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Turbo Channel Estimation and Equalization for a Superposition

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					                                                                                                                            1



 Turbo Channel Estimation and Equalization for
   a Superposition-based Cooperative System
                      Yu Gong, Zhiguo Ding, T. Ratnarajah and Colin F. N. Cowan

                                                         Abstract


         This paper investigates the superposition-based cooperative transmission system. In this system, a key

     point is for the relay node to detect data transmitted from the source node. This issued was less considered in

     existing literature as the channel is usually assumed to be flat fading and a priori known. In practice, however,

     the channel is not only a priori unknown but subject to frequency selective fading. Channel estimation is

     thus necessary. Of particularly interest is the channel estimation at the relay node which imposes extra

     requirement for the system resources. In this paper, we propose a novel turbo least-square channel estimator

     by exploring the superposition structure of the transmission data. The proposed channel estimator not only

     requires no pilot symbols but also has significantly better performance than the classic approach. The soft-

     in-soft-out MMSE equalizer is also re-derived to match the superimposed data structure. Finally computer

     simulation results are shown to verify the proposed algorithm.

         Index Terms - cooperative diversity, channel estimation, turbo equalization, superimposed training.



                                                   I. I NTRODUCTION

  Multipath fading is a main detrimental factor that damages the reliability of wireless communications,

causing dramatic fluctuation in signal power at the receiver [1]. It is well recognized that spatial diversity

achieved by multiple-input-multiple-output (MIMO) technology can effectively combat the multipath fading.

However, because the MIMO requires multiple antennas at the transmitter and receiver, it is not always

possible in practice due to, for example, the limit in size and processing complexity for mobile handsets.

Cooperative transmission thus becomes an attractive alternative to achieve spatial diversity as it can form a

virtual MIMO system by allowing mobile users to “help” each other in data transmission, even if every user

only has one antenna ([2], [3], [4]). Many cooperative communications approaches have been proposed, and

   Y. Gong is with the School of Systems Engineering of Reading University, Reading, RG6 6AY, UK. Tel.:+44 +44 0118 378 8581.
Email: y.gong@reading.co.uk. Z. Ding is with the Department of Communication Systems of Lancaster University, Lancaster, LA1
4WA, UK. Tel., +44 015245 10399, Email: z.ding@lancaster.ac.uk. T. Ratnarajah and C. Cowan are with ECIT, Queen’s University
of Belfast, Belfast, BT3 9DT, UK. Tel: +44 028 9097 1078. Email: T.Ratnarajah@ecit.qub.ac.uk.
                                                                                                                2


can generally be classified as orthogonal subspace or non-orthogonal subspace scheme [5]. Similar to some

classic multiplex methods in communications such as the TDMA, FDMA etc, the orthogonal cooperative

scheme allocates data transmission among cooperative users in orthogonal subspaces such as time, frequency

etc so that there is little inter-user interference. The orthogonal cooperative transmission is easy to implement

but at the sacrifice of spectral efficiency. The non-orthogonal schemes, on the other hand, can achieve high

spectral efficiency by allowing different users to share same subspace for data transmission. It has been

shown that non-orthogonal cooperative schemes achieve diversity-multiplexing tradeoff, a common index to

compare different MIMO or cooperative approaches, dominant to that of non-cooperative schemes [5].

  In this paper, we focus on a particular non-orthogonal cooperative scheme based on superposition modu-

lation [6], [7]. In this scheme, there are two source nodes, each transmitting data in turn to the destination

node. At any time slot, the transmitting node transmits a superposition of its own data and the data received

from the other source node during the previous slot. Both source nodes work in the decode-and-forward

mode such that the data received at a source node are decoded first before they are superimposed with

the local data and relayed to the destination node. Unlike the orthogonal subspace schemes such as the

selection relay scheme ([2]) where every transmitting time slot is divided into several sub time slots, the

superposition scheme does not further divide the time slots so that full spectrum efficiency can be retained.

The original scheme with two source nodes can be extended to more general multiple source scenario by

grouping the source nodes with two each as was shown in [7].

  Successful implementation of the superposition cooperative system depends heavily on reliable detection

of the transmission data at the relay node. This issue was less considered in the original protocol as

the channels were simply assumed to be flat fading and a priori known to the receivers [6]. In practice,

unfortunately, the channels are not only a priori unknown but often subject to frequency selective fading.

Channel estimation and equalization are thus necessary for both the relay and destination nodes. Usually,

channel coefficients are estimated with the help of the pilot or training symbols. As the channel estimation

is always required at the destination node no matter whether and how the cooperative transmission is

applied, of particular interest is the channel estimation at the relay node which imposes extra requirement
                                                                                                             3


for the system resource and may compromise the spectrum efficiency associated with the non-orthogonal

cooperative transmission. Therefore in this paper, we focus on the channel estimation and equalization at

the relay node.

  It is interesting to observe that pilot symbols for channel estimation can be saved at the relay node by

exploring the superposition structure of the transmission data that part of the data is known to the receiver.

This seems to fall into the area of the superimposed training, where the key issue is to separate the known

training data from the unknown information data (see [8] and the references therein). While most existing

algorithms about the superimposed training separate the information and training data by exploring some

periodic properties of the training data, they are not suitable for the case of the superposition cooperative

transmission. This is because in the superposition cooperative system, we have little control of the “training

data” which is in fact the information data of a source node and rarely periodic.

  In this paper, we propose a novel turbo least-square (LS) channel estimator by using the a priori

information fed back from the decoder to iteratively improve the channel estimation. We also re-derive

the soft-in-soft-out (SISO) MMSE equalizer described in [9] for this particular superimposed data structure.

The proposed turbo LS estimation has significantly better performance than the classic approach without

the turbo structure. This verifies that pilot symbols are not necessary for the relay nodes, which further

guarantees the high spectrum efficiency achieved by the superimposed cooperative transmission.

  The rest of the papers is organized as follows: Section II describes the system model of the superposition

cooperative transmission, where we particularly highlight how the data are superimposed and slightly

compare the outage performance of the superimposed cooperative transmission with other approaches;

Section III proposes the structure of the turbo LS channel estimation; Section IV re-derives the SISO

MMSE equalizer for the superimposed data structure; Section V verifies the proposed turbo LS estimation

through numerical simulations, where both static and random channels are considered; Finally Section VI

summarizes the paper. For simplicity and clarity of exposition, we assume BPSK modulation in this paper.
                                                                                                                                                 4


                                                                II. S YSTEM M ODEL

A. Basic Scheme

   In the superposition cooperative transmission, there are two source nodes, namely A and B , each

transmitting data to the destination node D alternatively. As is illustrated in Fig. 1, we assume without

losing generality that at time slot i, the source A transmits and B listens, and at time slot (i + 1), B

transmits and A listens.

                            √
                  xA, i =       1 − γ 2 · sA, i + γ · sB, i−1
          A                                                                               A
                                                                                           6


                                                    q
                                                        D                                                                       1    D
              ?


          B                                                                               B                   √
                                                                                                  xB, i+1 =    1 − γ 2 · sB, i+1 + γ · sA, i

                        (a) Time slot i                                                            (b) Time slot (i + 1)

Fig. 1.   The superposition-based cooperative relay scheme



   To be specific, at time slot i, if A successfully decodes sB,                     i−1   which is the data vector transmitted from

B at the time slot i − 1, it then transmits a packet of M superimposed symbols as:


                                                  xA, i =         1 − γ 2 · sA, i + γ · sB,    i−1 ,                                           (1)


where sA, i is the current data vector for A and γ is a constant factor that determines how the data are su-

perimposed and generally satisfies 0 < γ 2 < 0.5. For later use, we express xA, i = [xA, i (n), · · · , xA, i (n −

M + 1)]T , and similarly for other vectors whenever necessary. On the other hand, if A fails to decode

sB,   i−1 ,   it only transmits its own data packet, i.e. xA, i = sA, i .

   Also at the time slot i, the source node B receives the data from A as:


                                                            yB, i = Hi · xA, i + ni ,                                                          (2)


where Hi the Sylvester channel matrix from A to B , and ni is the noise vector at B . The task for B is to,

without knowledge of the channel, detect sA,                      i   and relay it to the destination at the next time slot. As this
                                                                                                                  5


paper mainly considers the relay node, the received signal vector at the source D is not shown here.

   Similarly, at time slot (i + 1), B transmits a packet of M superimposed symbols to the destination:


                                        xB,   i+1   =   1 − γ 2 · sB,   i+1   + γ · sA, i ,                     (3)


if B successfully decodes sA, i . Otherwise B transmits its own data packet. Without losing generality, the

rest of the paper considers the time slot i so that the time index i is dropped whenever no confusion is

caused.



B. The selection of γ

   The parameter γ in (1) is an important parameter which determines how the data from nodes A and

B are superimposed. Since the BPSK is considered in this paper, a transmitted symbol from A (xA (n))

only has 4 possible values, each corresponding to a pair of particular choices of sA and sB . This forms

a constellation map for xA (n) which is illustrated in Fig. 2, where the 4 possible constellation points are

labeled as E, F, G and J respectively.

                                x      o       x                  o                  x        o      x      -

                                E      −1      F                  0                  G        +1     J
                             sA = −1        sA = −1                               sA = 1           sA = 1
                             sB = −1        sB = 1                                sB = −1          sB = 1


Fig. 2.   The constellation map of a transmitting symbol xA (n) from source A.



   As the constellation points should be separated as far as possible for reliable transmission, the optimum

γ maximizes the minimum of the adjacent constellation distances such that


                                              γopt = max{min(EF, FG, GJ)},                                      (4)
                                                        γ



where EF, FG and GJ refer to the distances shown in Fig. 2 which are given by


                                 EF = GJ = 2γ               and       FG = 2 1 − γ 2 − 2γ                       (5)


respectively. In general, we have 0 < γ 2 < 0.5 that when γ = 0, the system reduces to the traditional
                                                                                                                       6


BPSK scheme, and when γ 2 = 0.5, F and G merge at the origin 0. For 0 < γ 2 < 0.5, EF, GJ and FG are

all monotonic functions of γ 2 . Specifically, when γ 2 is increased from 0 to 0.5, both EF and GJ increase

monotonically from 0, and FG decreases from 2. Then according to (4), the best γ must make “EF=FG=GJ”,

            2
leading to γopt = 0.2. This result well matches a statement in [6] that 0.075                          γ2     0.2.

  We highlight that the optimum γ obtained here is from the symbol detection point of view. When the

whole cooperative system is considered, the choice of γ becomes much more involved as it also depends

on other system factors such as the condition of the relay channels. The detail of this issue is beyond the

scope of this paper.



C. Outrage Performance

  The diversity gain achieved by the superimposed cooperative transmission can be well revealed by the

outrage probability which is obtained as

                                                                          Iout
                                                       Pout =                    q(I) dI,                            (6)
                                                                      0


where q(I) is the density function of the mutual information (or the maximum data rate I ), and Iout is the

targeted data rate.

  We have shown in [7] that the density function of the sum rate ISP for the superposition cooperation is

given by1
                                                              e2ISP
                                           2e2ISP                            e2ISP − x          x−1   1
                              qSP (ISP ) = 2                          f                     f           dx,          (7)
                                          ρ λ1 λ2         1                    ρλ1 x            ρλ2   x


where ρ is the SNR, f (·) denotes the PDF function of exponential distribution, λ1 and λ2 are two constants

determined by the coefficient matrix.

  For the purpose of comparison, the density function of data rate for the direct transmission and selection

relay scheme ([2]) are also shown below. Since the maximum data rate, or mutual information, for the direct

  1
      The density of the individual data rate can also be found in [7] but is less relevant in this paper.
                                                                                                                   7


transmission can be easily obtained as ID = log 1 + ρ|h1 |2 , the density function of ID is given by

                                             ∞
                                                                                    eID −       eID −1
                     qD (ID ) =                  δ (ID − ln(1 + ρx)) f (x) dx =        e          ρ
                                                                                                         ,      (8)
                                         0                                           ρ


where δ(·) denotes the Delta function. The second equation follows from the property of the Delta function
                     δ(x−xi )
that δ(f (x)) =   i |df /dx|xi   where xi is the ith root of f (x). Similarly, the density of the mutual information

for the selection relay scheme in [2] can be shown as


                                                    2e2ISR   e2ISR − 1       −   e2ISR −1
                                  qSR (ISR ) =                           e           ρ
                                                                                            .                   (9)
                                                      ρ          ρ


Substituting (7), (8) and (9) into (6) gives the outage probabilities for the superimposed cooperative, direct

and selection relay transmissions respectively.

  For better illustration, we plot the density functions qD , qSR and qSP for SNR=5dB and SNR=15dB

in Fig. 3 (a) and (b) respectively. It is clear from (6) that the outage probability is determined by the

density function q(I) and the targeted data rate Iout . But Iout is typically set much smaller than the

AWGN channel capacity which is given by log(1 + SNR). This is due to the use of channel coding and

deteriorating effects of multipath fading. For the examples shown here, the targeted data rates are about 1

and 2.5 bits/s/Hz for SN R = 5dB and 15dB respectively. Thus it is clearly shown in Fig. 3 that, when

I < Iout , the density for the superimposed transmission is significantly smaller than those for the direct and

relay selective transmission. This ensures the superimposed scheme to achieve lowest outage probability, or

the best reception robustness, among the three transmission schemes.


                                       III. T URBO LS C HANNEL E STIMATOR

  As was shown in the previous section, at time i, it is essential that the source node B can detect sA , the

information data transmitted from A. This makes it necessary to have channel estimation and equalization at

node B due to the frequency selective fading nature of the channel. In this paper, the LS channel estimation

is considered. We also assume the channel is quasi-static (slow fading) that it remains unchanged within

one packet and there is no inter-packet interference due to the guarded interval.
                                                                                                                                                                     8


                1.4                                                                         1.4
                                                           Direct transmission                                                          Direct transmission
                                                           Selection relaying                                                           Selective relaying
                1.2                                        SP cooperative                   1.2                                         SP cooperative

                 1                                                                           1



                0.8                                                                         0.8
         q(I)




                                                                                     q(I)
                0.6                                                                         0.6



                0.4                                                                         0.4



                0.2                                                                         0.2



                 0                                                                           0
                  0   1   2      3      4       5      6        7      8         9            0   1    2      3      4       5      6        7      8         9
                              Mutual Information in bit/s/Hz                                               Mutual Information in bit/s/Hz

                          (a) PDF of I at SNR 5dB                                                     (b) PDF of I at SNR 15dB

Fig. 3. Density functions of the mutual information for the schemes of direct transmission, selection relaying transmission and
superposition cooperative transmission.



  In general, the LS channel estimation is given by


                                                                ˆ
                                                                h = (CCH )−1 · C yB ,                                                                             (10)


where C = [c(n), · · · , c(n − NyB + 1)]T which is the input data matrix, c(n) = [c(n), · · · , c(n − NL + 1)]T

                                                                                 ˆ
which is the input vector at time n, NyB and NL are the vector lengths of yB and h respectively. Note that

yB is the received vector at source B which is given by (2), from which we have NyB = M − Nh + 1 where

Nh is the channel length2 . We also note that, in practice, the channel length Nh must also be estimated by,

for instance, classic order selection criterions such as the the Akaike’s Information Criterion (AIC) or its

variants [10]. In this paper, like many other approaches in the literature, we assume the channel length is a

priori known in order to focus on our main investigation of channel estimation and equalization. The joint

process with channel length estimation is beyond the scope of this paper and some references can be found

in [11], [12], [13].

  As we consider the case that A transmits and B listens, the task is for node B to estimate the channel

coefficients from A to B given the current received data vector yB at B and the data vector sB transmitted

by B during the previous time slot. The simplest method to explore the superimposed data structure for the

channel estimation is to regard sB as training sequence and sA as interference, or to let c = γ · sB in (10).

  2
      Extra zeros need to be padded to c when NL > Nh .
                                                                                                                 9


The performance is, obviously, severely limited to the “co-packet interference” from          1 − γ 2 sA .

  Since an equalizer is required for a selective fading channel, similar to the decision feedback equalizer

(DFE), we may feed back the hard decision of the equalizer output, ˜A , to the channel estimator to suppress
                                                                   s

the co-packet interference so that

                                          c = γ · sB +     1 − γ 2 · ˜A .
                                                                     s                                       (11)


Specifically, the channel estimator and equalizer operate in an iterative way. Initially, ˜A = 0 and only sB
                                                                                         s

is used for the channel estimation. The estimated channel coefficients are then used by the equalizer to

generate ˆA , the estimate of sA . After ˆA passes through the hard decision, it then feeds back to the channel
         s                               s

estimator for the next iteration. Although ideally such iterative approach converges to the case as if both

sB and sA are known to the LS estimator, it suffers from error propagation especially when the channel

SNR is low or the co-packet interference is high. In general, an ideal input to such iterative LS approach

has the form of

                                        c = γ · sB +     1 − γ 2 · f (ˆA ),
                                                                      s                                      (12)


where it is desirable that f (ˆA ) → sA when ˆA is close to sA and f (ˆA ) → 0 as otherwise.
                              s              s                        s

  Since the original superposition cooperative transmission belongs to the general decode-and-forward

scheme, a decoder is usually followed after the equalizer. Inspired by the excellent performance of the

turbo equalizer, we propose a so-called turbo LS estimator so that


                                        c = γ · sB +     1 − γ 2 · E[sA ],                                   (13)


where E[sA (n)] = 1 · P(sA (n) = 1) + (−1) · P(sA (n) = −1),


                                             1 + sA (n) · tanh(LLR(sA (n))/2)
                               P(sA (n)) =                                    ,                              (14)
                                                              2


and LLR(sA (n)) = ln[P(sA (n) = 1)/P(sA (n) = −1)] which is the log-likelihood fed back from the de-

coder. The overall structure of the turbo channel estimator is illustrated in Fig. 4, where initially LLRex (sA (n)) =

0 for all n.
                                                                                                                            10




              yB                                +           LLRex       De-               LLR               MAP
                            -                   -                   -                                   -             -
                                 Equalizer                              Interleaver                         Decoder
                                                    −   6
                                      6     6

                                                                                                −
                                                                                                ?
                           ˆ ˆ                                          Interleaver                +
                                                                                                    
                          (h, σ 2 )
                                                    LLR                               LLRex

                                            ?

                            -
                                 Channel
                                 Estimator


Fig. 4.   The turbo LS channel estimator.



   Because only the extrinsic information LLRex is fed back from the decoder, and also due to the deinter-

/inter-leaver, the error propagation can be effectively suppressed. To be specific, when SNR → ∞, we

have LLRex (sA (n)) → ∞ and E[sA (n)] = sA (n). When SNR → −∞, on the other hand, we have

LLRex (sA (n)) → 0 and E[sA (n)] = 0. Therefore, (13) is a good realization of ideal case of (12).

   In most cases, the noise power is also unknown and can be estimated as

                                                     ˆ              ˆ
                                                    (H · c − yB )T (H · c − yB )
                                             σ2 =
                                             ˆ                                   ,                                        (15)
                                                               NyB

      ˆ                                                                              ˆ
where H is the estimated channel matrix. It is obvious that (15) depends on not only H but also c. Thus

                                                              ˆ
if only sB is used for the channel estimation, then even with H = H, the noise power estimation is still

limited to the co-packet interference from sA . The turbo channel estimator, on the contrary, can solve this

                                                          ˆ
problem well because it has not only better estimation of H, but also less co-packet interference.


                            IV. SISO MMSE E QUALIZER WITH S UPERIMPOSED DATA

   In this paper, we are particularly interested in the linear SISO MMSE equalizer due to its simplicity

and nature connection to the turbo structure [9]. After the channel estimation, the known data sB must

be removed either before or after the equalization, which are, for clarity of exposition, denoted as “pre-

cancellation” and “post-cancellation” respectively. Although it looks straightforward, the “pre-cancellation”

approach suffers performance loss in SNR. To illustrate this phenomena, we first assume the channel is
                                                                                                                           11


perfectly known. Then if sB is removed before the equalization, the equalizer input is given by yB =

yB − γH · sB =         1 − γ 2 H · sA + n, and the equivalent channel SNR becomes


                                                                    1 − γ2
                                                         SNR =             .                                             (16)
                                                                      σ2


On the contrary, if the equalizer directly operates on yB and removes sB after the equalization, the channel

SNR is 1/σ 2 . This clearly reveals the SNR loss from the “pre-cancellation” approach, where the exact

amount of loss depends on the choice of γ . When the channel is not perfectly known, the analysis is

more complicated since the channel estimation error becomes another source of “noise”. However, when

the SNR is large enough, the proposed turbo channel estimator has small error and the above conclusion

still approximately holds. When the SNR is low, on the other hand, the BER performance deteriorates

seriously, making it little different between the “pre-” and “post- cancellation” approaches. Therefore sB

should always be removed after the equalization. This makes it necessary to re-derive the SISO MMSE

equalizer to fit the superimposed data structure of the equalization input.

   The structure of the equalizer is shown in Fig. 5, where w(n) is the equalizer vector, b(n) is a DC term,

                                     ˆ                                                 ˆ
∆ is the decision delay, ysB (n) = γ H · sB which corresponds to the sB part in yB and H is the estimated

                               ˆ
channel matrix. In particular, xA (n − ∆) is the equalizer output, or the estimation of xA (n − ∆), subtracting

                             ˆ
which by wH (n)ysB (n) gives sA (n − ∆), the estimation of sA (n − ∆). Finally the LLR generator calculates

the extrinsic information, LLRex (sA ), based on the Gaussian assumption.



                                                    LLR(sA ) from the decoder


                                       ?
            yB (n)                                  ˆ
                                                    xA (n − ∆) +         ˆ
                                                                         sA (n − ∆)
                          -
                              wT (n)yB (n) + b(n)              -                       -   LLR         -   LLRex (sA )
                                                                                           generator       to decoder
                                                                  −6
                                       6


                                                                       wT (n)ysB (n)
                                                                                SISO MMSE Equalizer

                                           ˆ
                                           h(n) and σ 2 from the channel estimator
                                                    ˆ


Fig. 5.   The SISO MMSE equalizer with superimposed data structure.
                                                                                                                   12


  It is clear from (1) that, for a known sB (n), xA (n) can only take two values: XA1 =          1 − γ 2 + γ · sB (n)

and XA0 = − 1 − γ 2 + γ · sB (n), corresponding to sA (n) = ±1 respectively. Then we have


                                     xA (n) = XA1 · P(sA (n) = 1) + XA0 · P(sA (n) = −1),
                                                                                                                (17)
                                              2                     2
                                 E[x2 (n)] = XA1 · P(sA (n) = 1) + XA0 · P(sA (n) = −1),
                                    A


where P(sA (n)) is calculated according to (14) and a = E[a] for any vector a. Then using (17), setting

LLR(sA (n − ∆)) = 0, and with similar procedures as those in [9], we obtain the equalizer tap-vector and

output as3


                                                                                ˆ ˆ       ˆ
               w(n) = (1 − γ 2 ) · {Cov(yB (n)) + [(1 − γ 2 ) − Cov(xA (n − ∆))]H∆ HH }−1 H∆ ,
                                                                                    ∆
                                                                                                                (18)
        ˆ                                                                                  ˆ
        xA (n − ∆) = γ · sB (n − ∆) + w (n)[yB (n) − yB (n) + (xB (n − ∆) − γ · sB (n − ∆))H∆ ],
                                                  H




                    ˆ                             ˆ
respectively, where H∆ is the (∆ + 1)th column of H and Cov(a) = E[aaH ] + E2 [a] for any vector a.

Note that Cov(yB (n)) and yB (n) can be easily further decomposed in term of channel parameters and

LLR(sA ).

                             ˆ
  The mean and covariance of sA (n − ∆) for a given sA (n − ∆) = SA are obtained as


                                  s                          x
              µsA , i (n − ∆) = E[ˆA (n − ∆)|sA (n − ∆)] = E[ˆA (n − ∆)|sA (n − ∆)] − E[wH ysB (n)]

                                 = γsB (n − ∆) +                           ˆ           ˆ
                                                          1 − γ 2 SA wH (n)H∆ − γwH (n)HsB (n)                  (19)

                  2                                                      ˆ       ˆ
                 σsA (n − ∆) = Cov[ˆA (n − ∆)|sA (n − ∆)] = (1 − γ 2 )wH H∆ [1 − HH w(n)],
                                   s                                               ∆


                                                                                            ˆ
where µsA , i corresponds to SA = ±1 for i = 1, 0 respectively. Note that the covariance of xA (n − ∆) and

ˆ
sA (n − ∆) are the same. Finally, with (19) and the Gaussian assumption, we obtain LLRex (sA ).

  Before leaving this section, we particularly highlight that during the first time slot or the initial transmission

of source A, the signal received by source B does not contain any superimposed “training” signals. Therefore,

the source A needs to send a “normal” training sequence to start up the cooperative transmission. At this

time, while it is not necessary to decode the data at all, the source B simply applies the classic LS channel

estimation and deactivates the proposed turbo channel estimation and equalization. Once the communications

  3
      The detail of the derivation is omitted due to the space constraint of this paper.
                                                                                                             13


starts up with the training symbols during the first time slot, no other training is required until a relay node

fails to decode and the whole process needs to start up again. As a comparison, the classic approach needs to

constantly send training symbols every other time, where the duration of the training time interval depends

on how fast the channels vary. Therefore, the proposed turbo approach is not only significantly more spectral

efficient but also better in tracking channel variations.


                                       V. N UMERICAL S IMULATIONS

A. Simulation Setup

  For simulations in this section, we assume each packet contains 128 symbols, and every symbol is encoded

by a half rate convolutional code with coding vectors of [1 0 1]T and [1 1 1]T . We compare four approaches,

i.e. that only sB is used for the channel estimation, the proposed turbo LS channel estimator, that both sB

and sA are assumed to be known for the channel estimation, and the perfect knowledge about the channel

information, which are denoted as “LS-sB ”, “LS-turbo”, “LS-both” and “Known-channel” respectively. For

fair comparison, the turbo equalization is applied for all approaches and the iteration number is set as 6.



B. Static Channel

  In the first experiment, we consider a static channel that the channel coefficient vector is fixed at h =

[0.1 0.3 1 0.3 0.1]T . The equalizer has length of 10. The results are obtained by averaging over 5, 000

independent runs.

  First we set γ 2 = 0.2 which the optimum γ derived in Section II-B and let sB be removed after

the equalization (i.e. “post-cancellation”). Fig. 6 (a) shows the mean squared error (MSE) of the channel

estimation which is defined as
                                                          ˆ
                                                        E|h − h|2
                                               ˆ
                                           MSE(h) =               .                                       (20)
                                                           |h|2


It is clearly shown in Fig. 6 (a) that, in the working SNR range (e.g. SNR > 5dB), the MSE performance

of the proposed turbo LS estimation is close to that when both sB and sA are assumed to be known. On

the other hand, when SNR is low (e.g. SNR < 2dB), the error propagation can be effectively suppressed
                                                                                                                                               14


and the turbo LS estimator works like a traditional LS estimator 4 .

   Fig. 6 (b) compares the BER performance for the 4 approaches, where it shows that the BER performance

with the turbo LS estimation is close to that of the ideal case with perfect channel knowledge, and

significantly better than that when only sB is used for the channel estimation. For example, about 3dB

gain in SNR can be observed at BER = 10−5 between the approaches of “LS-turbo” and “LS-sB ”.


                  0                                                              0
                 10                                                             10
                                                          LS−sB                                                         LS−sB
                                                          LS−turbo                                                      LS−turbo
                                                                                 −1
                                                          LS−both               10                                      LS−both
                  −1
                                                                                                                        Channel−known
                 10
                                                                                 −2
                                                                                10
        MSE(h)




                                                                          BER
                  −2                                                             −3
                 10                                                             10


                                                                                 −4
                                                                                10
                  −3
                 10
                                                                                 −5
                                                                                10


                  −4                                                             −6
                 10                                                             10
                       0      5           10         15              20               0   2    4     6         8   10        12         14
                                         SNR                                                             SNR


                                           ˆ
                                   (a) MSE(h)                                                 (b) BER performance

Fig. 6.           Static channel: MSE and BER performance for γ 2 = 0.2 and sB being removed after the equalizer.



   Next, we compare the cases for sB being removed before and after the equalization, i.e. “pre-cancellation”

and “post-cancellation” respectively. It has been mentioned earlier that we usually have 0.075                                     γ2        0.2.

But only for better exposition of the simulation results, here we deliberately set γ 2 = 0.45. This is because,

according to (16), the larger the γ 2 is, the bigger the difference between the two cases appears.

   Fig. 7(a) shows the output SNR of the equalizer which is obtained as

                                                                          E[µ2A , i (n)]
                                                                             s
                                                      Output SNR =             2
                                                                                                                                             (21)
                                                                           E[σsA (n)]

                       2
where µsA , i (n) and σsA (n) are given by (19). For better exposition, only the results for the proposed LS

turbo estimation and the approach with perfect channel information are presented, where the SNR advantage

of the “post-cancellation” over the “pre-cancellation” approach is clearly shown.

   Fig. 7 (b) compares the BER performance for different approaches. It is shown that the best BER

  4
      The MSE of the noise power estimation is similar to that shown in Fig. 6(a), but is not shown here to save space.
                                                                                                                                                  15


performance comes from the approach with perfect channel information and “post-cancellation”, and the

performance for the turbo LS estimation with “post-cancellation” is close to the best performance. On

the other hand, the approach of “LS-sB ” with “pre-cancellation” gives the worst BER performance. There

is about 3dB difference in SNR at BER = 10−5 between the best and worst cases. It is interesting to

observe that the performance for “LS-sB ” with “post-cancellation” is close to that for “LS-turbo” with “pre-

cancellation”, because the performance loss suffered by the two cases are due to the neglect of                                         1 − γ 2 sA

at the channel estimation and the neglect of γsB at the equalization respectively. But with γ 2 = 0.45, the

powers of                             1 − γ 2 sA and γsB are similar. This observation verifies that the information of sB and sA

should be used as much as possible by the channel estimation and equalization, which is the philosophy

behind the proposed approach of this paper.

  Finally we highlight the conclusion that sB should always be removed after the equalization does not

change for a different selection of γ .

                                                                                      0
                             20                                                      10
                                                                                                                             LS−sB
                                        LS−turbo
                                                                                                                             LS−turbo
                                        Channel−known                                 −1
                                                                                     10                                      Channel−known
                             15

                                                                                      −2
      Equalizer output SNR




                                                                                     10
                             10
                                                                               BER




                                                                                      −3
                                                                                     10

                              5
                                                                                      −4
                                                                                     10


                              0                                                       −5
                                                                                     10


                                                                                      −6
                             −5                                                      10
                                  0        5             10   15      20                   0   2    4     6         8   10        12         14
                                                        SNR                                                   SNR


                                        (a) Equalizer output SNR                                   (b) BER performance

Fig. 7. Static channel: the Output SNR and BER performance for γ 2 = 0.45. Dash lines: “pre-cancellation”; Solid lines: “post-
cancellation”.




C. Random Decaying Channel

  In this experiment, we consider a more practical frequency selective Rayleigh fading channel with 5

multipaths, i.e. the channel has 5 taps. The average powers of the 5 taps are exponentially decaying from

one tap to the next. The results are obtained by averaging over 5,000 independent runs, and every run

applies a random realization of the channel. We also let γ 2 = 0.2 and sB be removed after the equalizer
                                                                                                                                         16


(i.e. “post-cancellation”). The filter length for the equalizer is set as 20 as we are now dealing with a

“tougher” channel than the static channel used in the previous experiment.

   Fig. 8 (a) shows the MSE for the channel estimation which are similar to those in the previous experiment

shown in Fig. 6 (a). Fig. 8 (b) compares the BER performance for different approaches. It is clear that,

although the proposed Turbo LS estimator is inferior to the approaches for “LS-both” and “Channel known”,

it is still significantly better than the approach when only sB is used for the channel estimation.


                −1                                                            0
               10                                                            10
                                                       LS−sB                                                        LS−sB
                                                       LS−turbo                                                     LS−turbo
                                                                              −1
                                                       LS−both               10                                     LS−both
                                                                                                                    Channel known

                −2                                                            −2
               10                                                            10
      MSE(h)




                                                                       BER
                                                                              −3
                                                                             10


                −3                                                            −4
               10                                                            10


                                                                              −5
                                                                             10


                −4                                                            −6
               10                                                            10
                     0   2    4     6        8   10   12          14               0   2    4     6        8   10       12          14
                                     SNR (dB)                                                      SNR (dB)


                                          ˆ
                                  (a) MSE(h)                                               (b) BER performance

Fig. 8.         Random decaying channel: MSE and BER performance for γ 2 = 0.2 and sB being removed after the equalizer.




D. Random Channel

   In the last experiment, we further test the proposed algorithm with another kind of frequency selective

Rayleigh fading channel which also consists of 5 multipaths but has same average power for every path.

Again, we have 5,000 independent runs, and each run generates a random realization of the channel. All

other parameters are same as those in the previous experiment. Fig. 9 (a) and (b) plot and compare the

channel estimation MSE and BER performance for different approaches. The results are similar to those

for the “random decaying channel”. This further verifies the effectiveness of the proposed algorithm.


                                                           VI. C ONCLUSION

   In this paper, we proposes a novel turbo LS channel estimator for the relay nodes in the superposition

cooperative transmission. The soft-in-soft-out MMSE equalizer is also carefully re-derived to match the
                                                                                                                                         17



                −1                                                            0
               10                                                            10
                                                       LS−sB                                                        LS−sB
                                                       LS−turbo                                                     LS−turbo
                                                       LS−both                                                      LS−both
                                                                              −1
                                                                                                                    Channel known
                                                                             10
                −2
               10
      MSE(h)




                                                                       BER
                                                                              −2
                                                                             10


                −3
               10
                                                                              −3
                                                                             10




                −4                                                            −4
               10                                                            10
                     0   2    4     6        8   10   12          14               0   2    4     6        8   10       12          14
                                     SNR (dB)                                                      SNR (dB)


                                          ˆ
                                  (a) MSE(h)                                               (b) BER performance

Fig. 9.         Random channel: MSE and BER performance for γ 2 = 0.2 and where sB being removed after the equalizer.


superimposed data structure. We have thoroughly tested the proposed approach by extensive numerical

simulations under different scenarios including the static, random decaying and random channels. All of

results show consistent performance improvement of the proposed approach over the classic approach that

only the known part of the received data is used for the channel estimation. The exact amount of the

improvement depends on specific scenarios and is difficult, if not impossible, to be quantitatively analyzed.

For example, for the static channel used in the simulation, the proposed algorithm has very close performance

to the approach with perfect channel knowledge. For the random channel, however, the proposed algorithm

is inferior to that with perfect channel knowledge, though it is still significantly better than the classic

approach.

   Finally we point out that it would also be interesting to explore the superposition data structure at the

destination node so that the information data can be well detected at the destination eventually. This is left

as a future topic.


                                                      ACKNOWLEDGEMENT

   The authors would like to thank the Editor, Prof. Rashvand, and five reviewers who provided invaluable

comments for us to improve this manuscript.


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