CESSB isn't the same thing as compression, or TX equalization (such as
attenuating the low frequency audio). Yes these things can all get you an
increase in average power etc. but are independent of CESSB.

CESSB removes the overshoots in the SSB envelope that are mathematically
inherent in the SSB modulation process, without introducing or increasing
distortion. This increases the average transmit power. Or more accurately
it avoids the reduction in average power your transmitter ALC system will
otherwise inflict, as it must to avoid RF clipping which causes splatter.

CESSB therefore provides you an additional 4-5dB (typical) of distortion
free gain relative to SSB, in addition to anything you have done to the
baseband audio such as compression or TX equalization.

73 Hans G0UPL
http://qrp-labs.com
https://buymeacoffee.com/g0upl
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