We assume that each base station and all users are equipped with N = nt= nr antennas.
This assumption enables us to use Corollary 3.6 and to select users such that interfer- ence alignment is possible. However, if nt ̸= nr feasibility of interference alignment can
u2
H
u1 uK uK−1 S1 ST bs1 limited feedback: VT = (ˆv k,T)k∈S x1 xT y1 noise y2 yK−1 yK bsT limited feedback: V1= (ˆv k,1)k∈SFigure 4.1: Distributed system architecture. Each scheduled user um, m ∈ S, feeds back
quantized channel state information ˆvm,l= µm,lvm,l to base station l. Based
on local channel state information Vl= (µ
m,lvm,l)m∈S each base station l com-
putes the precoders π(m), m ∈ Sl, and transmits dedicated pilots to initiate
the next iteration step.
will see, the distributed algorithm makes efficient use of quantized channel state informa- tion. Figure 4.1 depicts an example of the distributed system architecture. The minimum interference algorithm with user selection and quantized channel state information is sum- marized in Algorithm 4. At the beginning of each transmission frame, orthogonal common pilots are transmitted so that all users m can measure the channel matrices Hm,b for all b.
Knowledge of the channel matrices Hm,b is required by the users to compute the effective
channels; common pilots must be retransmitted in intervals depending on the coherence time of the channel. Next, the user selection is performed and then the iterative opti- mization of transmit precoders and receive filters is executed. In each iteration, after the transmit precoders have been optimized every base station transmits dedicated pilots on orthogonal resources such that each user m ∈ S can measure the effective channels Hm,lπ(m), for all m∈ Sl and all l = 1, . . . , T . Based on the measured effective channels
each user optimizes its receive filter and feeds back the effective channels (Hm,l)Hum,
Algorithm 4 minimum interference algorithm with user selection Begin of transmission frame:
Transmit common pilots to all users and make an estimate Hm,b, with b = 1, . . . , T and
m∈ U.
Each base station b = 1, . . . , T selectsSb ⊆ U according to (4.7).
Set π(m) = 1/√nt(1, 1, . . . , 1)T for all m∈ S.
repeat
Each base station b = 1, . . . , T transmits dedicated pilots. for b = 1, . . . , T do
Compute receive filter matrix um, for all m∈ Sb, according to (3.6).
Quantize and feed back the effective channels ˆvm,l, for all m∈ Sb, l = 1, . . . , T ,
according to (4.6). end for
for b = 1, . . . , T do
Compute beamforming vectors π(m), for all m∈ Sb, according to (4.8).
end for
until termination condition is satisfied (e.g. maximum number of iterations, minimum residual interference, . . . ).
End of transmission frame
User Selection
If the channel coefficients are independent and drawn from a continuous distribution we can apply Corollary 3.6 to ensure that interference alignment is feasible. According to Corollary 3.6 spatial interference alignment is feasible if the number of active users per base station is bounded by
|Sb| ≤
1
T (2nt− 1) , b = 1, . . . , T. (4.7) Therefore, at the beginning of every transmission frame each base station selects⌊1
T (2nt− 1)⌋
users. The users can be selected according to some metric, e.g, maximum fairness, maxi- mum channel gain or the user selection can be defined by higher layers.
Receive Filter Optimization Based on Perfect Channel State Information
The receive filters are optimized by the users in a distributed manner. During the receive filter optimization all transmit precoders π are fixed. In the first iteration it is assumed that each base station transmits with a beamforming vector 1/√nt(1, 1, . . . , 1)T. Based on
dedicated pilots each user m∈ S measures the effective channels Hm,lπ(m), l = 1, . . . , T,
l ̸= b, m ∈ Sl. and computes the receive filter ρ(m) according to (3.6). Next each user
Feedback of Effective Channels
In each iteration step all scheduled users m∈ S feed back quantized channel state infor- mation to all base stations. For fixed receive filters ρ(m) each user m∈ S quantizes and feeds back the effective channels ˆhm,l to base station l. Such that in each iteration base
station l gets an update of the local channel state information Vl(4.6). Based on Vl base
station l updates the precoders π(m) for all m∈ Sl.
Distributed Transmit Precoder Optimization Based on Quantized Channel State Information
The transmit precoder optimization is performed by each base station independently and is based on quantized local channel state information Vl. Similar to the centralized algorithm
(Section 3.3.2) the transmit precoders are computed in two steps. First, the transmit subspace which causes minimum out-of-cell interference is determined. Second, the intra- cell interference is canceled by a zero forcing step.
Consider the reciprocal network, the reciprocal precoded channel from user m in cell b to base station l is given by ←v−m,l = ˆvm,l. For base station b the receive subspace (in the
reciprocal system) which causes minimum out-of-cell interference is given by Πb = ν |Sb| min ←− Θb ∈ Cnt×|Sb|, where νN
min(X) is defined as the eigenvectors corresponding to the N smallest magnitude
eigenvalues of the Hermitian matrix X. The interference covariance matrix←Θ−b is defined
as ←− Θb = T l=1 l̸=b m∈Sl ←v− m,b(←v−m,b)H.
Finally, the intra-cell interference is canceled by an additional zero forcing step. The zero forcing precoder wm,b ∈ C|Sb| for user m in cell b is chosen from the null space of the
effective channels vH
k,bΠb, with k∈ Sb and k̸= m such that the transmit precoder for user
m∈ Sb is given by
π(m) = Πbwm,b. (4.8)
After each base station l has determined the precoding vectors for all m∈ Sl it transmits
dedicated pilots such that the next iteration can be initiated by the users.
In the next section we are going to analyze the rate loss gap of the distributed interfer- ence alignment algorithm with respect to the centralized solution with full channel state
information. In Section 4.5 we compare the algorithms numerically and demonstrate that the distributed algorithm significantly reduces the feedback load.