For a complex signal of length $N$, I know that the discrete Fourier transform (DFT) gives a resulting complex transformed signal of length $N$. Furthermore, the DFT is a linear transformation which, with the correct scaling, can be represented as an orthogonal matrix. That is, the DFT corresponds to changing to a different orthonormal basis.
For a real signal of length $N$, this article says that the DFT transforms the signal into complex and real parts of length $N/2 + 1$ each. Thus, the resulting frequency space has $N+2$ dimensions (with respect to $\mathbb{R}$). Therefore, this transformation can't be represented as an orthonormal change-of-basis, since the dimension has increased. Is there some way to remove the redundancy and get a real orthonormal basis out of the DFT?