IHALAINEN et al.: CHANNEL EQUALIZATION FOR MULTI-ANTENNA FBMC/OQAM RECEIVERS 2073
Fig. 2. (a) OQAM modulator. (b) OQAM demodulator.
Fig. 3. MIMO FBMC/OQAM system model.
symbols a
k,l
. This complex-to-real mapping (indicated by the
abbreviation C2R) can be expressed as
d
k,2l
= {a
k,l
} and d
k,2l+1
= {a
k,l
} (3)
where {·} and {·} denote operators, which extract the real
and the imaginary part of a complex-valued argument, respec-
tively. Symbols d
k,n
and d
k,n+1
can therefore be interpreted
to carry the in-phase and quadrature (I/Q) components of
a complex-valued symbol a
k,l
from a QAM alphabet with a
relative time offset of T/2. The resulting sequence is multiplied
by θ
k,n
= e
j(π/2)(k+n)
= j
k+n
, which introduces a phase map-
ping between the real valued data sequence and the complex-
valued input samples of the synthesis bank.
1
The OQAM
symbols at the input of the kth synthesis filter can then be
expressed as v
k,n
= d
k,n
θ
k,n
. The OQAM demodulation [see
Fig. 2(b)], which is carried out in the receiver side, performs the
inverse operations in the reversed order. The abbreviation R2C
stands for a real-to-complex mapping that is expressible as
˜a
k,l
=
˜
d
k,2l
+ j
˜
d
k,2l+1
(4)
where two successive real-valued samples are used to form a
complex-valued received data estimate ˜a
k,l
.
1
Note that, although the signs of the sequence θ
k,n
can arbitrarily be chosen,
the pattern of real and imaginary samples has to follow the given definition [7]
to obtain the orthogonality of subcarriers.
B. MIMO Model
Let us consider a MIMO FBMC/OQAM system with N
T
transmit and N
R
receive antennas. The block diagram of such
a system is depicted in Fig. 3. The discrete-time baseband
signal, at the output of the M-subchannel synthesis bank of the
FBMC/OQAM transmitter that corresponds to the ith transmit
antenna, can be expressed as
s
i
m
=
k∈M
u
∞
n=−∞
v
i
k,n
g
k,m−n
M
2
. (5)
Here, v
i
k,n
= d
i
k,n
θ
k,n
is the output of the OQAM modula-
tor for the kth subchannel at the ith transmit antenna, and
M
u
denotes the set of active subchannels. We assume that
the transmitted symbols are uncorrelated in space, time, and
frequency, i.e., they satisfy E{v
p
k,m−α
v
q
l,m−τ
∗
} = δ
pq
δ
kl
δ
ατ
σ
2
v
,
where δ
xy
stands for the Kronecker delta, σ
2
v
= σ
2
a
/2, and
E{·} denotes a statistical expectation. The signals s
i
m
,for
i =1, 2,...,N
T
, are then transmitted through a frequency-
selective MIMO channel, which is assumed to be time invariant
over the duration of a transmitted FBMC/OQAM frame. A
quasistatic block-fading model is therefore assumed throughout
the forthcoming analysis. Moreover, let us denote by h
ji
p
the pth
tap of the multipath impulse response h
ji
=[h
ji
0
,h
ji
1
,...,h
ji
ν
],
which characterizes the channel between the ith transmit and
the jth receive antenna. Without loss of generality, we assume
that all channels have the same length ν +1. Throughout this
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