  We consider a constant $K$-user Gaussian interference channel with $M$
antennas at each transmitter and $N$ antennas at each receiver, denoted as a
$(K,M,N)$ channel. Relying on a result on simultaneous Diophantine
approximation, a real interference alignment scheme with joint receive antenna
processing is developed. The scheme is used to provide new proofs for two
previously known results, namely 1) the total degrees of freedom (DoF) of a
$(K, N, N)$ channel is $NK/2$; and 2) the total DoF of a $(K, M, N)$ channel is
at least $KMN/(M+N)$. We also derive the DoF region of the $(K,N,N)$ channel,
and an inner bound on the DoF region of the $(K,M,N)$ channel.
