> dealing with serial data and not parallel
In books, it is convenient to imply a 16-bit bus.
On PCB, it is a LOT cleaner to run one wire.
The trade-off is that this wire has to be 16 times faster.
But at ~~50KHz sample rate, for wires from an inch to a foot, the ~~1MHz datarate is not a hard problem.
And for wires over a few feet, parallel is not only costly, but small path-length differences mean all 16 bits and the clock don't arrive at the same instant, "skew". Not so bad for 6-foot 8-bit printer cables working well under 1MHz. Cost for 50 feet of 12-conductor is brutal, and even near 100KHz you can have skew errors and pattern-dependent errors.
8, 16, even 32 flipflops to shift-register parallel-serial-parallel are, on modern chips, far cheaper than 8 16 32 contact connectors. Not to mention copper wire.
> serial data could be either PDM or serially-coded PCM.
Could be ANY thing.
But more than likely, the next chip would like simple sequential 16-bit (or whatever) words, as if reading an MS WAV file. And is willing to provide a register to take the data on one pin to de-serialize for its internal use.
Serial gets into problems of framing and clocking. In a run of apparently random bits, where do "words" start? This is an old-old-old problem with lots of solutions. In Teletype there are 6 data bits and 2 stop-bits. Internally there is a finger which jams a gear if it stops for more than 1.5 bit-times. In random data it could catch randomly, but when getting TeleType data it will tend to catch the 2-bit stop gap, then be in-sync for the next character-word. Now that "gears and fingers" are dots on silicon, far trickier framing and sync is possible. And cheap.
> taking apart delay pedals that I noticed
Audio generally today moves serially, except maybe from RAM through CPU to sound-card. Or inside processor chips where you want to crunch 16- or 24-bit numbers between each sample. (There have been "1-bit" processors but awful slow to gobble wide data.)
FWIW, "all" networking (telegraph, TeleType, 2400 phone modem, AppleTalk, original EtherNet, Token-Ring, T1, cheapernet, UTP hubbed "ethernet") is serial data.
So are essentialy all disk drives. The ST-506 interface presented 8-bit words to the CPU bus, but it collected serial bits from the head-decoder into a register.
RAM is often built and studied parallel, and parallel makes sense for very fast very short data-paths. Skew is an issue, annoying to the PCB layout. Some things we handle as "funny RAM such as thumb-drives are clearly serial at the USB jack, could be serial inside for all we know or care.
The standard digital interface from CD-ROM drives is serial, and is the model for stuff like SP/DIF audio interfaces. And USB, though USB has grown far beyond that.
Seeing as how there is a parallel register inside and a serial path in/out, and it isn't a HIGH-spec converter, "Delta Sigma" seems like just buzz-words. Why do we care if it is SAR, D/S, 65 thousand flash-comparators, ramp-integration, or a teeny elf with a fast voltmeter?