Of course it can't know the load impedance actually connected, but it knows what you tell it that it is. So when I tell it the load is 50ohms, and to output a 1Vpp signal, what it does is calculate that the open-circuit voltage must be 2Vpp, and then it outputs a 2Vpp signal (which will be pulled down to 1Vpp if it's attached to a 50ohm load). So what I was saying is how good is this calculation? Is its firmware actually doing the proper calculation? That's what I was calling into question, was if the firmware's math functions were working correctly.
No.
You have an idea in your mind that is simply wrong, and I am trying to help you get around it.
Look, here's the text from the DG1022Z manual, you should read it and understand what you are saying is NOT what it does. Specifically look at the bolded text (my bolding). Most important is the last line that is bolded.
If you cannot understand this then I will let some other member take a crack at explaining it.
DG1022Z manual p. 2-88
2. Output Impedance
The output impedance setting influences output amplitude and DC offset. The
instrument has a 50Ω fixed serial output impedance for the [CH1] connector at
the front panel.
If the actual load is different from the specified value, the
voltage level displayed would not match the voltage level of the device under
test. To ensure correct voltage level, the load impedance setting must match the
actual load.
Press Utility Channel Set Output Set Imped to select “HighZ” or
“Load”. The default is “HighZ”. If “Load” is selected, use the numeric keyboard to
set specific impedance value. The default is 50Ω and the available range is from
1Ω to 10kΩ. Impedance setting will be displayed on the screen.
The generator will adjust the output amplitude and offset voltage automatically
once the impedance setting is changed. For example, the current amplitude is
5Vpp. At this point, change the output impedance from 50Ω to HighZ and the
amplitude
displayed on the screen will double to 10Vpp. If the output impedance
is changed from HighZ to 50Ω, the amplitude will reduce to half of the previous
value (2.5Vpp).
Notice that only the displayed values change with the parameter
and the real output from the generator does not change.