Errors
In this section, we move on to the second perspective of combating synchronization errors in cooperative MIMO communications. We will view the synchronization errors, not as impairments which cause performance degradation, rather as potential sources of diversity gains.
Through careful design of a BICM-based scheme and an iterative receiver, it will be shown that dramatic increase of diversity gain is actually achieved when severe synchronization errors exist.
3.1 System Model
We adopt the decode-and-forward protocol for this section, and BICM-OFDM [24] are chosen as the transmission technique. Fig. 13 shows a generic block diagram of a system employing BICM-OFDM, at the source side the information bits denoted b are first encoded by the outer convolutional encoder and the encoded bits are denoted by c∈C, C being the codeword set. The interleaver ∏ operates on K OFDM symbols of encoded bits with the output denoted by
c’, then the inner differential precoder with recursive structure [34]
is deployed to enhance overall performance and its output is denoted by d. The resulting bits are mapped into QAM or PSK symbols. The set of constellation points is denoted by χ, as γ bits are mapped into one of 2γ constellation points according to the mapping rule. After loading the modulated symbols onto active subcarriers, OFDM signal x is generated via N-point IFFT and CP is inserted. The performance depends on the size of interleaver that is γKN bits. Note that the encoded bits are interleaved across several OFDM systems and it is called time-frequency interleaving. The time-domain transmitted signal at Relay Node α can be written as
1
2 where Xα(n) is the modulated symbol at the n-th subcarrier, N is the OFDM symbol length, Ng isthe length of CP, k is the sampling index, α∈{1,2,..,M} is the relay node index, and M is the number of relays. Assume the CP is longer than the largest channel delay spread plus timing error so that ISI can be ignored.
Convolution
encoder
precoder Modulator IFFT CPb c c' d X x
Fig. 13. The block diagram of the proposed scheme for asynchronous cooperative communications
Time varying multipath Rayleigh fading channels are considered, and the discrete time baseband equivalent received signal at the k-th sampling time can be expressed as
2 1
where εα and ηα represents the normalize CFO and the timing error between destination node and the α-th relay node. Let hα(k,l) represents the l-th path gain of the multipath Raleigh fading channel from the α-th relay to the destination. The wide-sense stationary uncorrelated scattering (WSSUS) channel is assumed with
2 autocorrelation ( r(0)=1 ), and δ(l-l’) is the Kronecker delta function. Moreover, assume the paths( ) 0(2 ( ) )
r q J f
k m T
(14) where J0(〃) denotes the zero-order Bessel function of the first kind and f is the Doppler frequency of the α-th relay node, T represents one OFDM symbol time, xα(k) is the transmitted signal of the α-th relay node, and w(k) is additive noise, which is independently and identically distributed (i.i.d.) complex Gaussian random variable with zero mean and variance z2. Consider the model in frequency domain by taking the N-point DFT to y(k) in (2). The p-th OFDM symbol in the frequency received signal can be written be
, ,
F
Hz(k) are the frequency domain transmitted data and additive noise, respectively, where X
p andZ
p are an KN×1vector. Since FFT is unitary, the entries of Zp are still white complex Gaussian variables with mean zero and variance 2z.3.2 The Receiver Algorithm
For the receiver, both MSE and MLSE equalizers can be used. Here we focus on the design of MLSE receiver in the frequency domain and the overall receiver. The iterative receiver consists of a Soft-Input Soft-Output (SISO) MLSE demapper/equalizer and Maximum A Posterior (MAP) decoders for both the precoder and the convolutional encoder. The soft outputs are typically represented by the log-likelihood ratio (LLRs). The signal detection in the demapper/equalizer is carried out with MLSE.
The task of the equalizer is to estimate the transmitted X based on the received observations
R. more specifically, the maximum likelihood sequence estimation is to choose that sequence of
symbols X={x1,x2,…,xK} that maximizes the likelihood of the received sequence of observationR={R
1,R2,…,RK}, i.e., maximizes the joint conditional probability the P(R|X). the obtainedsequence is the optimal solution and procedure is referred to as MLSE. There exist basic approaches to implement an MLSE equalizer in [35].
Start with the states at the k-th stage of the associated trellis diagram that are related to the Q-1 most recent transmitted symbols, i.e.,
1 1
( , ,..., ,..., )
N N N
k D k D k k D k
s x
x
x x
(16) Thus, each state corresponds to one of the 2γ(Q-1) possible vectors that can be formed from Q-1 symbols. There are 2γ allowable transitions that emerge from a state sk and terminate at 2γ different states sk+1, leading to a total of 2γQ transition branches connecting two successive states (sk→sk+1). Each transition is associated with a cost, contributing to the total cost of a path along the states. The cost of the i-th transition between skand s
k+1 exists transition probabilities is called a branch metric, connecting two specific consecutive states (sk→sk+1), is given by2
Notice that each state has 2γ incoming branches except a few stages in the beginning and in the end. Each incoming branch is due to the advent of a new symbol. Of the 2γ incoming branches, only the one connected, and the new symbol metric Γ(sk) is calculated that formulation represent constellation point x of the value at the n’-th bit. That retained path is referred to as survivor path.
After all states of the trellis have been gone through, the smallest state metric be found and trace back that the ˆx
is obtained.
A soft decision as the log-likelihood ratio is obtained by (ˆ 1 | ) Thus MLSE output bit LLRs is transformed by symbol metric.
1 point set of n-th bit is b ∈{0,1}.The inner and outer decoder are adopting a maximum a posterior probability (MAP), output are the bit log likelihood ratio and log-MAX algorithm is usually applied for lower computational complexity. A trade-off between complex and performance can be achieved by different choices of D, K, and γ.
The computational complexity of MLSE is 𝒪 (IN2Q) where I is number of iteration and for the MMSE receiver it is 𝒪 (N3) in one OFDM symbol. The MLSE is much more expensive than the MMSE receiver when high order modulation or large D is used.
3.3 Simulation Results and Discussion
To demonstrate the effectiveness of the MLSE receiver, Monte Carlo simulations are carried out, and we compare the Bit Error Rate (BER) performance between the MLSE equalizer and MMSE equalizer. Notice that both receivers effectively harvest the extra diversity gain provided by synchronization errors.
We consider a BICM-OFDM system with N = 64, CP length = 8, and 4-QAM modulation. A two-path wide-sense stationary uncorrelated scattering (WSSUS) Rayleigh fading channel (generated using Jakes Model) between any relay nodes and each relay are equal power, the convotional code uses G(D)=(1+D2,1+D+D2) as the generator polynomial, and G(D)=1/(1+D2) is the generator polynomial for the precoder. One frame consists of 10 OFDM symbols.
Furthermore, perfect estimations of MCFOs and channel matrices are assumed.
Fig. 14 shows the BER performance versus SNR for the comparison between conventional MMSE equalizer, traditional 1-tap equalizer and the MLSE equalizer in synchronous impairments. For the simulation, normalized Doppler frequency fd=0.001 is employed at both relays, the normalized MCFOs are 0.2 and -0.2. With the large MCFOs, the 1-tap equalizer suffers an obvious error floor.
0 2 4 6 8 10 12 14 16 18
Fig. 14. BER comparison between MMSE equalizer, 1-tap equalizer and MLSE equalizer in the cooperative communication
On contrast, the MLSE equalizer not only successfully compensates for the ICI but also obtain an SNR gain about 3dB. The benefits of SNR gain, we can via SINR to explicit explanation and the derivation in appendix. The optimal solution is joint processing of demodulation and decoding is considered, which lead to approach low bound. Notice that with both equalizers the system achieves full diversity.
Fig. 15 shows the results for the two relay nodes and three relay nodes. It can be seen from the figure that as the number of relay increases in the systems, the diversity order of distributed BICM-OFDM increases up to the maximum diversity of min{M×rT×L,dfree}. It can be observed that the tree relays case has a diversity order of 5 and the BER curve is steep.
0 5 10 15 10-5
10-4 10-3 10-2 10-1 100
SNR
BER
M=2, CFOs=[0.2 -0.2] Iteration=5 M=3, CFOs=[0.2 0 -0.2] Iteration=5
Fig. 15. The BER curves compared with difference number of relay nodes
In Fig. 16, all the realistic synchronous impairments are considered. The timing errors is [0 3], normalized Doppler frequency is 0.1 for both relays and MCFOs is [0.2 -0.2]. In our proposed the performance show efficiently collects the diversity form time diversity due to the Doppler effect, frequency diversity due to timing error and special diversity converted to time diversity due to MCFOs. It observed that the diversity is more than four.
0 2 4 6 8 10 12 14 16 18
10-5 10-4 10-3 10-2 10-1 100
SNR
BER
CFOs=[0.2 -0.2] Fd=0.001 Timing error=[0 0]
CFOs=[0.2 -0.2] Fd=0.1, Timing error=[0 3]
Fig. 16. The BER for cooperative communication under time error = [0 3], normalize Doppler frequency = 0.1, MCFOs = [0.2 -0.2]
In summary, BICM has the potential to improve performance with relatively ease in many OFDM wireless communication systems. It is shown that, with proper receiver design, the BICM-OFDM can be effective to combat synchronous errors as well as harvest potential diversity gain in cooperative communications. Typical BICM-OFDM systems suffer error floors due to ICI caused by MCFOs and Doppler effects. To deal with such a problem, we propose an
MLSE-based frequency domain equalizer combined with a turbo decoder to break the error floor.
The proposed approach has excellent BER performance, and it is flexible in a way that extension to more relays for improvement in diversity gain is straightforward. The complexity is a big problem in the receiver if D is greater than three, and future research in the complexity reduction will be considered.