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Out of habit, I'm always choosing the smallest channel bandwidth in my WiFi setups to optimize range rather than throughput. Now I keep reading that the channel width does not affect range.

A long time ago, I've learned that effective transmit power depends on the bandwidth of your signal, as you're spreading your transmit power over a greater frequency range. For example SSB, single side band - essentially half an AM signal, is used in amateur radio to achieve greater ranges than with wider voice modes (AM/FM). The craziest of contacts are still made using "continuous wave" with Morse code, for that same reason.

If my assumption is correct, throughput over distance would be a mixed calculation, but the lowest bandwidth should provide the greatest possible range.

Is that a correct assumption, or are routers compensating for the higher power requirements? Or maybe there's something I'm missing?

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Those channel width settings only set the maximum channel width, not the minimum. Your Wi-Fi devices should automatically fall back to simpler modulation schemes (and narrower channels, if it would make a difference) as needed to optimize rate-at-range.

Also, I seem to recall that the regulatory power limits are measured across 1MHz or narrower slices, not the sum of the whole channel. So if the rule is that you can only use 1W in the 2.4GHz ISM band, it's that the power of any 1MHz-wide slice of your 20-, 40-, 80-, or 160MHz-wide channel may not exceed 1W. But don't quote me on that.

Channel frequency can have a much bigger effect in range than width, as radios using band-edge channels often use lower power to meet regulatory limits on out-of-band power bleedover. Also, in many regulatory environments, different sub-bands of the 5GHz band can have drastically different power limits. The regulations have changed over time, so it's hard to know for sure which rules your devices follow without testing them with proper RF lab equipment.

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