Possibility of Doing a 2048 point DFT?

[CamJackson]

Does anyone have experience doing a 2048 point DFT on an Mega 2560?

[Grumpy_Mike]

Given you need a float data type for an FFT, that is 4 bytes, then  a 2048 point array will consume all the 8K memory the meg has.

[robtillaart]

You might extend the RAM of the MEGA with FRAM - https://www.adafruit.com/product/1895

OK, performance will be less than you want but it works, and you can tweak the I2C to at least 600KHz

Check - https://github.com/RobTillaart/Arduino/tree/master/libraries/FRAM

[CamJackson]

What do you think of using a USB card, rather than FRAM due to the cost and memory size differences?

[Grumpy_Mike]

What do you think of using a USB card, rather than FRAM due to the cost and memory size differences?

Not much. I assume you mean USB memory stick, not card. That requires a USB host shield.

While you can use FRAM, you don't need it to be non volatile so simple SRAM will do, something like a 23K256 SPI interfaced chip.