EL REY AZÚCAR Y OTROS MONARCAS AGRÍCOLAS
LA CRISIS DE LOS AÑOS TREINTA:
In the SM-OFDM system, the columns in the channel matrix are not orthogonal to each other, i.e., the transmitting symbols from different transmitting antennas are interfering with each other at the receiving antenna. As discussed in Section 4.2.2, the MF receiver explores the MIMO diversity by combining the received energy from all transmitting antennas, however, it suffers from massive level of interfer- ence. The ZF receiver removes the interference completely, however, it does not explore the diversity benefits and suffers from large noise enhancement. The MMSE receiver makes the compromise between the interference and noise, however, it does not explore the benefits of MIMO diversity either. Therefore, in order to explore the MIMO diversity, remove the interference, and avoid noise enhancement at the same time, we apply the iterative parallel interference cancelation approach to per- form data detection. The receiver is shown in Fig.4.3, with a space-time processing module performing MMSE detection, a parallel interference cancelation approach for the SM-OFDM system, and a MAP decoder that performs data decoding while also providing feedback information for the iterative detection technique. The
4.3 Iterative Receiver for MIMO-OFDM 87 switch is shown to initially allow the selection of the MMSE detector for the first iteration and then select the interference canceler in subsequent iterations. In the following, we present an iterative algorithm for the detection of an OFDM received signal from multiple antennas, which performs data detection and decoding, while canceling interference. Iterative/ Final Channel Estimator Equalization/ Demapper Deinterleaver Decoder Decoded information bits Interleaver BPSK/ QPSK/ 16QAM mapping OFDM Demodulator OFDM Demodulator MMSE Detector Interference Canceller Initial Channel Estimator Preamble/ Pilot OFDM data symbol Received signal reconstruction Interference cancelation Transmitting antenna multiplexing Mean/ Variance estimator Canceller output for channel estimator Canceller output for demapper
Figure 4.3: Iterative receiver for SM-OFDM systems
In the first iteration, MMSE filtering is performed to reduce the error to a reasonable level for further processing in the later iterations. Assuming that the channel state information is perfectly known and the transmitted data symbols are uncorrelated, the output of the MMSE filter for the signal transmitted by all transmitting antennas is given by:
\
YM M SE
i,m =Fi,mM M SEYi,m, (4.44)
where the MMSE filter is given by: FM M SE
i,m = [HHi,mHi,m+ σ2 w Ed INR] −1HH i,m, (4.45)
where Ed is the channel data symbol energy. The mean and covariance matrix of
MMSE filter output is obtained as:
µM M SE
CM M SE
i,m =σw2[HHi,mHi,m+ σ2
w EdINR]
−1. (4.47)
Hence, the demapper outputs LLRλe
1 of the kth coded bitsck for MMSE filter output as:
λe
1(ck(Xi,m)) = ln
P
Sj∈Uk+(ANT)
P(Xi,m =Sj|Y\M M SEi,m ,Hi,m, λe2)
P
Sj∈Uk−(ANT)
P(Xi,m =Sj|Y\M M SEi,m ,Hi,m, λe2)
, (4.48)
where the conditional probability P(Xi,m = Sj|Y\M M SEi,m ,Hi,m, λe2) is proportional to its Gaussian p.d.f, i.e.
P(Xi,m =Sj|Y\MM SEi,m ,Hi,m, λe2) ∼exp{−12(Y\i,mM M SE−µi,mM M SE)H(Ci,mM M SE)−1
·(Y\M M SE
i,m −µM M SEi,m )} Q l6=k
P(cl(Xi,m)). (4.49)
From the second iteration onwards, to separate the desired signal from the interference signal, we define the NT ×1 interference cancelation vector as
et= [0,0, . . . ,0,1,0, . . . ,0]T (4.50)
such that all elements in et are zeros except the tth element is 1. The interference
vector can be constructed as:
Xti,m = Xi,m−Xi,mt et
= [X0
i,m, Xi,m1 ,· · · , Xi,mt−1,0, Xi,mt+1,· · · , Xi,mNT−1]T, (4.51)
where the symbol of interest Xt
i,m is removed. Then the parallel co-antenna inter-
ference canceler is performed as
Yt,ICi,m = Yi,m−Hi,mXti,m
= Yi,m−Hi,m(Xi,m−Xi,mt et)
= Ht i,mXi,mt | {z } desired signal +X k6=t Hk
i,m(Xi,mk −Xdi,mk )
| {z }
residual interference
+Wi,m, (4.52)
where Xdk
i,m is the soft symbol of Xi,mk . The residual interference term in equation
(4.52) will vanish if the soft symbol is estimated perfectly, i.e. Xdk
4.3 Iterative Receiver for MIMO-OFDM 89 that case, the interference canceler will have interference free performance, i.e.
Yi,mt,IC =Ht
i,mXi,mt +Wi,m, (4.53)
where a linear combiner (matched filter) as discussed in Section (4.3.1) provides the optimal detection as:
Yi,mt,IC−M F = FM F i,m Yi,mt,IC
= (Hti,m)HHti,mXi,mt + (Hti,m)HWi,m
=
NXR−1
r=0
|Hi,mr,t|2Xt
i,m+ (Hti,m)HWi,m. (4.54)
Generally speaking, the reliability of the soft data symbols improves over iter- ations, which means the co-antenna interference can not be removed completely in the first few iterations. Hence, the residual interference has to be taken into the consideration. Similar to the first iteration, data detection is performed in the single-tap demapper by assuming the output of space-time processing module is Gaussian distributed as N(µt,ICi,m ,Ci,mt,IC). The mean of the interference canceler output is obtained by taking the expectation of equation (4.52) as:
µt,ICi,m = E{Ht
i,mXi,mt + X
k6=t Hk
i,m(Xi,mk −Xdi,mk ) +Wi,m}
= E{Hti,mXi,mt }+X k6=t Hki,m(E{Xi,mk } −Xdk i,m | {z } = 0 ) +E{Wi,m} = Hti,mXi,mt . (4.55)
To compute the covariance matrix Ci,mt,IC, firstly we compute the mean of the inter- ference vector Xti,m, which is given by:
E{Xti,m} = E{Xi,m−Xi,mt et}
= [Xd0
i,m,Xdi,m1 ,· · · ,X[i,mt−1,0,X[i,mt+1,· · · ,X\i,mNT−1]T, (4.56)
and the covariance matrix of the interference vector Xti,m is given by: Ct,ICi,m = E{(Xti,m−E{Xi,mt })(Xti,m−E{Xti,m})H}
= diag(γ2
where
γi,m,k2 =|Xi,mk |2− |Xdk
i,m|2. (4.58)
As shown in Chapter 2 Section 2.5, the energy of soft data symbol is computed as: |Xdt i,m|2 = Ed|E{Xdi,mt }|2 = Ed| X Sj∈A Sj·P(Xdi,mt =Sj)|2, (4.59)
where A is the signal constellation set with 2log2M signal points. For equal energy
constellation, like BPSK and QPSK, the average signal power of the data symbol is given by:
|Xt
i,m|2 = |E{Xi,mt }|2
= Ed, (4.60)
For unequal energy constellation, like 16QAM or 64QAM, by using Jensen’s in- equality [106], the average signal power of of data symbols is given by:
|Xi,mt |2 = |E{Xi,mt }|2 ≤ E{|Xbt i,m|2} = Ed X Sj∈A |Sj|2 ·P(Xdi,mt =Sj). (4.61)
Hence, the covariance matrix Ci,mt,IC can be obtained by: Ci,mt,IC =Hi,mC
t,IC
i,m HHi,m+σ2wINT, (4.62)
With the Gaussian distribution N(µt,ICi,m ,Ci,mt,IC), the demapper outputs LLR λe
1 of the kth coded bitsc
k for MMSE filter output as:
λe
1(ck(Xti,m)) = ln P
Sj∈Uk+(A)
P(Xt
i,m =Sj|Ydi,mIC,Hi,m, λe2)
P
Sj∈Uk−(A)
P(Xt
i,m =Sj|Ydi,mIC,Hi,m, λe2)
, (4.63)
where the conditional probability P(Xt
4.3 Iterative Receiver for MIMO-OFDM 91 Gaussian p.d.f, i.e.
P(Xt
i,m=Sj|Y[i,mt,IC,Hi,m, λe2) ∼exp{−12(Y[
t,IC
i,m −Hti,mXi,mt )H(Ci,mt,IC)−1
·(Y[i,mt,IC−Ht
i,mXi,mt )} Q l6=k
p(cl(Xi,mt )), (4.64)
Furthermore, the post interference cancelation process can be performed to reduce the equalizer/demapper complexity. The IC-MF process is obtained by taking the linear combiner or MF on the output of the interference canceler, i.e.
Yi,mt,IC−M F = Fi,mM FYt,ICi,m
= (Hti,m)HHti,mXi,mt +X
k6=t
(Hti,m)HHi,mk (Xi,mk −Xdk i,m)
+(Ht
i,m)HWi,m. (4.65)
Again, we compute the mean µt,ICi,m −M F and covariance matrix Ci,mt,IC−M F as:
µt,ICi,m −M F = E{(Ht
i,m)HHti,mXi,mt + X
k6=t
(Ht
i,m)HHki,m(Xi,mk −Xdi,mk )
+(Ht
i,m)HWi,m}
= E{(Ht
i,m)HHti,mXi,mt }+ X
k6=t
(Ht
i,m)HHki,m(|E{Xi,mk {z} −Xdi,mk}
= 0
) +E{(Ht
i,m)HWi,m}
= (Hti,m)HHti,mXi,mt , (4.66)
and
Ci,mt,IC−M F = (Hti,m)H(Hi,mCt,ICi,m HHi,m+σw2INT)H
t
i,m. (4.67)
Another post interference cancelation process is the IC-MMSE, which is ob- tained by taking the MMSE filtering on the output of the interference canceler, which is given by:
Yi,mt,IC−M M SE = (Fi,mt,M M SE)HYt,IC
i,m . (4.68)
The mean µt,ICi,m−M M SE and covariance matrix Ci,mt,IC−M M SE can be obtained as:
µt,ICi,m−M M SE = (Fi,mt,M M SE)HHt
and
Ci,mt,IC−MM SE = (Fi,mt,M M SE)H(H i,mC
t,IC
i,m HHi,m+σ2wINT)F
t,M M SE
i,m . (4.70)