This Democratic Republic, our form of representative democracy.....
There are Parliamentary democracies
Even Monarchies that have democratic forms of governing.
There is no Pure Democracy Nations on Earth.... every one of the world's democratic nations, are a form of representative democracies.
The American system is "a compound Republic". A combination, of central and State Republics under which: "The different governments will control each other while within each Republic there are two safeguarding features: (a) a division of powers, as well as (b) a system of checks and balances between separate departments: "Hence a double security arises to the rights of the people."
See Federalist #51 by Madison.
By far the greater quantity and variety of power was retained by the government of each State when the United States Constitution was framed and adopted in 1787-1788. Only a comparatively small part of each State's power was delegated by its people to the new central, or Federal, government ,chiefly the powers concerning "war, peace, negotiation and foreign commerce" (See Federalist # 45 by Madison).
This delegated-power government, the central Republic, was granted few and limited powers; while each State's government is a full-power Republic under the State Constitution, subject to its restrictions, also to that grant, and to the few restrictions specified expressly in the United States Constitution as applying to the governments of the States.