Secure DoF of MIMO Rayleigh block fading wiretap channels with No CSI anywhere

We consider the block Rayleigh fading multiple-input multiple-output (MIMO) wiretap channel with no prior channel state information (CSI) available at any of the terminals. The channel gains remain constant in a coherence time of T symbols, and then change to another independent realization. The transmitter, the legitimate receiver and the eavesdropper have n
t
, n
r
and n
e
antennas, respectively. We determine the exact secure degrees of freedom (s.d.o.f.) of this system when T ≥ 2 min(n
t
, n
r
). We show that, in this case, the s.d.o.f. is exactly (min(n
t
, n
r
) − n
e
)
⁺
(T − min(n
t
, n
r
))/T. The first term can be interpreted as the eavesdropper with n
e
antennas taking away n
e
antennas from both the transmitter and the legitimate receiver. The second term can be interpreted as a fraction of s.d.o.f. being lost due to the lack of CSI at the legitimate receiver. In particular, the fraction loss, min(n
t
, n
r
)/T, can be interpreted as the fraction of channel uses dedicated to training the legitimate receiver for it to learn its own CSI. We prove that this s.d.o.f. can be achieved by employing a constant norm channel input, which can be viewed as a generalization of discrete signalling to multiple dimensions.