Both of these terms T1 and DS1 have their origination in the T-carrier system, to "trunk" lines between "telephone" company offices. DS1 is an aggregate of 24 DS0 streams while E1 carrier combines 32 or 33 such "clear" channels. While the original T-Carrier DS1 channel did not affect audio, the in-band signalling shared the LSB (least sig. bit) of the voice so it restricted the data to 7bit or 56kbps channels. Later DS1 carriers were adapted to use both methods of signalling in or out of band so they could offer both 56kbps or clear 64kbps channels used in E1 carriers.
When the DS1 is connected to a trunk wire circuit, it is called a T1 circuit.
Bell introduced the North American specs, but it was AT&T who adopted the T1 carrier name as distinct from L-carrier and N-carrier from their alphabetical lineage of carriers. These are now replaced with Federal Standards for multi-sourcing of technology.
Later the European Conference of Postal and Telecommunications Administrations (CEPT) improved it with the E-carrier system. This has now been adopted by the International Telecommunication Union Telecommunication, Standardization Sector (ITU-T). This is now widely used in almost all countries outside the USA, Canada, Japan & Korea and is centered in France.
Aside comments... It's a miracle they all talk to each other given all the regional differences.
T and E carrier systems set the benchmark for quality and reliability on communication. Calls around the world were like you were next door even though data had to be converted from µ-Law to A-Law by the American equipment it still provided the best telephony quality with 12bit resolution in an 8 bit sample with 99.99% call reliability and cheap monthly bills. But now with the demand for mobility, that was all compromised now to support more revenue with millions of cell phones to satisfy the urgent needs for mobile users even if it was trivial communication. The sharing of space, time slot, spectrum and bandwidth were compromised to provide the growth path for video and data services, at the expense of one or more other variables.
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When you talk about T1 interfaces, it is usually the physical interface lines and the dozen or so various signal codes such as Bi-Phase AMI etc. The terms get ambiguous on marketting brochures of hardware to expand the compatibility features. Perhaps the ambiguity of DS meaning "Data Signals" or "Digital Stream" caused some people to simply say T1 when they meant DS1.
When you talk about DSx unique protocols and specs, you are mainly talking about the data session layer such as the payload structure and managing of the payload thru session layer protocol and various software application layers for maintenance. You may be referring to µ-Law vs A-Law compander methods or binding channels ADSL or multiplexing various synchronous or asynchronous, symmetrical or asymetrical, bit streams, all of which are a subset of the Pleisiochronous Digital Heirarchy ( which means "almost" synchronous Mux carriers ) There is a Stratum-1 cesium clock for the highest speed synchronous networks and then the allow slop bits to add or rob bits for combining (almost) synchronous channels to prevent frame slip.
The original T-carrier signals transport includes all of the above but discussion of such is normally around hardware specific interfaces and the function of that hardware. It may include pseudo random sequence generators (PRSG) inside the equipment to test the bit error rates (BER) on the T1/E1/J1 lines and up and use eye patterns or stats to measure margin of the ac analog signal that combines clock and data in one of a dozen or more "codes". It may define the actual wire and connectors used on the physical wire.
When we say T1 we are either using it as a legacy marketing telemetry terminology or discussing physical wire or signal analog performance specs.