Oscilloscope Total Record Samples Vs Signal Frequency

There are really two things going on. Sampling into the memory buffer for each channel, and rendering for display.

Scope Paramters

The parameters you quote seem to be confusing Mega and Giga when it comes to memory depth, and also partially differ from the datasheet, which shows 2GSps (1GSps) with 2Mpt (1Mpt) memory depth for 1ch (2ch) mode. I base this answer on the datasheet presented values.

Acquisition Parameters

You are correct at 2GSps, you will only capture ~6.7 samples per cycle of your 300MHz waveform.

You are also correct that to capture a full cycle of a 1Hz waveform, your scope will have to drop the sample rate to compensate for memory limits, giving a max sample rate of 2MSps to get a full 1 second capture.

If you were to download the raw data from the oscilloscope and look at it, you will find 2Mpts for the 1Hz capture. You will find probably much more than 7 samples for the 300MHz capture, because although only the one cycle is displayed on-screen, the scope probably recorded the full 2Mpts buffer length so will have many cycles worth.

You are also correct that the oversampling factor is plenty to capture the full information from a 300MHz bandwidth-limited signal as indicated by the Nyquist theorem.

Display

However this is where things get interesting. How does the oscilloscope display the waveform on-screen? The scope display resolution is after-all fixed at a few hundred to a couple thousand pixels wide. From the datasheet again, your scope has a 640x480 pixel display, so it can show at most 640 samples on-screen.

• For your 1Hz waveform with 2Mpts of data, there is no way that every single sample can be rendered on-screen.

  Instead the scope will decimate the data (either intelligently with a multi-rate filter, or by simply keeping every Nth sample).

• For your 300MHz cycle, there are much fewer samples than can be displayed on-screen. So somehow the scope needs to convert from samples to pixels with interpolation.

  A low-end scope will do this with simple linear interpolation. Basically space the few samples it has equally across the screen, then draw a straight line between them. This will as you predict look very low resolution and nothing at all like your real waveform. Welcome to the world of pushing cheap test equipment to the limit.

  A higher-end scope is much more intelligent. Remember Nyquist? Assuming a bandwidth limited signal, we have all the information we need to recovery the real signal exactly (not quite, that statement neglects quantisation errors and noise). From the few samples the scope has, it can use more advanced interpolation methods to bump up the sample rate and recover the time-continuous signal. A common method is using sinc(x) interpolation, a.k.a. Whittaker-Shannon interpolation. By increasing the sample rate high enough to get one sample per display pixel, you get a very nice smooth, albeit frequency limited, view of the original signal.

  If there is any frequency content beyond the bandwidth limit of the scope and interpolation, this information is lost (either through aliasing or low-pass filtering depending on the scope and signal amplitudes), so you will not see any of this high frequency content - e.g. a 300MHz square wave will look very sinusoidal.

Looking at your scope datasheet again, it does support what it calls sin(x)/x interpolation for display. This is the same thing as sinc interpolation. It also supports the lower end linear interpolation presumably through some sort of display setting.