D-domain vs Time-domain (Digital Linear sequential ciruits)
There are two applications that I know of specifically (although I'm sure there are many more):


Linear feedback shift register analysis.  This would directly use the form you have in the question, i.e. XORs with a stream of bits.  LFSRs are used to implement CRC in hardware or to provide non-cryptographic pseudorandom sequences.
If you widen the notation to mean addition instead of XOR and allow each x value to have more than one bit of information, it is effectively equivalent to a z-transform.   Z-transforms are used in all sorts of digital signal processing applications, such as digital filters and correlators.