Good morning, all! I've been poring over Rick Lyon's paper on the phased method demodulator for SSB AM, specifically this particular block diagram:

I've recreated this block diagram in Gnu Radio, with this being my particular flowgraph:

Quick digression: HUGE shout-out to Rick for reminding me that the Hilbert transform is a -90 degree phase shift (not a +90). Saved me a LOT of aggravation!

Question: Is the purpose of the Hilbert transform and add after the lowpass filters to allow for handling an independent sideband (ISB) signal? The only other reason I can see for it is to improve the signal by 3 dB. Whereas, if you don't mind the 3 dB hit AND you're dealing just with a SSB signal, you could simply add the I & Q together at the outputs of the LPFs. At least, that's what I'm seeing when I prototype this in the flowgraph above.

Thoughts?

If I look at it from a Hamradio perspective it makes sense. You really want a single signal because there are so many signals near the one you want.
(...and the extra 3dB S/N is a wanted plus).

Fair enough. I've not dealt much with HF comms (which is where the SSB predominates).

"(...and the extra 3dB S/N is a wanted plus)"

That right there is probably enough to warrant it!

Inspecting your diagram shows that whether you want USB or LSB you still need the two branches to be added/subtracted.

I've now tried a modified flowgraph by adding the two signals coming out of the LPFs, and I'm able to recover either LSB or USB (so long as they're the only signal there). Mind you, that must be the ONLY signal within the passband. If you have two SSBs side-by-side (as pa3a suggested) or an ISB, then it won't work. But a SSB all by its lonesome? It will work. Although you do lose the 3 dB of extra signal boost from the add / subtract.

I cannot really read the GNU Radio figure in your question. But the original (Rick's) flowgraph works out in math, I checked it out a long time ago.
In GNU RAdio I'm skipping all this using complex numbers instead of floats. Makes it a lot easier to construct.