In this section, we consider our novel downlink constellation preforming scheme and apply it in a MIMO scenario. The aim of our MIMO preforming is to enable the mul- tiplexing of multiple users’ signals where the number of available transmit antennas is less than the number of component user streams.
5.3.1
Introduction
The key design principle for MIMO NOMA Constellation Preforming with Spatial Mul- tiplexing and Diversity (NCPf-SMD) is to enable spatial multiplexing ML independent multi-user signals on to just L transmit antennas, compared to the required ML transmit antennas in traditional MIMO SM. This is achieved by preforming the users’ component streams according to the transmit antennas. The component signals, each with modula- tion set Qm, are preformed with the objective that the dminof the superposed composite constellation points are maximized and fully decodable.
Figure 5.3: System Model of the proposed MIMO NOMA constellation preforming with spatial multiplexing and diversity scheme. The figure shows the eNB with L antennas broadcasting
preformed superimposed data to multiple M users equipped with Km antennas. The channel
and noise are also illustrated. It is assumed the eNB transmits on N orthogonal subcarriers.
5.3.2
System Model
Consider a multi-antenna system of M active users each with Kmantennas communic- ating simultaneously with a eNB equipped with L antennas on N orthogonal subcarri- ers as illustrated in Figure 5.3. The signal transmitted from each eNB antenna, subject to eNB total power constraint, is a composite signal composed of a set of independent streams from each of the M users. The users employ non-linear JML detection for signal recovery and extraction. The received signal at the users as
ym= Hm¯x + zm (5.5)
where ym is the received signal at the m-th user, Hm∈ CKm×Lis the m-th user channel matrix whose entries are assumed to be independent and uncorrelated zero mean unit
variance complex fading coefficients, ¯x ∈ CL×1 is the transmitted signal vector from the LeNB antennas. zm∈Km
×1
is zero mean, variance σ2
m, independent and identically dis- tributed (i.i.d) complex AWGN noise vector of the m-th user. The overall MIMO channel is expressed is thus
H= [H1, . . . , Hm, . . . , HM]T (5.6)
where H ∈ CK×Lis the channel matrix, with K = P
mKmas the number of total receive antennas.
We employ spatial multiplexing to our MA preforming scheme where M users have L independent streams to transmit on L eNB antennas. Traditionally, this requires at least MLtransmit antennas. However, for the proposed NCPf-SMD, we preform all the multi- user streams onto just L eNB transmit antennas i.e the l-th eNB antenna is composed of a subset of L independent streams of the M users.
Let sm= [s (1)
m , . . . s(l)m, . . . , s(L)m ]T; s(l)m ∈Qmdenote the L input streams of the m-th user, each from modulation set Qm. Furthermore, let s(l) = [s
(l) 1 , . . . , s
(l)
m, . . . , s(l)M]T denote the l-th stream of M users to be preformed for transmission on the l-th eNB antenna, the preformed composite signal at the l-th antenna can then be expressed as
¯x(l)= M X
m=1
w(l)ms(l)m s(l)m ∈s(l) (5.7)
where ¯x(l)is the composite signal of the l-th stream of the M users, w(l)
m is the preforming weight that maximizes the dminbetween the constellation points of ¯x(l). Therefore, each symbol ¯x(l)belongs to one of U(l)possible composite constellation points. The received
signal at the k-th antenna of the m-th user is then defined as ymk = L X l h(l)mk¯x(l)+ zmk k ∈ Km (5.8)
where h(l)mk is the channel from the l-th eNB antenna to the k-th receive antenna of the m-th user and zmk is the additive white Gaussian noise at the m-th user antenna k with variance σ2
mk.
As our MIMO preforming is as any MIMO SM system but with each stream a com- posite of component signals, we can employ any linear or non-linear detection scheme so long as channel state information and the designed composite constellation U = [U(1). . . U(l). . . U(L)]T are known at the receiver.
5.3.2.1 Maximum Likelihood Joint Detection and recovery
The nonlinear JML detection for the combined received signals is employed at users. It is optimal in minimizing the error probability by searching for the most likely transmitted signals when compared with the pre-known designed constellation. We define
˙
U= UHm (5.9)
as the reference composite constellation normalized by the channel. Employing the min- imum distance criterion, the estimated signal at the users can be expressed as
ˆx = arg [min
U (kym
where kym−Uk˙ 2is the distance between the received signal and the possible transmitted vector ¯x
5.3.2.2 Zero Forcing ML Detection
The suboptimal linear ZF receiver with ML decoding can be employed at the users to null-out the subsequent interfering composite streams by employing the Moore-Penrose pseudo-inverse of the channel matrix Gm= H†mwhere H
† m= (H
∗
mHm)−1H∗m. When Hm is square and invertible, then the output of a ZF receiver with perfect CSIR is thus given by ˆx = arg minU Gmym−U 2 = arg " minU H∗mHm¯x H∗mHm + H∗mHmzm H∗mHm −U 2!# (5.11)
where ˆx ∈ CL×1 is the detected signal vector. The joint decoding is decomposes the received signal into L streams which eliminates the MUI. However, this comes at the cost of noise enhancement as the noise term is also inverted in the detection process i.e. zmGm