how to connect a AD9225ARS 12-Bit, 25 MSPS A/D Converter with arduino

Your first problem with this chip is providing a proper clock drive to it.
The second is that its a very complex high speed device designed for performance at the expense
of simplicity - the datasheet is a long complex document full of stuff that is important to get proper
performance from the chip. Nowhere is the idea of running it slower than 25MSPS even mentioned
AFAICT.

You need something much simpler like a SAR ADC with fast SPI interface.

Aim for more like ~100kSPS sample rate on an Uno or Mega, it has no DMA hardware and very little
RAM, so its not designed for high speed data acquisition.

And you'll need to think about front-end analog processing (anti alias filtering, attenuation, protection).

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