Math Bridge: Antenna Gain and Repetition

A technician must decide whether coverage gap left for repetition is safe before changing comparison antenna gain on the real device. The result is unresolved until the rule and units are checked. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is comparison antenna gain. The middle card applies this page's rule. The green card is coverage gap left for repetition. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This model keeps those stated values fixed and changes only comparison antenna gain, so the numeric fixture does not switch without explanation.

Walk the arrows. More installed gain leaves a smaller coverage gap for repetition to fill.

Derive the baseline in four named moves

1
Name the input. The chapter baseline is 4 dBi.
2
Name the relationship. remaining gap = 20 dB target - antenna gain
3
Substitute with units. 20 - 4 = 16.0 dB
4
Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change comparison antenna gain

Try Predict the direction of remaining gap = 20 dB target - antenna gain. Test another comparison antenna gain, then compare coverage gap left for repetition.

Observe More installed gain leaves a smaller coverage gap for repetition to fill. Reset comparison antenna gain to 4 and compare coverage gap left for repetition.

Explain More installed gain leaves a smaller coverage gap for repetition to fill.

Check yourself

What should you do before trusting a moved-control result?

Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.

What does this small model leave out?

Answer: Only comparison antenna gain moves here. Field effects named in the technical boundary stay fixed.

1. Use one decibel ledger

Antenna gain raises EIRP. Repetition lets the receiver combine several copies. Both can add link margin. They spend different resources: beam shape versus radio-on time. The radio stays on for each copy.

Radio Remi: Equal decibel entries do not make the two levers equal in practice.

2. Name the algebra moves

Add antenna gainEIRP=Pt+G.

Find the missing marginΔrepeat=Δtarget−G.

Undo the logarithmRideal=10^(Δrepeat/10).

Select a supported stepRound up, then compare energy in proportion to repetition count.

3. Distinguish ideal and standard counts

10^((20−4)/10)=39.8 ideal repeats → next supported step 64

Rounding down would miss the target in this model. The supported 64-repeat step is half the 128-repeat baseline, so the repetition-dominated radio-energy share is 50%.

4. Try one controlled change

Rideal=10^[(Δtarget−G)/10]; Rstandard=next supported step ≥ Rideal

TryChange only antenna gain. The 20 dB target, 23 dBm transmitter, 128-repeat baseline, and supported power-of-two steps stay fixed.

ObserveAt 4.00 dBi, EIRP is 27.0 dBm and repetition must supply 16.0 dB. The ideal count is 39.8, the next supported step is 64, and the repetition-dominated energy share is 50% of the 128-repeat baseline.

ExplainA 4 dB gain does not divide 128 by a fixed arithmetic amount. The logarithm maps decibels to a power ratio, then the protocol's discrete step rounds the result up.

Technical boundaries.

The combining formula is an idealized upper bound.

Combining gain: Real channels, interference, correlation, and receiver algorithms reduce it
Antenna gain: Depends on installed direction, efficiency, mismatch, and regulatory EIRP
Energy: Only the repetition-dominated transmit share scales directly here

Use modem traces, actual coverage mode, delivery, latency, and current measurements.

5. Reproduce the chapter values

128 repeats have ideal gain 10log10(128)=21.1 dB, consistent with the chapter's 20 dB comparison. Adding 4 dBi leaves 16 dB for repetition. 10^1.6=39.8, so the next shown standard step is 64. Then 128/64=2.00× and 64/128=50%.

6. Carry the evidence forward

Record band, conducted power, installed gain and orientation, EIRP limits, serving cell, RSRP/SINR, enhancement mode, actual repetition count, payload, delivery, latency, current trace, and movement constraints.

7. Check yourself

Why round 39.8 up to 64?

Answer: The model must meet or exceed the needed gain using a supported discrete step; rounding down would underfund it.

Does 4 dBi guarantee half the whole-device energy?

Answer: No. It halves only this idealized repetition-dominated radio share; search, receive, processing, sleep, and retries remain.

Why might a fixed sector use gain more reliably than a mobile device?

Answer: A fixed sector can aim its pattern; an arbitrarily oriented or moving device may not keep a gain lobe pointed at the serving cell.

Honesty boundary.

The page explains an ideal gain-versus-repetition trade, not a guaranteed NB-IoT or LTE-M mode decision.

20 dB: Chapter coverage-extension target
64 repeats: Illustrative next power-of-two step after 39.8
50%: Repetition-dominated radio share only

Operator configuration, radio conditions, regulation, movement, and modem implementation decide the deployed outcome.