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Intuitively I get it that my SSB power has to be half of DSB power but mathematically I am unable to get this. Say working for tone modulation.

$$m(t)=mA\cos(w_mt)$$ where m is modulation index.

Modulated signal for DSB case is $$mA\cos(w_mt)\cos(w_ct)$$ Power is $$(ma)^2/4$$.

Modulated signal for SSB case is $$mA\cos((w_m-w_c)t))$$ Power is $$(ma)^2/2$$.

This seems total opposite to what appears intuitively . I used BP Lathi's Book on Communication where the formula for SSB is ssb signal=$$m(t)cos(w_ct)+m_h(t)sin(w_ct)$$ where $$m_h(t)$$ is the Hilbert transform on m(t).

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