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 Authors : Tinuola Olayinka Coker, Joulani Shadi Muhammad Jamal, Biniyame Mulatu Yilma And Redon Dimroci
 Paper ID : IJERTV2IS1144
 Volume & Issue : Volume 02, Issue 01 (January 2013)
 Published (First Online): 30012013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
A Distributed Polarizing Transmission System for Frequency Selective Fading Channels
Tinuola Olayinka Coker, Joulani Shadi Muhammad Jamal, Biniyame Mulatu Yilma and Redon Dimroci
Abstract Motivated by Arikans channel polarization that shows the occurrence of capacityachieving code sequences, we address the scheme design issues by switch ing to polarizing frequency selective fading channels while transmitting information symbols in a source relaydestination MIMOOFDM relay communication system. A simple polarandforward (PF) MIMO relay scheme, with source node polar coding and relay nodes polar coding, is proposed to provide an alternative solu tion for transmitting with higher reliability than the con ventional decode andforward/amplifyandforward (DF/AF) relay schemes. In the proposed scheme, OFDM modulator is implemented at source node, some simple operations, namely time reversion, complex conjugation and polarization, are implemented at relay nodes, and the cyclic prefix (CP) removal is performed at destina tion node. It is divided into two symmetrical polarizing relay systems, i.e., the downpolarizing system and the uppolarizing system, which result in different capacities for the polar system. We analyze the bit error rate (BER) performance with the fixed polar system equipped with four OFDM blocks, which is an idea approach to se lect signal sequences that tend to polarize in terms of the reliability under certain combining and splitting the transmitted OFDMs in the frequency selective fading (FSF) channels. The polar system has a salient recursive ness feature, and thus the transmitted information signals embedded in the polar code can be decoded with a low complexity decoder.
I. INTRODUCTION
The channel polarization shows an attractive construction of provably capacityachieving coding sequences [1][6]. It has provided an attempt method to meet this elusive goal for multi fold binaryinput discrete memoryless channels, where channel combining and splitting operations were applied to improve its symmetric capacity [1], [2]. Actually, the pola rization of multiple channels is a commonplace phenomenon and thus it is almost impossible to avoid as long as several channels are synthesized in a proper density with certain arrangements. During the past decade, the multipleinput multioutput (MIMO) communication system has been well
studied to promise significants of the increasing spectral eff ciency, channel capacity and link reliability [7][11]. It shows that the coding gain and diversity can be simulta neously achieved with su i t ab l e coding schemes. As the MIMO techniques grown up, researchers have been ex ploring new communication paradigms. A potential proposal is the socalled wireless relay system, which provides the reliable transmission, high throughput and broad coverage for wireless network [12], [19], and [20]. The eminent merits of a MIMO wireless system lie in its potential temporal diver sity gain, spatial diversity gain and multiplexing gain to en hance link reliability. This elegant technique can be further exploited to explore the potential spatial and temporal diver sity on flatfading or frequency selective fading (FSF) chan nels with some proper transmission schemes, such as space time (ST) coding [13], spacefrequency (SF) coding and spacetimefrequency (STF) coding [14][18]. It is shown that the coding gain and diversity can be simultaneously achieved with suitable coding schemes. Unfortunately, as the number of transmit antennas becomes large, the complexity of de coding increases, which makes the design of coding or modulating schemes difficult.
MIMO relay communications, together with the ortho gonal frequency division multiplexing (OFDM) techniques, present an effective way of increasing reliability as well as achievable rates in next generation wireless networks. Cooperative diversity is usually achieved through relay nodes that help the source node forwarding its information. Deploying proper relays between source node and desti nation node can not only overcome shadowing due to in evitable obstacles, but also reduce the transmit power from the source node. In the MIMO OFDM relay system, two or more nodes share and transmit jointly their information symbols in a multiantenna array, which enables the high data rate and diversity gain. A usual approach to share in formation is to tune in the transmitted signals and process the whole (or partial) received information in regenerative or nonregenerative way. The former employs a decodeand forward (DF) relay scheme in which each relay decodes the original information from the source and forwards it to the destination [12]. Unfortunately, since channels are usually noisy and fading, the processed information signals are not perfect. Therefore, we have to study a possible coding or
modulating strategy to improve its performance that makes a merit of relay system. In the latter scheme it exploits an am plifyandforward (AF) scheme to amplify and retransmit the scaled signals without any attempt to decode the original information [21], [22]. In the light of superiority of these relay strategies with the availability of CSI, we con sider the coding design of the MIMOOFDM relay system for the FSF channels with the fixed gain relaying scheme using the polar andforward (PF) relay technique, in which each relay node encodes and retransmits the partial signals with the fixed power constraint. A key feature of this scheme is that we do not require relays to decode. Only a simple processing operation is done at each relay, which makes the transmission much simple and hence can avoid imposing bottlenecks on the data rate.
Recently, significant efforts have been related to the in creasing capacity [23] or the optimal design of the relay sys tem

in terms of the DF/AF relay schemes based on a scenario equipped with single or multiple antennas. However, further improvement should be sought in these relay systems, in which the loss of the signal rate is boosted as the number of relay nodes along with antennas increases. While a key com ponent relay design is to optimize the precoding of source and relay in benefits of multiple antennas and multiple OFDM symbols, how to design the MIMOOFDM relay sys tem to achieve high reliability with lowdecoding complexity via a coding approach becomes a challenge.
The problem with the previous relay system is the data rate loss as the number of relay nodes increases. This leads to the use of polar coding sequences in MIMOOFDM system, where relay nodes are allowed to simultaneously transmit the same OFDM systems over the FSF channels. We consider a simple design of the relay system that achieves the fascinat ing symmetric capacity of the FSF channels based on polar coding with a successive interference cancellation (SIC) de coder at destination node, which is motivated by the fascinat ing Shanons channel coding theorem [25]. It is an extension of work where OFDM combining and splitting are used for recursive code construction with the SIC decoding, which are essential characters of the polar coding sequences [1][6]. This is an idea approach to construct code sequences as com bining and splitting OFDM for the FSF channels to increase its reliability.
Furthermore, we establish an analytical framework that illustrates the potential bit error rate (BER) performance to be achieved from the polar MIMOOFDM relay system. We argue that the present system may increase the symmetric capacity under a low computation complexity of the SIC de coding due to the fact that a large number OFD symbols may be equipped for the polarizing FSF channels that tend to
polarize under certain OFDM combining and splitting opera tions.
This paper is organized as follows. In Section II, we describe the polar MIMO relay system with two switch ing communication model, the downpolarizing system and the uppolarizing system. In Section III, we systemati cally study the design of an simple PF scheme with space timefrequency (STF) transmission for downpolarizing and uppolarizing FSF channels. Some simulation results are also depicted in order to show the BER performance be havior and robustness of this polar MIMOOFDM relay system. Finally, conclusions are drawn in Section IV.

Downpolarizing OFDM Blocks for Distributed System

Uppolarizing OFDM blocks for Distributed System

Fig. 1. The relay communication system based on the polarizing MIMOOFDM channels with two models: (a) denotes the downpolarizing communication model; and
(b) denotes the uppolarizing communication model.
Some notations are defined throughout this paper as fol lows:

: complex number field;

T: finite nonnegative integer set {0, 1, Â· Â· Â· , T 1};

Bold faced uppercase letters, such as A: matrices;

tr: trace of a matrix;

Bold faced downcase letters, such as a: column vec tors;
*
*
T H

Superscripts (Â·) , (Â·) , and (Â·) : transpose,
complex, conjugate transpose, complex conjugate, respectively;

F: Frobenius norm of a matrix;

E[]: expectation of variable ;

: the Kronecker product;

: the Hadamard product, i.e., the componentwise product;

In: identity matrix of size n Ã— n;

diag(d0 , , dN 1): a diagonal matrix with di

of the channels, and T,sk is the corresponding path delay. Each channel coefcient sk() is modelled as zero mean complex Gaussian random variables with variance 2,sk such
l l,sk sk
l l,sk sk
that L 1 2 1 . We also assume that () are i.i.d. ran
0
dom variables for any (k, ). Similarly, other two channel impulse responses k(t) from relay node Rk to destination node D are written as:
agonal entries d0, , dN 1.
L1
Kk ( t ) sk
l0
( l )( t
l ,sk ),
(2)

CHANNEL POLARIZATION: DOWN POLARIZING AND UPPOLARIZING MIMO OFDM RELAY SYSTEM
where rk(l) represents the channel coefficient of the lth path of the channels, and l,rk is the corresponding path delay. Each channel coefficient rk(l) is also modelled as zero mean complex Gaussian random variables with variance 2 l,rk
We consider the distributed wireless system based on
such that
L 1 2 1 . In addition, we denote the average
OFDM modulation with N subcarriers. There is one source
l
l,rk
0
node S, one destination node D, and two relay nodes R,
{R1,R2}, as shown in Fig.1. There is only one antenna at all nodes S, R and D, respectively. This assumption is applicable for any nodes equipped with multiple antennas. We consider a scenario where Ns OFDM symbols are transmitted for Ns =
2n. The design of the relay scheme that can mitigate relay
synchronization errors is considered. Each relay node Rk, k
{1, 2}, is assumed to be capable of processing the OFDM symbols independently and correctly. The average transmit power at source node S is pt. The relay scheme is halfduplex, meaning that S and R do not transmit and receive simulta neously. The Ns independent OFDM symbols are transmitted simultaneously from source node S to destination node D in two stages. In the first stage the initial signal OFDM symbols are polarized and transmitted from source node S to each relay node Rk, k {1,2}. In the second stage each relay node Rk forwards the (partial) signal vector received from source node S to destination node D while source node S keeps silent. We further assume that each singlelink between a pair of transmit antenna and receive antenna is frequency selective Rayleigh fading with L independent propagation, which experiences quasistatic and remains unchanged in certain blocks. Denote the fading coefficient from source node S to relay node Rk as hSRk = k and the fading coeffi cient from relay node Rk to destination node D as hRkD = k. Assume that k and k, k {1,2}, are independent zero mean complex Gaussian random variables. Two channel im pulse responses k(t) from source node S to destination node R are written as:
power for one transmission of each relay Rk as pr. The con straint on the total network power is p = pt+2pr. We also adopt the power allocation strategy suggested in [26], and thus have
pt = 2pr = p/2. (3)
The MIMOOFDM channel model, denoted by H 2Ã—2, is created between source node S and relay nodes R, and K 2Ã—2 between R and D. Here entries of H and K are assumed independent and identically distributed (i.i.d.) with distribu tion CN(0, 1). For the distributed MIMOOFDM relay wire less system with sourcerelaydestination triplet structure, it is equivalent to two partial MIMOOFDM wireless systems.
One part has the MIMO channel model H, i.e.,
H = diag(1, 2), (4) and another part K is given by
K = diag(1, 2). (5)
Based on the MIMOODFM relay channels H and K in

and (5), we design the polar system for the transmission of the signal vector x, in which we switch to the polar system in four consecutive time slots, i.e., downpolarizing and up polarizing communications in turn. The system has two transmission phases. In phase 1, the source node broadcasts four OFDM symbols that are first polarized at source node S to each relay node Rk. In phase 2, source node S stops the transmission and each relay node Rk that polarizes the re ceived symbols for the second time and retransmits the re sulting symbols to destination node D.
A. Downpolarizing MIMOOFDM Relay System
At source node S the transmitted information is modulated
L1
( t )
( l )( t ),
(1)
into complex symbols xij and then each N modulated symbol
k sk
l0
l ,sk
as a block are poured into an OFDM modulator of N subcar
where
th
()represents the channel coefficient of the path
riers. Denote four consecutive OFDM blocks by xi = (xi,0,
sk
xi,1, , xi,N1)T, i 4. We define xi + xj = (xi,0 + xj,0,
xi,1 + xj,1, , xi,N1 + xj,N1)T, i, j 4, for polariza
Therefore, the received signals at Rk, k {1, 2}, for four successive OFDM symbol durations can be given by
n
n
rk0 = pt 0 k + k0
tion calculation.
r pt
+
In the first time slot, four consecutive OFDM blocks are
k01=
1 k n k1
processed with the downpolarizing 4Ã—4 matrix Q4 at source
rk2 =pt 2 k + k2
n
n
node S, i.e.,
r pt
+
(10)
U = XQ4, (6)
where U = (u0, u1, u2, u3) denotes the polarized matrix of size
k3 =
1 k n k3
TABLE I
N Ã— 4, X = (x0, x1, x2, x3) denotes the signal matrix of size N
Ã— 4 corresponding to four OFDM blocks, the polar matrix Q4
is given by Q4 = I2 Q2. Here matrix Q2 is a down polarizing matrix defined as Arikans fashion [1], i.e.,
IMPLEMENTATION OF THE PF SCHEME FOR THE DOWN POLARIZED SYSTEM AT RELAY NODES. OFi DENOTE THE ith OFDM BLOCK.
Q2 = 1 1
0 1
. (7)
Therefore, we have u2k2 = 2k2 and u2k1 = 2k2 + 2k1, for k {1, 2}.
In the OFDM modulator, the four consecutive blocks are modulated by Npoint FFT. Then each block is precoded by a cyclic prefix (CP) with length lcp. Thus each OFDM symbol consists of Ls = N+lcp samples. Finally four OFDM symbols are broadcasted to two relay nodes. Denote by sd2 the overall relative delay from source node S to relay node R2, and then to destination node D, where the relative delay means it is relative to relay node R1. In order to combat against both frequency selective fading channels and timing errors, we assume that lcp maxl,k{l,sk+l,rk+sd2}. Denote four consecu
tive OFDM symbols by i, i 4, where i consists of
FFT(ui) and the corresponding CP.
At each relay Rk, the received noisy signals will be simply processed and forwarded to destination node D. Assume the channel coefficients are constant during four OFDM symbol
intervals.
1 0 2
1 0 2
We define two processed vectors = T , T T and 2 =
T , T T
1 3 which are polarized at R1 and R2, respectively.
Polar R1
Polar R1
Process R1
Process R2
OF0
r10
r20
(r10 )
0
OF1
r11
r21
0
r* 21
OF2
r10+r12
r22
(r10+r12)
0
OF3
r13
r23+r21
0
(r +r )*
23 21
Polar R1
Polar R1
Process R1
Process R2
OF0
r10
r20
(r10 )
0
OF1
r11
r21
0
r* 21
OF2
r10+r12
r22
(r10+r12)
0
OF3
r13
r23+r21
0
(r +r )*
23 21
where pt is the transmission power at source node S, k is an LÃ—1 vector defined as k = (sk(0), sk(1), , sk(L1)),
n
n
denotes the linear convolution, and ki, i 4, denotes
the corresponding additive white Gaussian noise (AWGN) at relay node Rk with zeromean and unitvariance, in four suc cessive OFDM symbol durations.
Then each relay node Rk polarizes, processes and forwards the received noisy signals as shown in Table I, where ( ) denotes the timereversal of the signals, i.e.,
(rki()) rki(Ls ), Ls, and hence (rki(Ls)) = rki(0),
k {1, 2} and i 4. Denote by 0 (r10), 1 (r10+ r12), 2 r21and 3 (r21+ r23). For the th subcarrier of i we also take the notations i, i(), N.
After performing the abovementioned processing
operations, each relay node Rk amplifies the yielded symbols with a scalar = pr/(pt + 1) while remaining the average
Namely, at source node S we have transmission power pr. In order to make the PF scheme avail
u1=
T , T T
0 3
able for the FSF channels, it is required that for each relay Rk it can only implement the time reversal operation ( ) or the
u2= ((0+1)T, (2+3)T)T, (8)
and consequently
1 = (FFT(0)T , FFT(2)T)T,
2 = (FFT(0+ 1)T , FFT(2+ 3)T)T. (9)
complex conjugation operation ( ) on the received OFDM symbols.
At destination node D, the CP is removed for each
OFDM symbol. We note that relay node R1 implements the time reversions of the noisy signals including both informa tion symbols and CP. What we need is that after the CP re
moval, we obtain the time reversal version of only informa tion symbols, i.e., (FFT(ui)), i 4. Then by using some
properties of FFT/IFFT, we achieve the feasible definition as follows.
Definition 2.1 According to the processed four
Denote by i = (i0, i1, , i(N1)), i 4, the received
signals for four consecutive OFDM blocks at destination node D after the CP removal and the Npoint FFT transfor mations. Namely, we have
y [ p FFT( ( FFT( u ))) ]
OFDM symbols at relay node R1 we can obtain
0 t 0
1 k1
n10 k1 n0
( ) (FFT(u ))
y1 [
pt FFT( ( FFT( u0 u2 ))) 1 k1 ( n10 n10 ) k1 ] n1
1 i
(13)
at destination node if we remove the CP as in a convention
al OFDM system to get an Npoint vector and shift the
y2 [
y [
pt FFT(( FFT( u1 ))*) f t 2 k2 n21 k2 ] n2
p FFT(( FFT( u u ))*) f 2 ( ) ] ,
last = l
1 + 1 samples of the N point vector as the 3 t
3 1 2 k2
n21
n23 k2 n3
1 cp
' ' '
first samples. Here is an N Ã— 1 vector defined as
Where 1 FFT( ( 1 )) , k1 FFT( k1 ) , 2 FFT((1 )*) ,
1 1
= (s1 (0), , s1 (L 1), 0, , 0),
FFT( k' ) , n FFT( n ) and n FFT( n ),k {1,2 ) and
1
1
1
1
1
and denotes the maximum path delay of channel
from
k2 2 ki
i 4.
ki i i
source node S to relay node R1, i.e., 1 = maxl{l,s1 }. In a similar way, we define another N Ã— 1 vector
1
1
= (r1(0), , r1(L 1), 0, , 0).
At destination node D, after the CP removal, the received four successive OFDM symbols can be written as
According to the properties of the wellknown FFT transforms for an N Ã— 1 point vector x, we have
(FFT(x)) = IFFT(x),
FFT((FFT(x))) = IFFT(FFT(x)) = x. (14)
Therefore, the formulas in (13) can be written in the polar
0
0
0
0
1
1
10
10
y = (pt(FFT(u ))( )+Â¯n
) +n
form on each subcarrier , N, as follows
1
1
0
0
y = (pt(FFT(u +u ))( )+ Â¯n +nÂ¯ ) +n
1 0 2 1 10 12 1 1
y = (pt(FFT(u ))t t +nÂ¯* ) +n
2 1 sd2 1 2 21 2 2
y = (pt(FFT(u +u )) t t +Â¯n* +nÂ¯* )
3 3 1
sd2 1
2 21 23
+ n , (11)
2 3
1
1
where tsd2 is an N Ã— 1 vector that represents the timing errors in the time domain denoted as tsd2 = (0,sd2 , 1, 0, , 0)T,
and 0
sd2
is a 1 Ã— sd2 vector of all zeros, and
is the
(15)
shift of samples in the time domain defined as t = (0 , 1,
1 1
which can be rewritten as
0, , 0)T. Since the signals transmitted from R2 will arrive at the destination sd2 samples later and after the CP removal,
1
1
1
1
the signals are further shifted by samples. The total
number of shifted samples is denoted by 2
= sd2
+ .
Here Â¯nki is the AWGN at relay node Rk after the CP removal, and ni denotes the AWGN at destination node D.
where 2 f 2 , * f 2
(16)
2
2
2
2
k k
k k
, f 2 exp(2 /N),
2
2
2
2
2
2
After that the received OFDM symbols are trans formed by the Npoint FFT. As mentioned before, because of the timing errors, the OFDM symbols from relay node R2 arrive at destination node sd2 samples later than that of sym bols from relay node R1. Since lcp is long enough, we can still
maintain the orthogonality between subcarriers. The delay
HI and HF are information generator matrix and frozen gene rator matrix defined, respectively, as
(17)
xI = (x0 , x1)T, xF = (x2, x3)T, xi is the th element
sd2
in the tim domain corresponds to a phase change in the
of xi, k, is the th element of k, nki, is the th element of
frequency domain, i.e.,
1
1
1
1
fsd2 = (1, e2sd2/N, , e2sd2(N1)/N)T, (12) where f = (1, e2/N, , e2(N1)/N)T and = 1. Similarly, the shift of samples in the time domain also corresponds to a phase change f , and hence the total phase change is f2.
nki, and ni, is the th element of ni, k {1,2} and i Z4. Two 4 Ã— 1 vectors e0 and e are the polarized noises given by e0=(e01,e02,e03,e04)T and e = (e01,e02,e03,e04)T, where e01 = n10, 1, + 0,, e02 = (n10, + n12,)1, + 1,, e03 = n21, 2,+2, and e04 = (n21, +n23,)2, +3,.
We note that subvector xI serves as the information
vector while subvector xF as the frozen vector for the
downpolarizing MIMO relay system, which can be derived from the Bhattacharyya parameter vector for the derivation of the reliability of the FSF channels, calculated in next section. The combined matrix H = (HI,HF ) has the same structure as Arikans 4 Ã— 4 polar matrix [1], [2]
,
In the OFDM modulator for the uppolarizing sys tem, four resulting consecutive blocks are also modulated by Npoint
FFT and are precoded by a CP with length lcp. Denote by
i
i
u ' , i 4 four consecutive OFDM symbols that consist
i
i
i
i
of FFT( u ' ) and the corresponding CP. At each relay Rk, the received noisy OFDM symbols, denoted by r ' , will be pola
rized, processed and forwarded to destination node D.
where P4 is a permutation matrix given by
Define two vectors
u' (u'T ,u'T )T
and
1 0
0 0
(18)
u' (u'T ,u'T )T such that
1 0 1
P4 = 0 0 1 0
0 1 0 0
'2 1
u
u
' 1
3
((x0
1
1
2
2

x )T , (x

x3
)T )T ,
(21)
0 0
0 1
u ' (xT ,xT )T .
2 1 3
B. Uppolarizing MIMOOFDM Relay System
In the next time slot, the four consecutive OFDM blocks are processed with the uppolarizing 4 Ã— 4 matrix
4
4
Q' at S, i.e.,
After performing N point FFT onto u ' , i 4
i
i
u1 = (FFT(x0 + x1 )T, FFT(x2 + x3 )T)T,
u2 = (FFT(x1 )T, FFT(x3 )T)T. (22)
Therefore, the received OFDM symbols at relay node
r
r
Rk can be written as
4
4
U ' XQ' ,
(19)
'
k 0
r '
ptu '
0
0
k
k
ptu '

nk 0 ,

n ,
TABLE II
k1 1
k k1
(23)
IMPLEMENTATION OF THE PF SCHEME FOR THE UP
' ptu '
k nk 2 ,
r
r
2
2
k 2
k 2
POLARIZED SYSTEM AT RELAY NODES OFi DENOTE THE ith OFDM BLOCK.
' ptu ' n ,
r
r
3
3
k 3
k 3
k
k
k 3
Polar R
Polar R
Process R
Process R2
OF0
1
r ' r1
10 12
1
r'
20
1
( r ' r' )
10 12
0
OF1
r'
11
r ' r' 21 23
0
( r ' r' )*
21 23
OF2
r'
12
r'
22
( r' )
12
0
OF3
r'
13
r'
23
0
(rr' )*
23
Polar R
Polar R
Process R
Process R2
OF0
1
r ' r1
10 12
1
r'
20
1
( r ' r' )
10 12
0
OF1
r'
11
r ' r' 21 23
0
( r ' r' )*
21 23
OF2
r'
12
r'
22
( r' )
12
0
OF3
r'
13
r'
23
0
(rr' )*
23
Then two relay nodes polarize, process and forward the re
ceived noisy OFDM symbols as shown in Table II. After that we obtain
' ' ' ' ' 1 ' '
v0 (r10 r12 ), v1 (r12 ), v2 (r21 r23 )*
and
1 '
v3 (r23 )*.
After performing the processing, each relay node Rk
ampli
fies the yielded signals with a scalar
forwards them to the destination node D.
pr /( pt 1) and
4
4
0 1 2 3
0 1 2 3
where U ' (u' ,u' ,u' ,u' ) denotes the uppolarized matrix of size N Ã— 4, X (x0 , x1, x2 , x3 ) denotes the initial signal matrix of size N Ã—4, the uppolarizing operation Q' is given
At destination node D, the CP is removed for each OFDM symbol before being depolarized to decode the initial information with high reliability. Then the received noisy OFDM symbols for four successive OFDM symbol durations
can be written as
by Q' I Q1 and Q1 is defined as
4 2 2
1
2
1 0
Q2 1 1
(20)
Therefore, we have u2k2= x2k2 + x2k1and u2k1 = x2k1, k {1, 2}.
y' ( p (FFT(u' u' ))
0 t 0 2
(' ) n n ) k' n
1 10 20 1 0
(28)
e
e
0
0
y' ( p (FFT(u' )) (' ) ' T
x
x
1 t 2 1
xF ( x0, x1 )
denote the frozen vector (bits),
1
1
n20) k' n1
(24)
' ( x2, x3 )T is the vector (bits), ' and e' =
I
I
' ' '
( e' ,e' ,e'* ,e'* )T
and
e' ( e'
,e'
,e'*
,e'* )T
where
y2 (
pt (FFT(u1 u3 )) *
01 02
03 04
01 02
03 04
' '

n*
n* ) k' n
e' ( n'10, n'12, ) , n , ,e' 02 n'12, ,
sd2 1
2 21 23 2 2
01
21,
21,
n'1, ,e' 03 ( n'
k1

n

n
'
23,
0 k1
)
)

n

n
,
,
k2, 2
y' ( p (FFT(u )) *
' '
3 t 3
sd2
and e 04 n23,k2, n1,.
' ' n* ) k' n
Next, we can design the polar MIMOOFDM relay
1 2 23 2 3
Consequently, after the CP removal and Npoint FFT transformations, the received four consecutive OFDM blocks can be given by
y
y
k
k
)
)
0
0
k
k
' [ pt FFT((FFT(u 0 u 2 ))) 1 1
system by switching to polarizing four OFDM symbols for the FSF channels. Based on the downpolarizing system to transmit the th subcarrier of four OFDM sybols, we have the received vector of size 4 Ã— 1 given by
(n10
n12 1
] n 0
y HI I HF F, e,
whereas deploying the uppolarizing system, we have
(29)
y'
H'
H' '
e'.
(30)
1
1
I
I
y' [
pt FFT((FFT(u 2
))) 1

k1
F,
I I ,
n12 k1
] n1
We can decode the initial information vectors xI and x' ,
y' [ p FFT((FFT(u u ))) f T2
(25)
respectively respectively, using the conventional the ML receiver or the ZF/MMSE receiver after depolarizing the
2 t 1 3
21
21
23
23
k k
k k
2 2 (n* n* ) 2 ] n 2
3 t 3
3 t 3
y' [ p FFT((FFT(u ))*) f T2
k k
k k
2 2 n*23 2 ] n 3
For each subcarrier , N, the formulas in (25) can be written in the uppolarizing structure as follows,
transmitted signals at destination node D. However, in this paper we introduce a polar receiver via the polar decoding under a low complexity SIC strategy that can bring out simi lar performance behaviors as that of ML decoding for small number of OFDM blocks, i.e., Ns = 4. Fortunately, due to the benefits of polar coding sequences for the large number Ns = 2n[1][6] it also shares the good BER performance be havior of polarizing FSF channels in terms of its capacity achieving properties [25] as nonnegative integer n goes to infinity.
which can be written as
(26)
(27)

DECODING OF THE POL AR MIMO
OFDM RE LAY SYSTEM
We consider all singlelinks of the FSF channel H from each pair of transmit antenna of source node S and receive anten na of relay node Rk , and K from relay node Rk to desti nation node D, which are independent complex Gaussian random variables with zeromean and unitvariance. Each singlelink channel, denoted by W, has the transition probabil ity W (yx), where x, y A. As a useful measurement of the reliability of the wireless network, there is a conventional channel parameter, the symmetric capacity I (W) with some
H
H
H
H
I
I
F
F
where ' and ' are information generator matrix and
frozen generator matrix defined, respectively, as
modulations [25]. We note that parameter I (W) is the highest rate at which the reliable communication is possible using inputs with equal probabilities.
Polarizations of the FSF channels are derived from the are derived from the Ns = 2n OFDM symbols polarization with
at source node S, ri ri, the received signals at relay node R,
generator matrix GN
{ Qn ,Q' n }, which is an operation
vi vi, the polarized signals at relay nodes R, and yi yi,
2 2 2
by which one manufacture out of Ns independent OFDM symbols W yields a second set of Ns splitting OFDM sym
Ns Ns
Ns Ns
bols {W ( i ) : i } that show a polarization effect in a
the received signals at destination node D, which are all cor responding to the th subcarrier of the ith OFDM symbol. For each subcarrier xi,, yi,, ui,, and ri, of xi, yi, ui and ri, we also
take the notations x (x , x , x , x )T, x (x ,
sense that, as Ns becomes large, the capacity terms { IW ( i ) :
0,
1,
2,
3,
k 2k2,
Ns x )T, y
(y , y , y , y
)T, y
(y , y
)T, u
i Ns } tend towards one or zero for all but a vanishing
2k1,
0,
1,
2,
3,
k 2k2,
2k1, k
fraction of indices i [1]. In this paper, we only consider pola rizations of an MIMOOFDM relay wireless system includ ing Ns = 4 OFDM symbols, i.e., the combination of four OFDM symbols yields a second set of four splitting FSF
channels {W ( i ) : i }. This channel polarization consists
(u2k2,, u2k1,)T, and rk (r2k2,, r2k1,)T, k {1, 2}
and N.
According to the downpolarizing system, we derive the OFDM downcombining operation of the polar FSF channels
4 4 in terms of the above simplified notations.
of two operations, i.e., OFDM combining and OFDM split ting.
OFDM Combining
While deriving the FSF channel combining operation of the polar system with four OFDM blocks, we combine four OFDM symbols, denoted by W, in a recursive manner to produce a multilevel structure channel W4. We consider
OFDM combining and splitting for the th subcarrier of each
x
x
I
I
OFDM block with the downpolarizing information bits xI or the uppolarizing information bits ' . Without loss of
generality, we only consider the OFDM combining and split ting operations of the downpolarizing system with four OFDM blocks while showing the feasibility of the polar sys tem. As for the OFDM combining and splitting operations of the uppolarizing system, we can achieve the similar results while referring to the polar coding processing in [1], [2].
2
2
Based on the downpolarizing system in (29), we define the downcombining equivalent FSF channel as H ( HI ,HF ) expressed in (17), respectively. We note that the OFDM combining operations have the similar struc ture fashion as Arikan codes generated from G4 P4Q2 ,
This process begins with the lowlevel of the recursion at Rk that combines two independent OFDMs with transition probability W, which results in the OFDM downcombining for the second level combining FSF channel W2 for Rk with transition probabilities
W2(ykrk) = W(y2k2r2k2) W(y2k1r2k2+ r2k1). (32)
Similarly, the OFDM downcombing for the FSF channel W2 at source node S can be obtained with transition probabilities
W2(u1xk)=W(u2k2x2k2) W(u2k1x2k2+x2k1). (33)
Furthermore, the third level of recursion for the MIMO relay system combines two independent FSF channels W2 to establish the high level FSF channel W4 with transition probabilities calculated from the recursive formula
W4(yx)=W2(y0x0) W2(y1x0+ x1, x2+ x3)
3
=W(y0x0)W(y1x0+x2)W(y2x0+x1)W(y3 xi ). (34)
i0
2
2
In the similar way, we calculate transition probabilities of OFDM upcombining for the second level FSF channel W'
at Rk with transition probabilities
i.e.,
W' ( y'  r' ) W' ( y'
 r'

r'
).W' ( y'
 r'
), (35)
2 k k
2k 2
2k 2
2k 1
2k 1
2k 1
1
G4 = 1
0 0 0
0 1 0
(31)
and transition probabilities of OFDM upcombining for the FSF channel W2 at S as
1 1
0 1
' ' '
' ' ' '
' ' '
1
0 0 1
W2( uk  xk ) W ( u2k 2  x2k 2 x2k 1 ) W ( u2k 1  x2k 1 ).
(36)
Then the third level OFDM upcombining for the FSF chan
It implies that the present polar system can be decoded via the depolarizing algorithm with a recursiveness feature.
Without causing undue prejudice or confusion for de
nel W' can be derived with transition probabilities
4
4
=W(y0x0)W(y1x0+x2)W(y2x0+x1)W(y3
scription, we especially take the simplified notations in this
W' ( y'  x ) W'( y'  x
x )W'( y'  x
3
)W'( y'  x )
4
section as follows. The notation xi xi, denotes the initial input signals at source node S, ui ui, the polarized signals
0 0 2 1 0
' '
' '
W ( y3  x0 x1 ).
2 i i0
According to (34) and (34), for the th subcarrie of
W(3)( y ,x ,x
 x ) 1 W(1)( y
,x  x )
4 , 1, 2, 3, 8 2 1,
2,
each OFDM symbol, we obtain the OFDM combining for the
W(1)( y ,x

x  x


x ).
(41)
4
4
third level FSF channels W4 and W' with transition proba
2 2, 0, 1, 2, 3,
bilities
W4(yx) = W(y0, x0, ) W(y1, x0, +x2, )
3
Similarly, the OFDM upsplitting operation that illustrates the relation of OFDM upcombining for two FSF channels W
2
2
and W' is
W(y2, x0+x1, ) W(y3,  xi ),
W'(0)( y'  r' ) 1 W' ( y'  r' r' )W' ( y'  r' ),
W' ( y'
 x ) W' ( y' ,  x , x
i0
) W' ( y'
 x )
2
'(1)
k 2k 2
' ' '
' 2
r
r
2k 2
1
2k 2
' '
2k 2
'
2k 1
'
2k 1
' '
2k 1
'
4 0,
0 0 2
1, 0
W 2 ( yk ,r2k 2  r2k 1 ) 2 W ( y2k 2  r2k 2 r2k 1 )W ( y2k 1  r2k 1 ).
3
W' ( y'  x
) W' ( y'
 x x
(38)
and the highlevel OFDM upsplitting operations that illu
2,
i0
i,
3, 0
1,).
strates the relation of two OFDM upcombining for the FSF channels W' and W' is given by
OFDM Splitting
Next, we consider the OFDM downsplitting operation for
2 4
W'(0)( y'  x ) 1W'(0)( y'  x' x )W'(0)( y'  x ),
the downpolarizing system, which splits the synthesized FSF
4 0 8 2
x1
1 0 1
2 2 1
channel W4 back into a set of equivalent singlelink FSF
W'(1)( y' ,x
 x ) 1 W'(0)( y'  x' x )W'(0)( y'  x ),
channels W( i ) , i 4. The downsplitting OFDMs can
4
(2)
0 1 8 2 1 0 1
(1)
2 2 1
4 W'
( y' ,x  x ) 1W' ( y' ,x x  x' x )
x
x
be used for the transmission of signals in the polar system
4 1 2 8 2 1 0 1 2 3
3
with high reliability in terms of transition probabilities, as
W'(1)( y ,x  x ),
4
4
well as downsplitting channel capacity I( W( i ) ) [25]. At
'(3) '
2 2 1 3
1 '(1) ' '
each relay node Rk, we define the OFDM downsplitting op
W 4 ( y ,x1,x2  x3 ) 8 W 2 ( y1,x0 x1  x2 x3 )
erations as oneone maps that illustrate the relation of the
W'(1)( y' ,x  x ).
(42)
transition probabilities of each subcarrier of two level OFDM
downcombining FSF channels W and W2 as follows
2 2 1 3
2 r 2
2 r 2
W( 0 )( yk  r2k 2 ) 1W( y2k 2  r2k 2 )W( y2k 1  r2k 2 r2k 1 );
2k 2
So far we have established the polar system based on the po larization of OFDMs. It is known that the channel capacity of
W(1)( yk ,r2k 2  r2k 1 ) 1 W( y2k 1  r2k 2 r2k 1 ).
(39)
OFDM splitting for the FSF channel W( i ) can be bounded by
2 2 4
Considering all nodes S, R and D for the whole polar system,
I(W( i ) ) 1 z(W( i ) ),
4
4
for any subcarrier we derive the highlevel OFDM downsplit 4 4
ting operations with the transition probabilities given by
where
z(W( i ) ) are the Bhattacharyya parameters [1] given
W(0)( y  x0 ) 1W(0)( y1  x0 )W(0)( y2  x0 x1 ),
by
x
x
4 8 2 2
1
z(W( i ) ) W( y ,x x  x ).
W(1)( y,x
 x ) 1 W(0)( y  x
)W(0)( y  x
x ),
4
yA4 x0, xi1,,Ai
xA
0, i 1,
4 0 1 8 2 1 0 2
2 0 1
Next, we analyze the reliability of the OFDM downsplitting
W(2)( y,x1  x2 ) 1W(1)( y1 ,x0  x2 )
( i )
x
x
4 8 2
3
W(1)( y2 ,x0 x1  x2 x3 ),
for the FSF channels with transmission probabilities W4 in

based on the Bhattacharyya parameter vector
2
W(3)( y,x ,x  x ) 1 W(1)( y ,x  x )
z4 = (z4,0, z4,1, z4,2, z4,3),
4 1 2 3 8 2 1 0 2
which can be calculated from the recursion formula [1], [2],
2
2
W(1)( y2 ,x0 x1  x2 x3 ).
(40)
Namely, for the th subcarrier of each OFDM symbol we achieve the transition probabilities as follows
W(0)( y  x0, ) 1W(0)( y1,  x0, )W(0)( y2,  x0, x1, ),
4 8 2 2
x1,
W(1)( y ,x0,  x1, ) 1 W(0)( y1,  x0, )W(0)( y2,  x0, x1, ),
4 8 2 2
W(2)( y ,x1,  x2, ) 1W(1)( y1, ,x0,  x2, )
4 8 2
x3,
2
2
W(1)( y2, ,x0, x1,  x2, x3, ),

The recursive downpolarization (b)The recursive uppolarization



Fig. 2. The tree process of the Bhattacharyya parameters for the recursive polarizing OFDMs.
i.e.,
frozen bits that provide assistance for the reliable transmis sions.

The Switching Polar Relay Communications with Space
z 2
for 0 j k 1;
(43)
TimeFrequency Codes
z2k , j
k , j ,
k
k
2zk , j k z 2, j k , for k j 2k 1,
for k {1, 2} starting with z1,0 =1/2, shown in Fig.3(a).
From scratch, we form a permutation 4 = (i0, i1, i2, i3) of
(0, 1, 2, 3) corresponding to entries of x = (x0, x1, x2, x3) T so that the inequality z4,ij z4,ik , 0 j < k 3, is true. Thus we have the reliability of OFDM splitting for the FSF channels given by
z(4) = (1/16, 7/16, 9/16, 15/16) (44)
which creates a permutation 4 = (0, 1, 2, 3). It implies that for each subcarrier of the source OFDM symbols x, the first two signals {x0, , x1, } can be transmitted with higher re liability than that of the last two signals {x2,, x3,}, as shown in (44). Therefore, for the reliable transmission of signals
while polarizing the FSF channels, we let {x , x } to be the
In what follows, we propose a highreliable MIMO OFDM relay system by rearranging a class of spacetime frequency code (STF) codes for four OFDM symbols over the FSF channels with the structure expressed in the stacked Alamouti code and the Jafarkhani code. Generally speaking, it is not difficult in practice to provide parallel transitions for an MIMOOFDM communication system, especially for sys tem with a large number of OFDM symbols. Therefore, a suitable STF may adapt itself to the transmission of multiple OFDM symbols with parallel transitions in the polar relay system.
According to the reliability of the switching polar relay system while calculating the Bhattacharyya parameters expressed in (44) and (46), for each subcarrier we obtain in
formation bits XI, and X ' and frozen matrices XF, and
0, 1, I
information bits that are required to be transmitted from relay nodes, and {x2,, x3,} to be frozen bits that provide assistance for transmissions. In practice, the frozen bits {x2,, x3,} are always be set zeros for the depolarizing for convenience, i.e.,
{x2, = 0, x3, = 0}. This property can be utilized for the flex ible transmission of signals on the FSF channels with high reliability [1].
In the similar way, we can derive the reliability of
4
4
upsplitting system for the FSF channel W( i ) with transmis
sion probabilities in (42) based on the Bhattacharyya parame ter vector
' in the downpolarizing system and the uppolarizing
X
X
F
F
system, respectively.

Switching Polar System with the Alamouti code: According to the abovemntioned OFDM polarizing for the FSF channels in two polar systems, i.e., the downpolarizing system and the uppolarizing system, we assume the trans formed OFDM symbols ui can be encoded with the orthogon al block code that combines spatial, temporal and multipath processing for grouping signals. Actually, we can construct the Alamouti code structure on each subcarrier if the length
of the CP lcp satisfies the constraints
z' ( z' ,0 ,z' ,1 ,z' ,2 ,z' ,3 ),
lcp max . .
4 4 4 4 4
k ,l l ,sk l ,rk sd 2
which can be calculated from [1], [2], i.e.,
To make the transmission processing clear, we consider an
' 2z'
z 2k, j 2
z'
k , j k
,
' 2

z

z
k , j k
, for 0 j k 1; for k j 2k 1,
(45)
MIMOOFDM relay system with one transmit antenna at source node S, each relay node Rk, and one receive antenna
k , j
for k {1, 2} starting with z1,0 = 1/2, shown in Fig.3(b). Consequently, we form a permutation 4 = (i0, i1, i2, i3) of (0,
1, 2, 3) corresponding to entries of x = (x0,, x1,, x2,, x3,)T
at each relay node Rk with an OFDM symbol matrix of size
4N Ã—2 in two time slots given by the stacked Alamouti code,
x x
x x
*
0 1
4 4 A( x ,x
) x1 x*
so that the inequality z' i , j z' i ,k 0 j < k 3, is true. The
(x) = (c0,c1)=
0 1
3
3
A( x ,x )
0 ,
(47)
reliability of OFDM splitting for the FSF channels can be
2 3
x2 x*
derived as
z(4) = (15/16, 9/16, 7/16, 1/16) (46)
which creates a permutation 4 = (3, 2, 1, 0). It implies that for x embedded in four OFDM symbols, the last two signals
{x2,, x3,} can be transmitted with higher reliability than that of the first two signals { x0,, x1,}. Therefore, for the reliable transmission of signals over each subcarrier for uppolarizing system, we let {x2,, x3,} to be the information bits that are required to transmit from relay nodes, whereas {x0,, x1,} are
x3 *
x
x
2
2
where A(x2k2, x2k1), k {1, 2}, denotes the stacked Alamouti code [7] with N pairs of variables
{(x2k2,, x2k1), : N}.
For each subcarrier, N, it is given by
A(x , x ) =
A(x , x ) =
2k2, 2k1,
x2k 2,
x2k 1,
*

x

x
2k 2,
x
x
*
2k 1,
. (48)
According to the reliability of OFDM splitting for the FSF channels in (44) and (46), we implement the PF relay scheme while transmitting the information bits with four
Next, we describe the polar MIMOOFDM relay system by
switching to polarizing OFDM symbols with the stacked Alamouti code. Based on the downpolarizing system in (29) to transmit (x) expressed in (47), we have the received ma trix of size 4 Ã— 2 in two time slots given by
Y = HIXI + HFXF + E, (49)
where and E = (e, e) is an equivalent AWGN vector of size 4
Ã— 2. While deploying the uppolarizing system in (30) to transmit the same (x) in (47), we have the received matrix
OFDM symbols via two polar systems, i.e., downpolarizing system and uppolarizing system. The polar system that are composed of downpolarizing system in (49) and up polarizing system in (50) with the Alamouti code structure in terms of spatial, temporal and multipath for the FSF channels in four time slots has the similar performance behaviors as that of the Alamouti code for the spacetime or space frequency transmissions.
1
1
2
2
x
x


Switching Polar System with the Jafarkhani code: We consider a 4 Ã— 4 matrix given by the Jafarkhani structure [8]

of size 4 Ã— 2 in the next two time slots given by
x0

x*
x* 3
x1
x* x*
x2
Y = H FX F + H IX I + E , (50)
J(x) C0 ,C1 ,C2 ,C3 )
x2
0

x*

3 .
x0 x*
(52)
x
x
2
2
3 1
where E = (e, e) is an equivalent AWGN matrix of size 4 Ã—
x3 *
*
x
x
x
x
1 0
2 for the uppolarizing system in (30).
The two column vectors c0 and c1 in (47) are mod ulated for four OFDM symbols and transmitted, respectively, in the first two time slots for the downpolarizing system. Therefore, information matrix XI = A(x0, x1) and frozen matrix XF = A(x2, x3), and hence extensive information matrix XI = A(x0, x1) and frozen matrix XF = A(x2, x3) are
embedded in two OFDM symbols for two time slots, respec
For the PF scheme, two column vectors c0 and c2 are trans mitted in the first time slot and the third time slot for the downpolarizing system, whereas other two column vectors c1 and c3 are in the second time slot and the fourth time slot for the uppolarizing system.
In the downpolarizing system at source node S for four successive OFDM symbols we take the notations
XI
tively. In the next two time slots, we switch to the uppolariz
X(x) ( c0 ,c2 ) X ,
(53)
ing system and achieve frozen matrix X = A(x , x ) and in
F
F 0 1
formation matrix XI = A(x2, x3) in the same two respective OFDM blocks as that of the downpolarizing system. The
where XI and XF are information matrix and frozen matrix given by
selections of information matrices {XI, XI} and frozen ma
x0
x*
x2
x*
3
3
1
1
XI
2 , XF
0 .
(54)
trices {XF, X F} for switching polar system are based on the calculation of the Bhattacharyya parameters, as shown in Fig.2.
Theorem 3.1: According to the downpolarizing
x1 x* x3 x*
In the uppolarizing system we use the notation
X'
X'(x) ( c1 ,c3 ) I ,
(55)
system in (49) for the first two time slots, we consider 4N
X'
F
F
F
signals for four OFDM symbols (x0, x1, x2, x3) embedded in
the stacked Alamouti code in (47) for the transmission and
where X and XI are frozen matrix and information matrix
switch to the uppolarizing system in (50) for the next two time slots. After being processed with the aforementioned
given, respectively, by
x* x
x*
x*
X 'F 1
3 , XF 3
1 .
(56)
transformations, the CP removal and the Npoint FFT opera
x*
x2 x*
x*
0 2 0
tions while switching the downpolarizing to uppolarizing system for four OFDM blocks in four time slots, the informa tion matrix can be depolarized and hence be decoded with high reliability at destination node D as follows
Based on the abovementioned matrices X and X, we switch to polarizing MIMOOFDM relay system in four time slots, i.e., the downpolarizing system for transmitting X and the uppolarizing system for transmitting X. After being depola
XI
022 A( x0 ,x1 )
022
rized and decoded at destination node D, we achieve
A( x )
'
,
(51)
022
XI 022
A( x2 ,x3 )
which is an orthogonal code that can achieve the full diversi ty.
The proof of Theorem 3.1 can be found in Appendix A.
x0
x1
0 x*
2
2

x

x
0
0
*
*
0 x* 0
4, is independently transmitted across Rk and a channel output y is obtained with transition probability W(y x ).
J(x)
3 .
(57)
i,
i,
i,
3
3
0 x*
x0 1
For each subcarrier in four OFDM symbols we misuse the
0 x* 0 x*
T T T
T T T
2 0
simplified notations xx= ( xI ,,xF , )
,yy= ( y1,,y2, ) for
Theorem 3.2: We consider 4N signals for four OFDM sym bols (x0, x1, x2, x3) embedded in (52) for the transmission in
the downpolarizing system, where y1
y1,
= (y0,
, y1,)T
both downpolarizing system and uppolarizing system. In
the first and third time slots we transmit X in (53) for the
and y2 y2, = (y2,, y3,)T. Similarly, we define x
downpolarizing system in (29), and switch to transmitting X
x' ( x'T
,x'T
)T and y'
y' ( yT ,yT
)T for each subcarrier
in (55) in the second and fourth time slots for the up
I ,
F ,
1, 2,
polarizing system in (30). After being processed with ab ovementioned processes, the CP removal and the Npoint FFT operations while switching the downpolarizing system to uppolarizing system for four OFDM blocks in four time slots, the received noisy information matrix can be depola rized and decoded with high reliability at destination node D, namely we can achieve the transmitted matrix of size 4 Ã— 4
given by decision matrix J ( x ) expressed in (57), which is a
in the uppolarizing system.
The SIC decoder of the downpolarizing system ob serves y and generates an estimate of x of x. We may visual ize the decoder as consisting of four decision elements for four respective OFDM symbols, each element xi for source
element xi , i 4.
The OFDM depolarizing algorithm of the polar system
begins with the ith decision element xi for the down
quasiorthogonal code equivalent to the Jafarkhani code. The proof of Theorem 3.2 can be found in Appendix B.
We can decode the initial signal vectors XI and XI from XI
polarizing system. It waits till receiving all previous deci sions xi 1 , and upon receiving them; it calculates the likelih
4
4
and XI, respectively, using the conventional Alamouti decod ood ratio (LR) Li as follows
ing or the Jafarkhani decoding with the ML receiver or the
ZF/MMSE receiver after depolarizing the transmitted signals
Li ( y,x x )
W ( i )( y,x1 xi1  0 )
4
4
4
4
1
1
(58)
4 1 11
4 1 11
,
,
at destination node D. However, in this paper we introduce a
W ( i )( y,x1 xi 1)
polar receiver via the polar decoding under a low complexity and generates its decision as
SIC decoding strategy that can bring out similar performance
0
if L( i )( y,x1 ,.xi1 ) 1;
behaviors as that of ML decoding. According to the reliabili ty of OFDM splitting for the FSF channels in (44) and (46),
xi
1
4
otherwise ,
(59)
we implement the PF relay scheme while transmitting the information bits with four OFDM symbols via switching to downpolarizing communication and uppolarizing commu nication, which has the similar performance behaviors as that of the Alamouti code and the Jafarkhani code.
which is then sent to succeeding decision element xi1 . The
complexity of the decoding algorithm is determined essen tially by the complexity of calculating LRs, which is N(1+log2 N) = 12 for computing one round. For the initial LRs, we calculate
It is shown in both (51) and (57) that the present PF MIMO
L( y
) W( yi  0 ) .
i
i
(60)
OFDM relay scheme with the STF code can achieve the simi
i W( y 1)
lar diversity gain as that of the Alamouti code and the Jafark hani code with OFDM polarizing for the FSF channels. Ac cording to Arikans statement, we should obtain the good
For the downpolarizing system, the low level of LRs are obtained with a simple calculation using the reclusive formu
las in (39), which gives
BER performance as long as the employed system is pro
vided with a large number of OFDM symbols while imple
L( 0 )( yk
) L( y
2k 2 )
(61)
2
2
menting the polarizing operations on source node S and relay
L(1 )( yk ,x2k 2
) ( L( y
2k 1
))12x2k 2 ,
2
2
nodes R over the FSF channels.

OFDM Depolarizing
In this subsection, we consider the SIC decoding for the proposed polar MIMOOFDM relay system with standard complex constellations, such as binary phase shift keying (BPSK) modulation constellation. Recall that each signal
xi,, N, in the th subcarrier of OFDM block xi, i
k {1, 2}. After that the high level of LRs can be calcu lated using the reclusive formulas in (41). A straightforward calculation yields
L( 0 )( y ) L( 0 )( y1 ),
4 2
L( 1 )( y, x0 ) ( L( 0 )( y2 ))12x 0 ,
4 2 (62)
L( 2 )( y, xI ) ( L( 1 )( y1 ,x0 ),
4
4
4
L( 3 )( y, xI
,x2
2
2
2
) ( L( 1 )( y2
.x0
x1
))12X 2 .
4
4
The LRs L( i ) in (62) can be derived from (41), which can be found in Appendix C.
However, for the uppolarizing system the low level
of LRs are given in [1] by
2
2
L' ( 0 )( yk )
L( y2k 2 )L( y2k 1 ) 1
L( y2k 2 ) L( y2k 1)
Fig. 3. Implementation of the successive cancellation decoder.
2
2
k
k
L' (1)( y
,x
2 k 2
) ( L( y
2k 2
))12x2k 2 L( y
2k 1 ),
(63)
Next we design an implementation of the SIC de
and the high level of LRs can be calculated as
( 0 ) 1 2
( 0 ) 1 2
L' ( 0 ) ( y )L' ( 0 ) ( y ) 1
L' ( y ) 2 2 ,
coder for switching polar system. There are 12 nodes corres
ponding to LRs for decision elements x . The depolarizing process carries out two crucial actions in the polar system,
4 L' ( 0 ) ( y ) L' ( 0 ) ( y )
2 1 2 2
L'( 1 ) ( y, x0 ) ( L'( 0 ) ( y1 ))12×0 L'( 0 ) ( y2 ),
i.e., activating and responding [1]

Step 1: It begins with the activating phase in the down
4 2 2
( 2 )
L'(1 ) ( y1 , x0 )L'(1 ) ( y2 , x0 x1 )) 1
polarizing system, in which the first decision element
L'4
( y, xF ) 2 1 )
2
(1 )
, (64)
x from the leftmost column activates node1 for the cal
2
2
4
4
L'(
( y1 , x0 ) L'2
( y2 , x0 x1 )) 1
4
4
L'( 3 )
( y, xF , x2 ) L'(1 )
( y1 , x0 ))12x 2
L'(1 )
( y2 , x0 x1 ).
culation of
L'( 0 ) ( y ) to decode
x1 of column vector x,
2
2
2
2
2
2
2
2
So far we have calculated LRs of the polar system. The ad
which in turn activates node2 and node5 to achieve a
vantage of this OFDM depolarizing algorithm is due to the
pair of LRs,
L( 0 )( y1 ) and L( 0 )( y2 ) . After that node2
relations of two level LRs in coordination with the formulas
activates node3 and node4, and node5 activates node6
and node7, respectively, for calculating two initiallevel
in (61,62,63,64). For example, two LRs
L( 0 )( yt )
pairs of LRs, L(0)(y2k2) ad L(0)(y2k1), k {1, 2}.
4
4
4 t 0
4 t 0
and L(1 )( y ,x
2
2
L( 0 )( y1 ) and
) are assembled from the same pair of LRs
2
2
L( 0 )( y2 ) , while the other two LRs

Step 2: In the responding phase, node3 and node4 com pute L(0)(y0) and L(0)(y1) using (60), respectively, and pass them to their leftside two neighbors, node2 and node9. Similarly, node6 and node7, compute and pass
L( 2 )( y ,x
) and
L( 3 )( y ,x x
) are from
L(1 )( y ,x
) and
the pair of LRs L(0)(y2) and L(0)(y3) to their leftside two
4 t 1
4 t 1 2
2 1 0
neighbors, node5 and node10, respectively.
2
2
L(1)( y2 ,x0 x1 ) due to the symmetry properties of the FSF

Step 3: In what follows, node2 and node5 compute
( 0 )
L( 0 )( y1 ) and L( 0 )( y2 ) using (61) and pass the resulting
channels. In addition, two LRs
L2 ( yk )
and 2 2
2
2
L( 0 )( yk ,x2k 2 ) are assembled from the LRs L(0)(y2k2) and L(0)(y2k1), k {1, 2}. This process proposes an elegant
pairs of LRs to its leftside two neighbors node1, respec tively. After that, node1 compiles its response L'( 0 ) ( y ) to calculate x 0 according to (62). Consequent
4
4
approach for an accurate count of the total number of LRs
ly, node1 sends
x 0 to its neighbor node8, which is
that are required for a full description of the OFDM depola
needed for calculating x1 . The yielded decision elements
rizing algorithm, shown in Fig.4.
x 0 and
x1 are both passed to node8 and node12 that
may generate
x 2
and x 3 . Fortunately, since
x 2 and x3
are the frozen bits that have lowreliability in the down polarizing system, it is not necessary to generate x 2 and
x 3 while directly setting x 2 = x 3 = 0.

Step 4: The depolarizing process switches to the up polarizing system while calculating x 2 and x 3 with high

reliability. In this phase
x 0
and x1 are the frozen bits
4
4
that are directly set x 0 = x1 = 0. Consequently, x 2 only
activates node11 for computing
L'( 2 ) ( y,x1 ) based on
(64), where
L' (1 )( y
,x x
) L' (1 )( y,x
) can be ob
2 2 0 1 2 1
tained from the response of node9 and node10 using (63). The decision element x 2 is then sent to node12 for x 3 .
The last decision element
x 3 activates node12 for x3
based on
L' ( 3 ) ( y,x1 ,x2 ) in (64) without activating
4
4
node9 and node10. The algorithm continues in this man ner until the receiver receives and decides the transmit ted vector x .
We note that in the downpolarizing system it is not ne cessary to generate x 2 and x 3 since they are frozen bits that have lowreliability while transmitting in the down polarizing channels. Similarly, in the uppolarizing system it is not necessary to generate x 0 and x1 since they are frozen bits with the lowreliability while transmitting in the up polarizing channels. In this way, the proposed depolarizing algorithm can be made while directly setting x 2 = x3 = 0 in the downpolarizing system and setting x 0 = x1 = 0 in the uppolarizing system. This decoding process continues until all information bits x are jointly decoded in the end. Next, we will show the BER performance behaviors of the polar sys tem with simulation results. Thus, we can obtain x from x in the polar system with the highreliability.
Fig. 5. BER performance comparison with ML decoding and polar decoding.



Simulation Results
According to the OFDM depolarizing algorithm with the SIC decoder for the polar system, we present some simula tion results and compare their BER performance behaviors. We present the BER performance as functions of the transmit power pt. We deploy the Alamouti code and the Jafarkhani code on each relay node Ri while implementing the OFDM depolarizing techniques for the polarizing FSF channels. Therefore we can use the ML symbolwise decoding, as well as the OFDM depolarizing algorithm in four time slots, where the data symbols in A are drawn from BPSK constella tion.
Fig. 4. BER performance behaviors with polar decoding.
In Fig. 4, we present the BER curves of the stacked Alamouti code for four OFDM symbols transmitted at source node S. We consider the polar MIMO relay systems provided with transmission power pt for reference in terms of the pow er allocation strategy in (3). For the present polar system, it shows that the slope of the BER performance curve of the proposed PF scheme with the Alamouti code for the polar system via the OFDM depolarizing algorithm approaches the direct relay system without OFDM polarizing via the ML decoding when power pt increases. It implies that the PF scheme can achieve full diversity with the depolarizing algo rithm. Furthermore, the BER performance behavior of the present polar system is a little better than that of the direct transmission approach which verifies our analysis of the transmission reliability of the polarized FSF channels. Simu lations demonstrate that the proposed PF scheme with the OFDM depolarizing algorithm has a similar performance as that of the Alamouti scheme with the ML decoding for large transmission power pt when the depolarizing is applied at the receiver. In Fig. 6 it implies that the PF scheme can achieve full diversity in terms of the ML decoding and the OFDM depolarizing decoding. In this case, the BER performance behavior is much similar as those of the ML decoding, as
0
0
stated in [11]. Fortunately, for the polar system, the BER
A( x0 ,x1
022
performance of the decoder provided with the depolarizing algorithm outperforms that of the direct approach using the
( x )
22
A( x2
,
,x3 )
(67)
ML decoding.

CONCLUSION
In this paper, we have presented a simple design of the PF scheme based on two polarizing systems over the FSF chan nels, i.e., the downpolarizing system and the uppolarizing system. There are two polar coding processes for each pola rizing system, including source polarizing and relay polariz ing OFDM sequences. The present polar system has a salient
where can be further decoded with the orthogonal STBC de
coding algorithms. This completes the proof of Theorem 3.1.
B. Proof of Theorem 3.2
For each subcarrier of four OFDM symbols (x0, x1, x2, x3) embedded in (52) for the transmission in both down polarizing system. We transmit X in (53) in the first and third time slots for the polarizing MIMO relay system in (29).
Namely, for each subcarrier N, we have
recursiveness feature and can be decoded with the SIC de
c ( x , x , x , x
)T and c
( x , x ,x , x
)T According to
3
3
2
2
coder, which renders the scheme analytically tractable and
0 0 1
2 3
3 2 3 0 1
provides a lowcomplexity coding algorithm while multiple OFDM symbols are equipped. We analyze the BER perfor mance and diversity of such systems based on the STF codes with the fixed size using the polarizing FSF channels, which tend to polarize with respect to the increasing reliability un der certain OFDM combining and splitting operations. Simu lations demonstrate that the proposed polar system has the similar BER performance behaviors as that of the STF codes, the stacked Alamouti code and the Jafarkhani code, but out
the tree process of the Bhattacharyya parameters for the re cursive polarizing OFDMs in Fig.2(a), we achieve the re spective information bits ( x0 ,x1 ) an ( x ,x ) in the first and third time slots after the CP removal and the Npoint FFT operations at destination node D. The frozen bits are given by ( x2 ,x3 ) and ( x0 ,x1 ) , respectively. After depo larizing the system we obtain the 2Ã—2 originally transmitted matrix with high reliability given by
performs these STF codes in terms of the BER performance
x
x
for large transmission power when the OFDM depolarizing
algorithm is applied at the receiver.
' 3
X
X
I x
1 .
x
(69)
2 0
APPENDIX
A. Proof of Theorem 3.1
We consider 4N signals for four OFDM symbols (x0, x1, x2, x3) embedded in (47) for the transmission in the downpolarizing
system for the first two time slots. For each subcarrier N,
Combining (68) and (69) in four time slots, we have the em bedded information matrix can be rewritten as After being depolarized and decoded at destination node D, we have

x

x

x

x
0
0
2
2
0
0
x0 0
T * *
* * T
J ( x ) x1
0
3 .
(70)
0
0
we have c0 ( x0 , x1 , x2 , x3 ) and c1 ( x1 , x0 ,x3 , x2 ) .
0 x3
0 x1
According to the tree process of the Bhattacharyya parame
2
x
x
0
0
0
x
x
ters for the recursive polarizing OFDMs in Fig.2(a), we achieve the respective information bits ( x0 ,x1 ) and
We note that the depolarized information matrix J in (70) is a quasiorthogonal matrices, which is equivalent to the Ja
( x* ,x* ) in two time slots after the CP removal and the N
1 0
3
3
,x
,x
2
2
point FFT operations at destination node D. The frozen bits are given by ( x2 ,x3 ) and ( x* * ) respectively. We do
farkhani code if and only if c0c1 c2c3 . In fact, it is easy to prove that
J(x)J H (x)
not need to decode frozen matrix XF since it has been trans
mitted on channels with low reliability in the down
 c0
2  c2 2
c0c c
2c 0
1
1
3
3
polarizing system. Therefore, after depolarizing the system
c c c c
 c1 2  c3 2 0
we obtain the 4 Ã— 2 originally transmitted matrix given by
1 0
0
3 2
0
 c1
2  c3 2
c2c
3
3
c
c
c0 1
x x*
0 0
c3c
c1c
 c 2  c
2
X ' 2
3 .
(66)
2 0 0
2
x
x
I x3
*
2
From the above analysis, it implies that the depolarized codes
Combining (65) and (66) in four time slots, we have
A
A
X diag( X A , X ' ) from which the embedded signal matrix
can be further decoded with the ML decoder or MMSE/ZF decoder for the quasiorthogonal STBC [8]. However, when
ever there is the constraint c0c c2c for J ( x ) , the yielded
can be rewritten as 1 3
codes have an orthogonal structure and hence have the simi
lar performance behaviors as that of the Alamouti code [7]. This completes the proof of Theorem 3.2.
C. Proof of (62) from (41)
In order to derive the LRs for the downpolarizing sys tem, we deploy the calculation approach for the uppolarizing
In a similar way, we derive the formulas in (61) while calcu lating the low level LRs L( i ) , i 2, from (39). Namely, we obtain the relationship of two level LRs L( i ) and LRs
2
2
2
2
L( i ) given by
system suggested by Arikan [1].
( 0 )
L
L
2
( yk )
W ( 0 ) ( yk  0 )
2
2
2
2
W ( 0 ) ( yk 1 )
For the simplicity of this proof, we consider the proof of the
W( y2k 2  0 )W( y2k 1  0 ) W( y2k 2  0 )W( y2k 1 1)
validity of the LRs L( i )( yt ), i 4, for decision element
W( y
2k 2
1)W( y
2k 1
1) W( y
2k 2
1)W( y
2k 1
 0 )
4
4
xi in the depolarizing algorithm to illustrate the relationship
L( y2k 2 )L( y2k 1 ) L( y2k 2 )
of two level of LRs
L( i )( yt
) in (62) and L( i )( yt
) in (61).
L( y
1 L( y2k 1 )
)
4
4
2
2
According to the defined LR in (58), we use the recursive of
4
4
the splitting transition probabilities W ( i ) i 4, and thus
2k 2
L( 1 )( y
,x
W ( 1 )
2
2
) 2
( yk ,x2k 2  0 )
achieve
2 k 2k 2
W ( 1 )( yk
,x
2k 2
1)
L
L
( 0 )
4
W ( 0 ) ( y  0 )
( y ) 4
4
4
W ( 0 ) ( y 1 )
W( y2k 2  x2k 2 )W ( y2k 1  x2k 2 0 ) W( y2k 2  x2k 2 )W ( y2k 1  x2k 2 1)
(73)
W ( 0 )
2
2
2
2
2
2
2
( y1  0 )W ( 0 )
( y2  0 ) W ( 0 )
( y1  0 )W ( 0 )
( y2 1)
( L( y
2k 1
))12×2 k 2
W ( 0 ) ( y1 1)W ( 0 ) ( y2 1) W ( 0 ) ( y1 1)W ( 0 ) ( y2  0 )
2 2 2 2
1 2 1
1 2 1
L( 0 ) ( y )L( 0 ) ( y ) L( 0 ) ( y )
2 2 2
2
2
1 L( 0 ) ( y2 )
2
2
L( 0 ) ( y1 )
This completes the proof of our statement of derivation of LRs in (62) and (61).
ACKNOWLEDGEMENTS
L
L
( 1 )
4
( y,x 0 )
W (1 )( y,x0  0 )
4
4
4
4
W ( 1 )( y,x0 1)
This work was supported by the World Class University R32 2008000200140 NRF, Korea, and Fundamental Research
W ( 0 )( y1  x0 )W ( 0 )( y2  x0 0 )
20100020942 NRF, Korea.
2 2
, (71)
W ( 0 )( y1  x0 )W ( 0 )( y2  x0 1)
2
2
2
( L( 0 )( y2
2
))12×0
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