receive-antenna effects without requiring the two sources to share the same geometry.

4 Interference-to-noise-density ratio J∕N₀

EM26 defined

(12)

The ratio of a total received interference power J to thermal-noise density is therefore

(13)

Because J has units W while N₀ has units W/Hz,

(14)

Consequently J∕N₀ is conventionally reported in dB-Hz:

(15)

Worked example: J∕N₀ from system temperature

Let

(16)

and let the received interference power be

(17)

The noise density is

N0,dBW∕Hz	= 10 log 10(kTsys)	(18)
	≈−203.16 dBW/Hz.	(19)

Hence

dB−Hz	= −150 − (−203.16)	(20)
	= 53.16 dB-Hz.	(21)

Thus

(22)

5 Connecting J∕S, J∕N₀, and C∕N₀

If S = C and all quantities are referred to the same plane and the same thermal-noise density N₀, then

(23)

Therefore

(24)

In logarithmic units,

(25)

The dB-Hz units cancel in the subtraction, leaving a dimensionless power ratio in dB.

6 Bandwidth converts J∕N₀ into J∕N

Thermal-noise power in equivalent noise bandwidth B_n is

(26)

Hence

	=	(27)
	= .	(28)

Thus

(29)

In decibels,

(30)

This is why J∕N₀ and J∕N answer different questions. J∕N₀ compares an integrated interference power with a noise density. J∕N compares interference power with the actual thermal-noise power admitted by a specified bandwidth.

7 Spectral overlap: the receiver only admits part of the interference

A total transmitter or antenna-terminal interference power is not automatically the interference power that reaches the detector. Let S_J(f) denote the interference power spectral density at the receiver input and let H(f) denote the receiver transfer function. The admitted interference power is

(31)

Similarly, for flat thermal-noise density N₀,

(32)

If |H(f)|² is normalized to unity at its peak, define the equivalent noise bandwidth

(33)

Then

(34)

If a noise-like interferer is approximately flat with density J₀ throughout the receiver passband, then

(35)

This special case permits especially simple density-ratio formulas.

8 Deriving C∕(N + J)

Suppose the detector sees desired carrier power C, thermal-noise power N, and admitted interference power J. If the interference contribution may be treated as an additive uncorrelated power for the receiver metric of interest, then the total undesired power is

(36)

The resulting carrier-to-noise-plus-interference ratio is

(37)

Divide numerator and denominator by C:

(38)

Alternatively, factor out N:

	=	(39)
	= .	(40)

Thus

(41)

This equation gives the receiver degradation caused by additive in-band interference.

9 Receiver degradation factor

Define the degradation factor as the ratio of the noise-only carrier-to-noise ratio to the carrier-to-noise-plus-interference ratio:

(42)

Substituting the result above,

(43)

In decibels,

(44)

This is a positive penalty. It is the number of decibels by which C∕(N + J) lies below the noise-only C∕N.

Three useful landmarks are

J∕N = −10 dB	D ≈ 0.414 dB,	(45)
J∕N = 0 dB	D = 3.010 dB,	(46)
J∕N = +10 dB	D ≈ 10.414 dB.	(47)

10 Using C∕N₀ and J∕N₀ directly

EM26 often provides C∕N₀ rather than C∕N. Since

(48)

we may write

	=	(49)
	= .	(50)

Therefore

(51)

The numerator and denominator on the right both have units of hertz, so the final ratio is dimensionless.

Worked example: C∕N₀, J∕N₀, and finite bandwidth

Suppose

(52)

First compute the noise-only value:

(C∕N)dB	= 60 − 10 log 10(105)	(53)
	= 60 − 50	(54)
	= 10 dB.	(55)

Next,

(J∕N)dB	= 55 − 50	(56)
	= 5 dB.	(57)

Thus

(58)

The degradation is

DdB	= 10 log 10(1 + 3.162)	(59)
	= 6.19 dB.	(60)

Therefore

(61)

11 Noise-like interference density J₀

For broadband noise-like interference that is approximately flat across the receiver passband,

(62)

Then

N + J	= N0Bn + J0Bn	(63)
	= (N0 + J0)Bn.	(64)

Hence

(65)

It is often convenient to define the effective carrier-to-noise-plus-interference density ratio

(66)

Relative to the thermal-noise-only value,

= .

(67)

Therefore the density-domain degradation is

(68)

Worked example: interference density below thermal noise density

Let

(69)

Then

(70)

The degradation is

DdB	= 10 log 10(1 + 0.2512)	(71)
	≈ 0.973 dB.	(72)

Thus even an interference density 6 dB below thermal-noise density raises the combined noise-plus-interference floor by almost 1 dB.

If the original

(73)

then

(74)

12 Equivalent interference temperature

Noise-like interference can also be represented by an equivalent temperature. If in-band interference power is J in bandwidth B_n, define

(75)

Since

(76)

we may write

N + J	= kTsysBn + kTJBn	(77)
	= k(Tsys + TJ)Bn.	(78)

Therefore

(79)

for the special case in which the interference is legitimately represented as additive noise over the bandwidth of interest.

Worked example: equivalent interference temperature

Suppose

(80)

is admitted over

(81)

Then

TJ	=	(82)
	≈ 72.4 K.	(83)

For a 290 K receiver system,

(84)

The associated degradation is

DdB	= 10 log 10	(85)
	≈ 0.97 dB.	(86)

This is the same result obtained from 1 + J∕N.

13 Worked example: broadband spectral overlap

Suppose a cochannel noise-like interferer has total received power

(87)

spread uniformly across

(88)

Assume an ideal rectangular receiver bandwidth

(89)

lies entirely inside the interference band. The admitted fraction is

(90)

Therefore

Jin,dBm	= −90 + 10 log 10(0.1)	(91)
	= −100 dBm.	(92)

At 290 K,

NdBm	= −173.98 + 10 log 10(2 × 106)	(93)
	≈−110.96 dBm.	(94)

Hence

(95)

The degradation is

(96)

This calculation demonstrates why total interference power without bandwidth information can be misleading. The receiver responds to the portion of the interference spectrum that survives its transfer function.

14 Worked example: a GNSS-style receiver-side calculation

Consider an illustrative received carrier power

(97)

and system noise temperature

(98)

From EM26,

(99)

so

(100)

Now suppose the receiver input contains an interfering signal with total admitted power

(101)

Then

(102)

and

(103)

If one models this interference simply as additive in-band power over

(104)

then

(C∕N)dB	= 44.08 − 10 log 10(2 × 106)	(105)
	≈−18.93 dB,	(106)

while

(107)

The additive-power penalty is therefore only

(108)

Hence the scalar model gives

(109)

This result is intentionally instructive: a large J∕S does not uniquely determine receiver degradation. In a spread-spectrum or correlation receiver, a narrowband, pulsed, swept, structured, or partially correlated interferer may behave very differently from broadband Gaussian noise having the same total power. Acquisition, tracking, automatic-gain-control behavior, and front-end dynamic range require additional receiver-specific models [4, 5].

15 When C∕(N + J) is not enough

The equation

(110)

is an additive-power model. It is highly useful, but it does not describe every interference mechanism. Several distinctions matter.

Narrowband versus broadband interference

A narrowband tone and broadband noise can have the same total received power J but produce different detector behavior. The correct first step is always to calculate the admitted interference

(111)

For modulation- or correlation-based receivers, the relevant transfer function may include more than the analog RF filter.

In-band versus out-of-band interference

A signal outside the final channel bandwidth may contribute little directly to J_in after filtering. Nevertheless, if it is sufficiently strong at an earlier receiver stage, it can affect automatic gain control, amplifier compression, mixer products, or analog-to-digital-converter dynamic range. These are nonlinear or implementation-dependent effects and cannot be predicted by the simple additive N + J formula alone.

Deterministic versus noise-like interference

Thermal noise is stochastic and is naturally described by N₀. A deterministic sinusoid is not thermal noise. Representing it by an equivalent J₀ is only appropriate when the receiver metric averages its effect in a way that makes a density model meaningful.

Waveform correlation

A receiver designed to correlate against a known signal can strongly suppress some waveform components and respond strongly to others. Consequently the same pre-correlation J∕S may correspond to different post-correlation performance. A scalar link metric should therefore be viewed as one layer of the analysis, not a complete receiver model.

16 A useful hierarchy of interference metrics

The different ratios answer different physical questions:

	: How strong is received interference relative to the desired signal?	(112)
	: How large is total interference relative to thermal-noise density?	(113)
	: How large is admitted interference relative to in-band thermal noise?	(114)
	: How large is noise-like interference density relative to thermal-noise density?	(115)
	: What additive carrier-to-undesired-power ratio remains at the chosen detector plane?	(116)

No single ratio replaces the others. The appropriate quantity depends on whether one is studying RF propagation, spectral occupancy, front-end loading, demodulation, or a particular tracking/detection process.

17 Summary

The desired-signal and interference powers are separate RF links that terminate at a common receiver. At a common reference plane,

(117)

Thermal-noise density gives

(118)

with dB-Hz units, while finite bandwidth converts this to

(119)

The receiver admits interference according to

(120)

When the admitted interference may be treated as additive uncorrelated power,

(121)

and the degradation relative to thermal noise alone is

(122)

For flat noise-like interference,

(123)

EM27 therefore extends the EM25–EM26 link-budget chain to

(124)

The next article can combine the complete transmit, propagation, antenna, receiver-noise, and interference models into a single end-to-end RF/ link budget.

References