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If you want to use a file system, such as FAT, so that you can read back the data on a PC, you will need a CPU.
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It's the regular way, but not absolutely required, in my opinion. You can also think of a low-level storage, that is compatible with FAT. Of course, some restrictions apply. The most basic method would assign the full media storage to a single continuous file. FAT management on the CPLD side reduces to determining the file's relative sector 0 address in this case.
Regarding CF specific handling, usually true IDE mode is utilized, which is identical to accessing an ATA/IDE hard disk.
t13.org has the respective documents. There have been also good OEM CF manuals from Sandisk, describing the protocol in detail.