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It seems most codec ICs typically use I²S for audio data but can also use SPI or I²C.

Should any DAC with an SPI interface such as this one be called a codec IC? If not, what makes such chip be called a codec IC?

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No, because that chip is just a DAC, not a codec.

A codec both codes and decodes, and for most parts, relates to streams or continuous signals that are being converted between analog and digital forms.

For the first part, the DAC you suggest has no ADC so it cannot be a codec.

The second part relates to the continuous signals. For example, I2S is an audio interface bus for continuous audio data streams, it has continuous clocks and a fixed relation between "master" clock, "bit" clock and "frame sync", also called the "left/right" clock.

The SPI bus natively isn't a streaming bus, so in order to provide a continuous stream, it needs to be handled externally on top of the SPI bus protocol. For instance you would need to handle sending data over SPI and then time the data transmission and handing the syncronization signal to latch in data at precise enough intervals, so that the samples have low enough jitter.

So while it is possible to use that DAC to build a codec for continuous data conversio n, by itself it is not a codec.

For same reason, if I put a R2R network into outputs of a 74HC595 shift register, I have built a DAC and it has by definition SPI interface, but it isn't a codec.

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CODEC is an abbreviation for "COder-DECoder", so the term is generally reserved for chips that handle full-duplex signals. They contain both a DAC and an ADC, along with optional other functions such as filtering and compression.

It really has nothing to do with the type of interface used on the digital side, although in my experience (in telecommunications and audio signal processing), the vast majority of codec chips do use a serial interface of some sort.

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