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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 softin-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:
[email protected]. Z. Ding is with the Department of Communication Systems of Lancaster University, Lancaster, LA1 4WA, UK. Tel., +44 015245 10399, Email:
[email protected]. T. Ratnarajah and C. Cowan are with ECIT, Queen’s University of Belfast, Belfast, BT3 9DT, UK. Tel: +44 028 9097 1078. Email:
[email protected].
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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 modulation [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
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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.
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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.
A
xA, i =
√
1 − γ 2 · sA, i + γ · sB, i−1
A 6
q
D
1
D
?
B
B
√ 1 − γ 2 · sB, i+1 + γ · sA, i
(b) Time slot (i + 1)
(a) Time slot i Fig. 1.
xB, i+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 =
p
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 superimposed 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
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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
=
p
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 sB = −1 Fig. 2.
sA = −1 sB = 1
sA = 1 sB = −1
-
sA = 1 sB = 1
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)}, γ
(4)
where EF, FG and GJ refer to the distances shown in Fig. 2 which are given by
EF = GJ = 2γ
and
p 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
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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 = 0.2. This result well matches a statement in [6] that 0.075 6 γ 2 6 0.2. leading to γopt
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 Z Pout =
Iout
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 2e2ISP qSP (ISP ) = 2 ρ λ1 λ2
Z
µ
e2ISP
f 1
e2ISP − x ρλ1 x
¶ µ ¶ x−1 1 f dx, ρλ2 x
(7)
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.
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¡ ¢ transmission can be easily obtained as ID = log 1 + ρ|h1 |2 , the density function of ID is given by Z qD (ID ) =
0
∞
eID − δ (ID − ln(1 + ρx)) f (x) dx = e ρ
³
eID −1 ρ
´
,
(8)
where δ(·) denotes the Delta function. The second equation follows from the property of the Delta function that δ(f (x)) =
P
δ(x−xi ) 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 qSR (ISR ) = ρ
µ
e2ISR − 1 ρ
¶ e
³ 2I ´ −1 − e SR ρ
.
(9)
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.
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1.4
1.4
Direct transmission Selection relaying SP cooperative
1.2
Direct transmission Selective relaying SP cooperative
1.2
0.8
0.8
q(I)
1
q(I)
1
0.6
0.6
0.4
0.4
0.2
0.2
0 0
1
2
3
4
5
6
7
8
9
0 0
1
Mutual Information in bit/s/Hz
2
3
4
5
6
7
8
9
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 = (CCH )−1 · C yB , h
(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 respectively. Note that which is the input vector at time n, NyB and NL are the vector lengths of yB and h 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 .
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The performance is, obviously, severely limited to the “co-packet interference” from
p 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, ˜sA , to the channel estimator to suppress the co-packet interference so that c = γ · sB +
p 1 − γ 2 · ˜sA .
(11)
Specifically, the channel estimator and equalizer operate in an iterative way. Initially, ˜sA = 0 and only sB is used for the channel estimation. The estimated channel coefficients are then used by the equalizer to generate ˆsA , the estimate of sA . After ˆsA passes through the hard decision, it then feeds back to the channel 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 +
p 1 − γ 2 · f (ˆsA ),
(12)
where it is desirable that f (ˆsA ) → sA when ˆsA is close to sA and f (ˆsA ) → 0 as otherwise. 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 +
p
1 − γ 2 · E[sA ],
(13)
where E[sA (n)] = 1 · P(sA (n) = 1) + (−1) · P(sA (n) = −1),
P(sA (n)) =
1 + sA (n) · tanh(LLR(sA (n))/2) , 2
(14)
and LLR(sA (n)) = ln[P(sA (n) = 1)/P(sA (n) = −1)] which is the log-likelihood fed back from the decoder. The overall structure of the turbo channel estimator is illustrated in Fig. 4, where initially LLRex (sA (n)) = 0 for all n.
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yB
-
+-
Equalizer 6
6
−
P 6
LLRex
-
DeInterleaver
LLR
-
−
ˆ σˆ 2 ) (h,
Interleaver LLR
? P
MAP Decoder
-
+
LLRex
? -
Fig. 4.
Channel Estimator
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
σ ˆ2 =
ˆ · c − yB )T (H ˆ · c − yB ) (H , NyB
(15)
ˆ is the estimated channel matrix. It is obvious that (15) depends on not only H ˆ but also c. Thus where H ˆ = H, the noise power estimation is still if only sB is used for the channel estimation, then even with H
limited to the co-packet interference from sA . The turbo channel estimator, on the contrary, can solve this ˆ , but also less co-packet interference. problem well because it has not only better estimation of H
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 “precancellation” 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 0 = perfectly known. Then if sB is removed before the equalization, the equalizer input is given by yB
yB − γH · sB =
p
1 − γ 2 H · sA + n, and the equivalent channel SNR becomes
SNR =
1 − γ2 . σ2
(16)
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 · sB which corresponds to the sB part in yB and H ˆ is the estimated ∆ is the decision delay, ysB (n) = γ H
channel matrix. In particular, x ˆA (n − ∆) is the equalizer output, or the estimation of xA (n − ∆), subtracting which by wH (n)ysB (n) gives sˆA (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)
-
wT (n)yB (n) + b(n) 6
xˆA (n − ∆) + -
P
sˆA (n − ∆)
−6
-
LLR generator
wT (n)ysB (n) SISO MMSE Equalizer ˆ h(n) and σ ˆ 2 from the channel estimator
Fig. 5.
The SISO MMSE equalizer with superimposed data structure.
-
LLRex (sA ) to decoder
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It is clear from (1) that, for a known sB (n), xA (n) can only take two values: XA1 =
p
1 − γ 2 + γ · sB (n)
p 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[x2A (n)] = XA1 · P(sA (n) = 1) + XA0 · P(sA (n) = −1),
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 ∆H ˆ H }−1 H ˆ ∆, w(n) = (1 − γ 2 ) · {Cov(yB (n)) + [(1 − γ 2 ) − Cov(xA (n − ∆))]H ∆
(18) ˆ ∆ ], x ˆA (n − ∆) = γ · sB (n − ∆) + w (n)[yB (n) − yB (n) + (xB (n − ∆) − γ · sB (n − ∆))H H
ˆ ∆ is the (∆ + 1)th column of H ˆ and Cov(a) = E[aaH ] + E2 [a] for any vector a. respectively, where H
Note that Cov(yB (n)) and yB (n) can be easily further decomposed in term of channel parameters and LLR(sA ).
The mean and covariance of sˆA (n − ∆) for a given sA (n − ∆) = SA are obtained as µsA , i (n − ∆) = E[ˆ sA (n − ∆)|sA (n − ∆)] = E[ˆ xA (n − ∆)|sA (n − ∆)] − E[wH ysB (n)] = γsB (n − ∆) +
p ˆ ∆ − γwH (n)Hs ˆ B (n) 1 − γ 2 SA wH (n)H
(19)
ˆ ∆ [1 − H ˆ H w(n)], σs2A (n − ∆) = Cov[ˆ sA (n − ∆)|sA (n − ∆)] = (1 − γ 2 )wH H ∆
where µsA , i corresponds to SA = ±1 for i = 1, 0 respectively. Note that the covariance of x ˆA (n − ∆) and sˆA (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.
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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 = MSE(h)
ˆ − h|2 E|h . |h|2
(20)
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
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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−both
LS−turbo LS−both Channel−known
−1
10
−1
10
−2
BER
MSE(h)
10
−2
10
−3
10
−4
10 −3
10
−5
10
−4
10
−6
0
5
10 SNR
15
20
10
0
2
6
8
10
12
14
SNR
ˆ (a) MSE(h) Fig. 6.
4
(b) BER performance
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 6 γ 2 6 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
Output SNR =
E[µ2sA , i (n)] E[σs2A (n)]
(21)
where µsA , i (n) and σs2A (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.
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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 “prep
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
p 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 Channel−known
10
15
−2
10 10
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Fig. 7. Static channel: the Output SNR and BER performance for γ 2 = 0.45. Dash lines: “pre-cancellation”; Solid lines: “postcancellation”.
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
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(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.
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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
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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. R EFERENCES [1] T. S. Rappaporat. Wireless Communications: Principle and Practice. Prentice-Hall, 1998.
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[2] J. N. Laneman, D. N. C. Tse, and G. W. Wornell. Cooperative diversity in wireless networks: efficient protocals and outage behavior. IEEE Trans. on Information Theory, 50(12):3062 – 3080, Dec. 2004. [3] A. Bletsas, H. Shin, and M. Z. Win. Outage analysis for co-operative communication with multiple amplify-and-forward relays. Electronics Letters, 43(6):51 – 52, March 2007. [4] S. Ahmed, Z. Ding, T. Ratnarajah, and C. F. N. Cowan. Cooperative transmission protocol with full diversity and iterative detection. IET Signal Processing, 2(4):361 – 368, Dec. 2008. [5] K. Azarian, H. El Gamal, and P. Schniter. On the achievable diversity-multiplexing tradeoff in half-duplex cooperative channels. IEEE Trans. on Information Theory, 51(12):4152 – 4172, Dec. 2005. [6] E. G. Larsson and B. R. Vojcic. Cooperative transmit diversity based on superposition modulation. IEEE Communications Letters, 9(9):778 – 780, Sept. 2005. [7] Z. Ding, T.Ratnarajah, and C. Cowan. On the diversity-multiplexing tradeoff for wireless cooperative multiple access systems. IEEE Trans. on Signal Processing, page 4627 4638, Sept. 2007. [8] M. Ghogho, D. McLernon, E. A.-Hernandez, and A. Swami. Channel estimation and symbol detection for block transmission using data-dependent superimposed training. IEEE Signal Processing letters, 12(3):226 – 229, March 2005. [9] M. Tuchler, A. C. Singer, and R. Koetter. Minimum mean squared error equalization using a priori information. IEEE Trans. on Signal Processing, 50(3):673 – 683, March 2002. [10] H. Akaike. A new look at the statistical model identification. IEEE Trans Automat. Contr, AC-19:716 – 723, Dec. 1974. [11] A. Mertins. Design of redundant FIR precoders for arbitrary channel lengths based on an MMSE criterion. In ICC IEEE International Conference on Communications (ICC), volume 1, pages 212 – 216, May 2002. [12] Y. Gong, V. Bhatia, B.Mulgrew, and C. F. N. Cowan. A non-parameteric ml estimator with unknown channel order. In 13th European Signal Processing Conference (EUSIPCO’2005), Antalya, Turkey, Sept. 2005. [13] X. Wang, H. Wu, S. Y. Chang, Y. Wu, and J. Y. Chouinard. Analysis and algorithm for non-pilot-aided channel length estimation in wireless communications. In IEEE Global Telecommunications Conference (GLOBECOM), pages 1 – 5, Nov. 2008.
