YAWN. I was specifically NOT talking about interpolation processing (that ruins the fluidity and real-time display of a complex, varying waveform).
No, it doesn't, unless your DSO is from the 90's. Pretty much every somewhat modern DSO reaches high waveform rates when using sin(x)/x, and that includes entry-level scopes like the Agilent/Keysight DSO-X2k/3k, or cheap ones like the Rigol DS2000 (all which aren't even new designs).
Wuerstchenhund: Asking for a 5GSa/s sample rate for 1GHz BW is nonsense.
A high sample rate is good for various reasons and one of them is to prevent alias terms from corrupting the displayed waveform. A typical 500MHz scope will have a graceful frequency rolloff with flat group delay so the amount of alias rejection will be limited if the sampling rate is only 2 or 3GHz.
That is true for low bandwidth DSOs (<=350MHz) which usually come with a Gaussian frequency response similar to analog scopes, however most 500+MHz real-time DSOs have a flat response with brick wall like sharp rolloff at the top end. A typical 500MHz DSO will have a -3dB point at somewhere between 700MHz and 1GHz, with a cut-off I'd expect at some where below 1.2GHz (which only requires fs>2.4GSa/s)
Therefore, if I had to choose a DSO to replace a 500MHz analogue scope and I wanted the same frequency (and flat group delay) response then I'd want a high sampling rate to minimise the influence of alias terms when looking at complex waveforms that have spectral content up in the GHz region.
I agree, however there is is no point in a sample rate that is higher than what's required to resolve the highest frequency component your scope can actually "see", which slightly over 2x f
0. Anything higher will not result in additional details (as clearly visible in the example screenshots I posted), all it does is filling up your sample memory more quickly.
On a DSO, rather than an excessive sample rate, it's important to have a large enough sample memory to avoid dropping your sample rate (and therefore your usable real-time bandwidth) on longer time bases.
So I think 5GHz sampling would be a good/safe choice. Go down to 2.5GHz sampling and I think that the alias effects will begin to degrade the display of the wideband waveform compared to what you would see on the analogue scope.
On a 1GHz BW signal? No, it wouldn't, as demonstrated.
I guess it depends on how accurate you want the display to be? Some people will be happy to live with the degraded display at the lower sample rate. They might not care if it isn't quite right and the degradation is negligible to them?
Assuming they even understand the real degradation and don't assume that, just because something is visible on the screen, that this is necessarily a true representation of the input signal.
Operating test gear in the border areas of its specifications or confirmed performance envelope isn't really a good idea.
The other option would be to choose a scope with a steeper filter rolloff (eg max flat response) as this will be less prone to alias issues. So you could use a lower sample rate and have less problems with alias terms. But this won't have the same frequency response characteristics as the analogue scope. It's up to the user to decide if this is a good/bad thing for the range of things they want to use the scope for.
Don't forget that the 7904 vs DSO discussion was about a 1GHz sine wave initially, and a true continuous-time 1GHz sine wave is, by nature, BW limited. And as demonstrated, there's zero use in sampling a 1GHz sine wave at more than 2.5GSa/s.
And as staed above, the other question is how much use is in using a scope, especially one with Gaussian response, beyond it's rated (specified) BW, considering how the vastly attenuated and potentially phase-shifted frequency components in the border areas of it's frequency response very likely results in a distorted waveform on the screen, something that is also true for analog scopes. Especially if the true performance beyond the specified 500MHz BW has never been properly measured/assessed, as it seems to be the case with GK's Tek 7904.