How to remove 20MHz SPI signal noise?
1V isn't just simple noise, something is really inducing it. Here are the steps I would take to solve it:

1) Take a look at the power supply to the DSP and whatever else is communicating and see whether they have any kind of noise on them. It's likely the noise you see is induced on VCC and this might be going to the DSP I/O.

2) if you have any kind of switching regulator, try and use an LDO and see if it makes a difference

3) Attempt to use a good clean lab power supply to the board if you can

4) Isolate other board parts, turn off everything you don't absolutely need. If the issue disappears, slowly bring up each part until you can nail down what it is that is causing it.

5) Take into account any other strange symptoms. Given the large voltage, it's unlikely that the SPI is the only thing affected.

We might be able to help more if you post a schematic and give more details to the parts and whether you have any kind of switching supply