Electromagnetic Waves, Antennas, and RF: End-to-End RF and GNSS Link-Budget Synthesis

Electromagnetic Waves, Antennas, and RF: End-to-End RF and GNSS Link-Budget Synthesis

The preceding articles developed the pieces of a radio link separately. EM24 introduced antenna gain, EIRP, and effective aperture; EM25 derived Friis and free-space path loss; EM26 developed thermal noise, system noise temperature, G∕T, and C∕N₀; and EM27 added received interference and receiver degradation. EM28 now assembles those pieces into one consistent end-to-end calculation [1, 2, 3, 4, 5].

The basic engineering chain is

• P_t
• EIRP
• path loss
• C
• G∕T
• C∕N₀
• C∕N
• interference margin

The purpose is not merely to collect formulas. A reliable link budget requires every term to refer to a stated physical reference plane, to use compatible units, and to avoid counting the same gain or loss twice.

1 The complete signal chain

Consider a transmitter and receiver separated by range r. The transmitter begins with power P_t at some stated reference plane. A transmit feeder or implementation loss L_t reduces the power before the antenna, while the antenna gain G_t redistributes the radiated power in angle. The resulting equivalent isotropically radiated power is

(1)

The wave then experiences propagation loss. For ideal free space,

(2)

and

(3)

Additional terms may include atmospheric absorption, rain attenuation, polarization mismatch, pointing loss, radome loss, ionospheric or scintillation allowances, and other modeled impairments. We collect these as L_other when their detailed separation is not needed.

At the receiver, the antenna gain in the desired-signal direction is G_r. The received carrier power at the stated receiver reference plane is therefore

(4)

2 Why the reference plane matters

The expression above is only correct if the terms are defined consistently. For example, if EIRP already includes transmit feeder loss, that same feeder loss must not be subtracted again. Likewise, if a quoted receive G∕T already includes a lossy cable ahead of the LNA through the system-noise-temperature calculation, that cable loss must not also be independently subtracted from C∕N₀ unless the reference plane is changed consistently.

A useful receiver reference plane is the antenna terminal. At that plane,

• G_r is the receiving antenna gain toward the transmitter;
• T_sys contains antenna noise plus all receiver noise referred back to that terminal;
• preamplifier cable or filter losses are included in T_sys through their equivalent noise temperature when appropriate.

The system figure of merit is then

(5)

or, in logarithmic form,

(6)

3 From received carrier power to C∕N₀

EM26 established the thermal-noise density

(7)

where

(8)

Thus

(9)

In dB units,

dB−Hz

= CdBW − 10 log 10k − 10 log 10Tsys.

(10)

Since

(11)

we obtain

(12)

Now substitute

(13)

Grouping G_r − 10 log ₁₀T_sys gives the compact end-to-end result

(14)

This is one of the most useful equations in satellite and GNSS link analysis. It combines transmitter strength, propagation, receiver antenna gain, and receiver thermal performance into one bandwidth-independent carrier-quality metric.

4 From C∕N₀ to finite-bandwidth C∕N

For an effective noise bandwidth B_n,

(15)

Therefore

(16)

In logarithmic form,

(17)

This distinction is essential for spread-spectrum systems. A GNSS signal may have negative pre-correlation C∕N across a wide RF bandwidth while still possessing a useful C∕N₀ because the receiver later exploits known signal structure and processing gain.

5 A complete RF link-budget table

An end-to-end dB budget can be organized so that every row has a sign and a physical interpretation:

A good practical check is to calculate the link in two independent ways:

1. compute the received carrier C first and then subtract N₀;
2. compute C∕N₀ directly using EIRP, path loss, and G∕T.

The two results should agree when the reference planes are consistent.

6 Worked example 1: complete microwave-style RF link

Suppose an RF link has

Pt	= 10 W = 10 dBW,	(18)
Lt	= 1.0 dB,	(19)
Gt	= 24 dBi,	(20)
f	= 5.8 GHz,	(21)
r	= 5.0 km,	(22)
Lother	= 2.0 dB,	(23)
Gr	= 20 dBi,	(24)
Tsys	= 500 K,	(25)
Bn	= 1.0 MHz.	(26)

The EIRP is

(27)

The free-space path loss is approximately

(28)

Thus

C	= 33 − 121.69 − 2 + 20	(29)
	≈−70.69 dBW.	(30)

The system noise density is

N0	= 10 log 10(kTsys)	(31)
	≈−201.60 dBW/Hz.	(32)

Hence

(33)

For B_n = 1 MHz,

(34)

This example is intentionally a strong terrestrial link; its purpose is to verify the bookkeeping before moving to the much weaker received powers typical of satellite navigation.

7 Worked example 2: GNSS-style end-to-end synthesis

Consider an illustrative L1-like link with

f	= 1.57542 GHz,	(35)
r	= 20,200 km,	(36)
EIRP	= 27.0 dBW,	(37)
Lother	= 2.0 dB,	(38)
Gr	= 2.0 dBi,	(39)
Tsys	= 400 K,	(40)
Bn	= 2.0 MHz.	(41)

These values are illustrative for learning the link-budget method; they are not intended as a specification for a particular GNSS spacecraft or receiver.

The free-space path loss is

(42)

The received carrier at the antenna-terminal reference plane is

C	= 27.0 − 182.50 − 2.0 + 2.0	(43)
	≈−155.50 dBW .	(44)

The receiver figure of merit is

G∕T	= 2.0 − 10 log 10(400)	(45)
	≈−24.02 dB/K .	(46)

Using the direct end-to-end formula,

C∕N0	= 27.0 − 182.50 − 2.0 − 24.02 + 228.60	(47)
	≈ 47.08 dB-Hz .	(48)

The same result follows from the carrier power. At 400 K,

(49)

so

(50)

Across a 2 MHz front-end bandwidth,

C∕N	= 47.08 − 10 log 10(2 × 106)	(51)
	≈−15.93 dB .	(52)

The negative wideband C∕N does not contradict useful GNSS reception; it simply shows why correlation and signal processing are necessary.

8 Adding received interference

Suppose an interfering signal produces received power J at the same receiver reference plane. The first useful ratio is

(53)

If desired and interfering signals are computed as separate links,

CdBW	= EIRPs − Ls + Gr,s,	(54)
JdBW	= EIRPj − Lj + Gr,j,	(55)

where all loss terms are expressed positively and subtracted. Therefore

(56)

The receiving antenna generally has different directional gains toward the desired and interfering sources. That angular discrimination belongs explicitly in the link budget.

For a broadband noise-like interferer, it is often more useful to work with interference spectral density J₀. Define

(57)

The effective carrier-to-noise-plus-interference density is then

(58)

Thus the interference degradation is

(59)

The degraded density ratio is simply

(60)

9 Interference margin from a required C∕N₀

Suppose the no-interference link provides

(61)

and the receiver requires at least

(62)

With noise-like interference density J₀,

(63)

At the threshold of acceptable performance,

(64)

Solving for allowable interference gives

(65)

If the available C∕N₀ margin is

(66)

then

(67)

and therefore

(68)

This is a receiver-side robustness margin. It states how much additional noise-like interference density can be tolerated at the chosen reference plane before the specified C∕N₀ threshold is reached.

10 Worked example 3: interference margin on the GNSS-style link

Continue the illustrative GNSS-style link with

(69)

Assume a receiver requirement of

(70)

The no-interference margin is

(71)

The largest noise-like interference density ratio consistent with that threshold is

(J0∕N0)max	= 105.08∕10 − 1	(72)
	≈ 2.22,	(73)

so

(74)

At 400 K,

(75)

so the corresponding receiver-plane interference-density threshold is

(76)

Now suppose the actual admitted noise-like interference density is

(77)

Its degradation is

DJ	= 10 log 10(1 + 10−8∕10)	(78)
	≈ 0.64 dB .	(79)

The effective density ratio becomes

(80)

The remaining margin above the 42 dB-Hz requirement is therefore

(81)

Over the same 2 MHz bandwidth, thermal noise is

(82)

and the interference power is

(83)

Since the carrier is −155.50 dBW,

(84)

This is an instructive result: the admitted interference can be several decibels stronger than the desired spread-spectrum carrier while still remaining well below the total thermal noise in a wide front-end bandwidth. Consequently J∕S alone does not determine receiver degradation; bandwidth, spectrum, correlation, and receiver architecture matter.

11 Link margin, implementation margin, and uncertainty

A link budget often contains several distinct margins. They should not be merged blindly.

Propagation or availability margin

This allowance covers fading, atmospheric variability, pointing uncertainty, or other propagation effects not represented by the nominal path model.

Receiver implementation loss

A practical receiver may perform worse than an ideal detector by an implementation loss L_impl. This can be handled either by increasing the required C∕N₀ threshold or by subtracting a clearly labeled implementation allowance from the available margin. It should not be counted both ways.

Interference margin

This is the room between the nominal thermal-noise-limited operating point and the requirement that may be consumed by additional interference. For noise-like interference, the nonlinear relation

(85)

should be used rather than simply subtracting J₀∕N₀ in dB.

Model uncertainty

Antenna pattern error, cable-loss uncertainty, temperature uncertainty, and calibration error may be carried as explicit uncertainty terms. A conservative design should state whether such terms are deterministic worst-case values, statistical standard deviations, or engineering reserves.

12 Common bookkeeping failures

Several errors recur in RF and GNSS link calculations.

1. Double-counting antenna gain. If EIRP already contains G_t, do not add G_t again.
2. Double-counting receive loss. A cable loss incorporated into T_sys or a quoted G∕T should not also be subtracted independently without changing the reference plane.
3. Mixing dBW and dBm. They differ by exactly 30 dB:

   (86)

4. Confusing C∕N₀ with C∕N. The former is in dB-Hz; the latter depends explicitly on bandwidth.
5. Adding powers directly in dB. Independent noise and interference powers must be added in linear units before conversion back to decibels.
6. Using J∕S as a complete receiver metric. It does not by itself include receiver bandwidth, spectral overlap, filtering, correlation, AGC behavior, or nonlinear front-end effects.
7. Using a scalar margin without a requirement. A margin is meaningful only relative to a stated threshold and reference condition.

13 Compact end-to-end workflow

A practical calculation can follow the sequence

EIRP	= Pt − Lt + Gt,	(87)
C	= EIRP − Lpath − Lother + Gr,	(88)
G∕T	= Gr − 10 log 10Tsys,	(89)
C∕N0	= EIRP − Lpath − Lother + G∕T + 228.60,	(90)
C∕N	= C∕N0 − 10 log 10Bn,	(91)
DJ	= 10 log 10(1 + J0∕N0),	(92)
(C∕(N0 + J0))dB−Hz	= (C∕N0)dB−Hz − DJ,	(93)
M	= (C∕(N0 + J0))dB−Hz − (C∕N0)req,dB−Hz.	(94)

Every line should be annotated with its reference plane and assumptions. That discipline is more important than any individual numerical formula.

14 Summary

EM28 combines the preceding RF topics into a single end-to-end link model. The transmitter is summarized by

(95)

propagation by

(96)

and the receiver by

(97)

These combine to give

(98)

with C∕N₀ in dB-Hz when the other terms use the stated logarithmic units.

Finite bandwidth gives

(99)

while additive noise-like interference produces

(100)

Finally, the receiver-side interference allowance associated with a C∕N₀ margin M is

(101)

The complete physical chain is therefore

(102)

This synthesis provides the foundation for detailed satellite-navigation receiver performance studies, measurement-based link validation, and later treatments of acquisition, tracking, processing gain, and integrity.

References