When an Antenna Is Worth 200 Repetitions

Ada re-derives this chapter’s own numbers step by step, at full precision

ADA · CALCULATION AUDIT

When an Antenna Is Worth 200 Repetitions

In this chapter’s metering rollout, 20% of the meters sit in basements at an RSRP of −123 dBm — deep enough that NB-IoT must repeat every message many times just to be heard, draining airtime and battery. An external antenna at street level lifts that same basement signal to −100 dBm, a 23 dB gain bought once in hardware. This audit asks what that antenna is really worth: how many repetitions of airtime does its 23 dB replace?

Companion to the chapter Lab: Coverage Planning — every number here comes from that chapter.

Ada: The section’s whole argument is that coverage and capacity are the same lever, and it hangs on the repetition-gain rule ~10 x log10(N) dB. Let me verify that rule on the chapter’s own numbers and then price the basement fix in repetitions.

• The stated case: 10 x log10(32) = 15.0515 dB, matching the chapter’s ~15 dB for 32 repeats.
• The deep-coverage band it quotes (64-128 repeats): 10 x log10(64) = 18.0618 dB up to 10 x log10(128) = 21.0721 dB – roughly 18-21 dB bought purely by repeating in time.

Now the worked fix. The external antenna lifts the basement from -123 dBm to -100 dBm, a gain of -100 - (-123) = 23 dB. How many repetitions would buy that same 23 dB instead? Invert the rule: N = 10^(23 / 10) = 10^2.3 = 199.5, about 200 repeats.

That is the equivalence the chapter is pointing at: 23 dB of hardware antenna gain replaces roughly 200 repetitions of airtime per message. The design meaning is that the link-budget lever and the capacity lever are literally the same decibels – spend 23 dB once in an antenna, or pay it forever in ~200x the airtime and battery, which is why good planning attacks the deep-coverage minority first.

Every number above is taken from the chapter’s own material and re-derived step by step.