Electrical Engineering C
Resume:
Multiuser MIMO degrees of freedom with no CSIT
Chiachi Huang and Syed A. Jafar Shlomo Shamai (Shitz)
Electrical Engineering and Computer Science Department of Electrical Engineering
University of California Irvine Technion-Israel Institute of Technology
Irvine, California, USA Technion City, Haifa, Israel
Email: {chiachih, syed}@uci.edu Email: *******@**.********.**.**
user MIMO interference channel under the same channel state
Abstract We provide the characterization of the degrees of
freedom (DOF) region for a 2-user fading MIMO broadcast chan- information setting. We also extend the outerbound of the DOF
nel when perfect channel knowledge is available to the receivers region to nd the capacity region for a speci c 2-user MIMO
and no channel knowledge is available to the transmitters. The
broadcast channel.
results are applied to nd the DOF region for some special cases
of a 2-user MIMO interference channel. We also extend the
outerbound of the DOF region to nd the capacity region for II. S YSTEM M ODEL
a speci c 2-user MIMO broadcast channel.
Consider the 2-user Gaussian MIMO broadcast channel
I. INTRODUCTIONS where the transmitter is equipped with M antennas and
receivers 1, 2 are equipped with N1, N2 antennas, respectively.
Multiple-input-multiple-output (MIMO) systems are capa-
The channel is described by the input-output relationship:
ble of signi cantly higher capacity compared to traditional
single-input-single-output systems. One of the key features of
Y[1] (t) = H[1] (t)X(t) + Z[1] (t) (1)
MIMO systems is the possibility of multiplexing signals in
Y[2] (t) = H[2] (t)X(t) + Z[2] (t) (2)
space. The ability of multiplexing signals in space is measured
by spatial multiplexing gain [1], which is called capacity
where at the tth channel use, Y[i] (t), Z[i] (t) are the Ni 1
prelog or degrees of freedom (DOF). For the point-to-point
vectors representing the channel output and additive white
MIMO communication systems, it has been shown that the
Gaussian noise at receiver i, H[i] (t) is the Ni M channel
availability of the channel state information at the transmitter
matrix corresponding to receiver i, i {1, 2}, and X(t) is
(CSIT) does not affect the spacial multiplexing gain [2].
the M 1 input vector. The elements of H[i] (t) and Z[i] (t),
Unlike the point-to-point case, in a network with distributed
i = 1, 2, are independent identically distributed circularly
processing units, it is well known [3], [4] that in the absence
symmetric complex Gaussian random variables with zero
of channel knowledge, spatial multiplexing gain is lost. For
mean and unit variance. We assume perfect CSIR, i.e., each
example, the DOF of the fading multiple-input-single-output
receiver has perfect knowledge of all channel matrices at each
broadcast channel with M antennas at the transmitter and 1
instant, and no CSIT, i.e., the transmitter has no knowledge of
antenna at all M receivers is M when perfect channel state
the instantaneous values taken by the channel coef cients. To
information is available at the transmitter (perfect CSIT) [5],
avoid cumbersome notation, we will henceforth suppress the
[6], [7]. However, the DOF of the same system is only 1 when
channel use index t where it does not cause ambiguity.
channel state information is not available at the transmitter
The transmit power constraint is expressed as:
(no CSIT) [3]. Further understanding about the availability of
channel knowledge and its effect on the degrees freedom of the
E[X2 ] P. (3)
networks can provide insights into the design and optimization
of the wireless networks. A natural goal is to extend the results
There are two independent messages W1, W2, associated with
in the previous example to a more general MIMO broadcast
rates R1, R2, to be communicated from the transmitter to
channel with arbitrary number of users and antennas. In this
receivers 1, 2, respectively. The capacity region C (P ) is the
paper, we make progress on this problem by studying the
set of all rate pairs (R1, R2 ) for which the probability of
DOF region of a 2-user MIMO broadcast channel where the
error can be driven arbitrarily close to zero by using suitably
transmitter is equipped with M antennas and receivers 1, 2
long codewords. The degrees of freedom region is de ned as
are equipped with N1, N2 antennas, respectively, under the
follows:
assumption of perfect channel state information at the receivers
(perfect CSIR) and no CSIT. An exact characterization of the
(d1, d2 ) R+ : (R1 (P ), R2 (P )) C (P ) s.t.
D
DOF region of the channel is given, and our result shows 2
Ri (P )
di = lim, i = 1, 2.}. (4)
that a simple time division between the two users is DOF-
P log(P )
region optimal. We then use the result of the MIMO broadcast
channel to nd the DOF region for some special cases of a 2-
= I (X; Y[1] H[1], H[2], U ) + o(log(P ))
I II. D EGREES OF F REEDOM OF MIMO BC WITH N O CSIT
min(M,N2 )
Theorem 1: The degrees of freedom region of the MIMO [2] [2]
+ I (X; Yi H[1], H[2], U, Y[1], Y(N1 +1:i 1) )
BC with no CSIT, as de ned in Section II is the following:
i=N1 +1
d1 d2 r log(P ) + o(log(P ))
D = (d1, d2 ) : + 1 .
R+
min(M,N1 ) min(M,N2 )
2
min(M,N2 )
(5) [2] [1]
+ I (X; Yi H[1], H[2], U, Y(2:N1 ) )
Proof: Without loss of generality, let us assume N1
i=N1 +1
N2 . The case where M N1 N2 is trivial, because in this
= r log(P ) + o(log(P ))
case the degrees of freedom region, even with perfect CSIT,
[1] [1]
+(min(M, N2 ) N1 )I (X; Y1 H[1], H[2], U, Y(2:N1 ) )
is given by:
r
r log(P ) + (min(M, N2 ) N1 ) log(P ) + o(log(P ))
D = {(d1, d2 ) R+ : d1 + d2 M } (6)
2 N1
min(M, N2 )
which is clearly achievable even without CSIT, by simple time- = r log(P ) + o(log(P )) (14)
N1
division between the two users.
Thus, for 0 r N1, an outerbound on the boundary of the
For the remainder of this section, we consider M N1 .
Since the MIMO BC with no CSIT, as de ned above, falls in degrees of freedom region is characterized as follows.
the class of degraded broadcast channels [8], [9], its capacity min(M, N2 )
(d1, d2 ) =
region C (P ) is the set of rate pairs (R1, R2 ) given by: N1 r, r, (15)
N1
R1 I (U ; Y[1] H[1], H[2] ) (7) which implies that
= I (X; Y H, H ) I (X; Y H, H, U ) (8)
[1] [1] [2] [1] [1] [2]
d1 d2
(d1, d2 ) R+ : + 1 .
D (16)
min(M, N2 )
2
R2 I (X; Y[2] H[1], H[2], U ) N1
(9)
Achievability of this outerbound follows trivially by time
where U X (Y[1], Y[2] ) forms a Markov chain. Since
division between the two users, and the proof of Theorem
the channel between the transmitter and receiver 1 cannot have
1 is complete.
more than min(M, N1 ) = N1 degrees of freedom, we have:
IV. D EGREES OF F REEDOM OF THE MIMO I NTERFERENCE
I (X; Y[1] H[1], H[2] ) N1 log(P ) + o(log(P )). (10) C HANNEL WITH N O CSIT
A. System Model
Let us de ne r as the degrees of freedom for the term
I (X; Y[1] H[1], H[2], U ), i.e., Consider the 2-user Gaussian MIMO interference channel
where transmitters 1, 2 are equipped with M1, M2 antennas,
I (X; Y[1] H[1], H[2], U ) = r log(P ) + o(log(P )). (11)
respectively, and receivers 1, 2 are equipped with N1, N2
antennas, respectively. The channel is described by the input-
From (11) we obtain the following useful inequality:
output relationship:
I (X; Y[1] H[1], H[2], U )
Y[1] (t) = H[11] (t)X[1] (t) + H[12] (t)X[2] (t) + Z[1] (t) (17)
N1
[1] [1]
= I (X; Yi H[1], H[2], U, Y(i+1:N1 ) ) Y[2] (t) = H[21] (t)X[1] (t) + H[22] (t)X[2] (t) + Z[2] (t) (18)
i=1
where at the tth channel use, Y[j ] (t), Z[j ] (t) are the Nj
[1] [1]
N1 I (X; Y1 H[1], H[2], U, Y(2:N1 ) ). (12) 1 vectors representing the channel output and additive white
Gaussian noise at receiver j, H[ji] (t) is the Nj Mi channel
Equation (12) implies that
matrix corresponding to receiver j, and X[i] (t) is the Mi
r
1 input vector, i, j {1, 2}. The following assumptions are
[1] [1]
log(P ) + o(log(P )) I (X; Y1 H[1], H[2], U, Y(2:N1 ) ).
N1 similar to those in Section II. The elements of H[ji] (t) and
(13)
Z[j ] (t), i, j {1, 2}, are independent identically distributed
Now we can write the upperbound (9) for R2 as:
circularly symmetric complex Gaussian random variables with
R2 zero mean and unit variance. We assume perfect CSIR and no
CSIT.
I (X; Y[2] H[1], H[2], U )
The transmit power constraint is expressed as:
[2]
= I (X; Y(1:min(M,N2 )) H[1], H[2], U )
E[X[i] 2 ] P, i = 1, 2. (19)
[2] [2]
+I (X; Y(min(M,N2 )+1:N2 ) H[1], H[2], U, Y(1:min(M,N2 )) )
There are two independent messages W1, W2, associated with
[2]
= I (X; Y(1:min(M,N2 )) H[1], H[2], U ) + o(log(P )) rates R1, R2, to be communicated from the transmitter 1 to
receiver 1 and from the transmitter 2 to receiver 2, respectively.
[2]
= I (X; Y(1:N1 ) H[1], H[2], U ) + o(log(P ))
The standard de nitions of the capacity region and the DOF
[2] [2]
+I (X; Y(N1 +1:min(M,N2 )) H[1], H[2], U, Y(1:N1 ) ) region are the same with those in Section II.
B. Main Results B. Main Result
Theorem 4: The capacity region of the MIMO BC with no
Theorem 2: If M1 N2 and M2 N1, the degrees of
CSIT, as de ned in Section V-A is the following:
freedom region of the MIMO interference channel with no
CSIT is the following: R1 R2
C= (R1, R2 ) R+ : + log(1 + P ) . (27)
2
N1 N2
d1 min(M1, N1 ),
Proof: The proof follows the similar lines in the proof of
D = (d1, d2 ) R+ : d2 min(M2, N2 ), (20)
2
Theorem 1 and we omit the parts that are the same with those
d1 + d2 min(N1, N2 )
given in Section III for brevity. Without loss of generality, let
Proof: Let a genie provide the transmitters with perfect
us assume N1 N2 . Following (7), we have
channel state information. Since giving CSIT does not hurt,
R1 I (X; Y[1] H[1], H[2] ) I (X; Y[1] H[1], H[2], U ), (28)
the converse argument is still valid. Then, the outerbound
follows directly from the results of [10]. Achievability of this
where U X (Y[1], Y[2] ) forms a Markov chain. Denote
outerbound follows trivially by receiver zeroforcing.
the capacity of the point-to-point link from the transmitter to
Theorem 3: If M1 N1 and M2 N2, the degrees of
receiver 1 as C1 and let
freedom region of the MIMO interference channel with no
= I (X; Y[1] H[1], H[2], U ). (29)
CSIT is the following:
Following (28), we have
d1 d2
D= (d1, d2 ) R+ : + 1 (21)
2
N1 N2 C1
R1
Proof: Let a genie provide transmitter 1 with W2 and
P
= EQ log I + H[1] Q( I)Q H[1]
transmitter 2 with W1 . Since the resulting channel is equivalent M
to a broadcast channel, the outerbound follows directly from [1]
P
H[1] E[QQ ]H
= log I +
Theorem 1. Achievability of this outerbound follows trivially M
by time division between the two users, and the proof is P
= log I + H[1] (M I)H[1]
complete. M
= log I + P H[1] H[1]
V. C APACITY R EGION OF A C LASS OF B ROADCAST
= N1 log(1 + P ) . (30)
C HANNELS WITH N O CSIT
Following (12), we have following useful inequality:
A. Models
Consider the 2-user Gaussian MIMO broadcast channel 1
I (X; Y[1] H[1], H[2], U )
where the transmitter is equipped with M antennas and N1
receivers 1, 2 are equipped with N1, N2 antennas, respectively. [1] [1]
I (X; Y1 H[1], H[2], U, Y(2:N1 ) ). (31)
M, N1, and N2 are assumed to satisfy
M
N1 (22) Now, following (9), we can write the upperbound for R2 as
follows.
M.
N2 (23)
R2 I (X; Y[2] H[1], H[2], U )
The channel is described by the input-output relationship:
[2]
= I (X; Y(1:N1 ) H[1], H[2], U )
Y[1] (t) = H[1] Q(t)X(t) + Z[1] (t) (24)
[2] [2]
+ I (X; Y(N1 +1:N2 ) H[1], H[2], U, Y(1:N1 ) )
Y[2] (t) = H[2] Q(t)X(t) + Z[2] (t) (25)
= I (X; Y[1] H[1], H[2], U )
where the notation usage for X[i] (t) and Y[i] (t), the assump-
N2
tion for the noise term Z[i] (t), and the assumption that the [2] [2]
+ I (X; Yi H[1], H[2], U, Y[1], Y(N1 +1:i 1) )
channel is equipped with perfect CSIR and no CSIT are the
i=N1 +1
same with those in Section II. However, different from the
N2
previous assumption, H[i] is assumed to be a time-invariant [2] [1]
+ I (X; Yi H[1], H[2], U, Y(2:N1 ) )
Ni M channel matrix with Ni orthonormal rows, i {1, 2}.
i=N1 +1
Note that this is possible only when N1 M and N2 M . Q
N2 N1
is an M M isotropically random unitary matrix, normalized +
N1
so that
N2
. (32)
E[QQ ] = M I, (26) N1
Thus, for 0 N1 log(1 + P ), an outerbound on the
where I is a M M identity matrix. The transmit power
boundary of the capacity region is characterized as follows.
constraint and the standard de nition of the capacity region
N2
are the same with those in Section II and we omit them for
(R1, R2 ) = N1 log(1 + P ),, (33)
brevity. N1
which implies that
d1 d2
(R1, R2 ) R+ : + log(1 + P ) .
C (34)
2
N1 N2
To provide the achievability of this outerbound, we rst prove
that (0, N2 log(1 + P )) is achievable. Denote the capacity of
the point-to-point link between the transmitter and receiver 2
as C2, and we have
P
= EQ log I + H[2] Q( I)Q H[2]
C2
M
P
H[2] E[QQ ]H[2]
= log I +
M
P
= log I + H[2] (M I)H[2]
M
= log I + P H[2] H[2]
= N2 log(1 + P ). (35)
Thus, (0, N2 log(1 + P )) is achievable. Using time division
between the two users, and the proof is complete.
VI. C ONCLUSIONS
In this paper, we explore the effect of the absence of
channel state information for MIMO networks. Throughout
the paper, we assume perfect CSIR and no CSIT. We provide
the characterization of the DOF region for a 2-user MIMO
broadcast channel. We then nd the DOF region for some
special cases of a 2-user MIMO interference channel by
using the broadcast channel outerbound. We also extend the
outerbound of the DOF region to nd the capacity region for
a speci c 2-user MIMO broadcast channel.
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