What is the simplest possible antenna that I could
 construct for use on 40M
  with my DSW-40?

1. A 50 ohm 5 watt resistor to ground.

2. A single piece of wire about 33 feet long stuck into the antenna socket.

3. #2 above with another 33' of wire to the antenna socket's 'ground.' 
 

Up the line!

 8. #7 above, replace dipole with 2el yagi

 9. #8 above, replace 2el yagi with 3el yagi.

 10. anything more and your yl/xyl replaces YOU.

SGC-2020 Portapack LDG Z-11 Tuner WWII era J-45 knee key Coleman folding table base Shade Grass PERFECT!

Apache Junction, Arizona

 4. #3 above with a length of 50ohm coax to get the feedpoint outside.

5. #4 above, replace coax with ladder line and a tuner.

6. #4 above, replace wire with aluminum tubing.

7. #6 above, put dipole on a rotator.
 

Outbacker  on an aluminum "counterpoise"