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Frequency domain equalization spacetime blockcoded CDMA transmission system
EURASIP Journal on Wireless Communications and Networking volume 2011, Article number: 80 (2011)
Abstract
In this work we propose a spacetime blockcoded (STBF) CDMA transmission system suitable for use with frequency domain equalization (FDE) algorithms. We illustrate the FDE by implementing the maximal ratio combining, the zero forcing and the minimum mean squared error single user detection algorithms. A diversity gain analysis is developed and some interesting results are pointed out. It is shown through computer simulations that the proposed transmission system exhibits good performance in terms of bit error rate when compared to previously proposed STBC CDMA transmission systems.
1 Introduction
Spacetime blockcodes (STBC) schemes have emerged as a powerful transmit diversity technique to combat fading in wireless communications. One of the most successful spacetime coding scheme was first proposed by Alamouti [1] for the case of two transmit and multiple receive antennas. Further developments for more than two transmit antennas were later reported on [2, 3], and it was shown that the Alamouti's scheme is the only existing complex orthogonal design, with full rate, full diversity and minimal delay. STBC was applied for directsequence code division multiple access (DSCDMA) transmission system in [4, 5]. All the aforementioned systems [1, 5], assume flatfading channels and suffer performance degradation in frequencyselective channels. Some STBC schemes to deal with frequency selectivity have been proposed for orthogonal frequency division multiplexing (OFDM) [6], singlecarrier (SC) timedomain equalization [7] and singlecarrier frequencydomain equalization (SCFDE) systems [8, 9].
The complexity and performance of SCFDE systems are comparable to that of OFDM systems while avoiding drawbacks associated with multicarrier (MC) implementation. On the other hand, SC systems cannot certainly offer the same flexibility as OFDM in the management of bandwidth and energy resources [10] and FDE does not represent an optimal solution to signal detection over frequencyselective channels due to intersymbol interference (ISI). In [11], adaptive algorithms to mitigate ISI effects for frequency domain equalization (FDE) in frequencyselective channels were proposed.
STBC CDMAbased transceivers for frequencyselective channels have been studied in [12–16], using a structure similar to the one proposed in [4, 5] for the case of flatfading channels. A different structure for STBC single carrier CDMA transmission system based on chipinterleaved blockspread (CIBS) CDMA [17] was proposed in [18]. The structure in [18], though promising excellent performance, incurs in a relatively high computational complexity to update the equalizer coefficients [19]. In [20], timereversal is used to provide FDE capabilities to STBC single carrier CDMA transmission system.
In this work a structure for FDE STBC CDMAbased transmission system is proposed. In this structure, transmit symbols are spread in a symbolbysymbol basis and the selfinterference in the receiver is avoided by the use of permutation matrices [21, 22] in the transceiver, which also allow us to decode each transmit symbol separately. We present the proposed transmission system in a general framework, which allows us to perform an unified analysis and to present a fair comparison between commonly used CDMAbased block transmission systems. Also, as we show through computer simulations, FDE algorithms used with single user detection results in a simple receiver design with good performance in terms of bit error rate (BER) when compared with previously proposed STBC CDMA transmission systems. A diversity gain analysis of the proposed transmission system is performed providing interest results.
This paper is organized as follows: Section 2 describes the baseband system model, addressing the definitions and properties of the employed matrices. In Section 3 we present and point out some properties of the receiver design, while in Section 4 different approaches for FDE are applied in the proposed scheme. Section 5 presents the results obtained through computer simulations and Section 6 gives some conclusions. A diversity and coding gain analysis is included as an appendix.
Notation. In what follows, I_{ k }represents a k × k identity matrix, 0_{m×n}, an m × n null matrix, (·)^{T}, (·) ^{H} , (·)* and (·)^{†} denote transpose, Hermitian transpose, complex conjugated and MoorePenrose matrix inverse, respectively, ⊗ is the Kronecker product, diag(x) is a diagonal matrix with the components of x as its nonzero elements, rank(·) is the rank of a matrix, det(·) denotes determinant, the operator E [·] stands for ensemble average and ℂ(ℝ) represents the field of the complex (real) numbers.
2 System model
Consider the discretetime equivalent baseband model of a synchronous downlink STBC CDMA system shown in Figure 1 (for convenience only one user is shown). For simplicity, in this paper we will focus on the case of two transmit antennas and one receive antenna, although this schema could be easily extended to configurations with more receive antennas, provided an appropriate combiner, such as maximal ratio combiner (MRC), would be used.
In the proposed system each of the K users transmits symbols s_{ k }(i) ∈ ℂ, which are first spread by the Mchip spreading code c_{ k }∈ ℂ ^{M} , c_{ k }^{2} = 1, and then grouped in vectors c_{ k }s_{ k }(i) of size M. It is assumed that symbols s_{ k } (i) are drawn from some constellation with zero mean and unit average symbol energy and they are independent and identically distributed (i.i.d.).
Then, the spread symbols are linearly precoded by the matrix G∈ ℂ ^{M×M} and fed into the Alamoutibased spacetime encoder [1] to get the spacetime symbol
where ${\stackrel{\u0304}{s}}_{k}\left(i\right)=G{c}_{k}{s}_{k}\left(i\right)$, G represents an arbitrary linear operation used to combat deleterious channel effects and to simplify equalizer designs. P_{tx} ∈ ℝ ^{M×M} is a permutation matrix whose design depends on the overall system and it is used to decouple, in the receiver, the transmitted symbols, as we will show later.
It should be noted that a guard interval is necessary to avoid interblock interference (IBI) in the received signal. The guard interval insertion is performed by the matrix T∈ ℝ ^{P×M} , where P = M + L_{ gi } and L_{ gi } is the length of the guard interval. For the most commonly used guard intervals, cyclic prefix (CP) and zero padding (ZP), the matrix T is defined as [23]:
The frequencyselective channel from the j th (j = 1, 2) transmission antenna to the receiver can be modeled using a finiteimpulse response (FIR) filter with L taps, whose gains are samples, taken at the chip rate, of the equivalent baseband complex channel impulse response. Assuming that during two symbol periods the impulse response of both channels remain constant, that is, h_{ j }(2i) = h_{ j }(2i + 1) = [h_{j,0}(2i) ... h_{j,L1}(2i)] ^{T} , E [h_{ j }(2i)^{2}] = 1, the transmission through the frequencyselective MIMO channel can be represented by a P × P lower triangular Toeplitz convolution matrix H_{ j }(2i), whose first column is [h_{j,0}(2i) ... h_{j,L1}(2i) 0 ... 0] ^{T} .
As we assume a downlink scenario, where the users experience the same channel condition, the received signal collected over two consecutive symbol periods is represented by the two Pdimensional vectors:
where n(i) is a complex white Gaussian noise vector with zero mean and covariance matrix E [n(i)n^{H}(i)] = N_{0}I_{ P }, N_{0} is the noise spectral density, η(i) denotes the IBI present in nonZP systems (η(i) = 0 in ZP systems) and
where ${\u1ebc}_{s}=\left({E}_{1}+\cdot \cdot \cdot +{E}_{K}\right)\u2215K$ is the mean received energy, with E_{ k } being the energy of the k user signal, ${\rho}_{k}=\sqrt{{E}_{k}\u22152{\u1ebc}_{s}}$, ρ= diag(ρ_{1}; ⋯ ρ_{ K } ), C= [c_{1} ··· c _{ K } ], s(i) = [s_{1}(i) ⋯ s_{ K } (i)]^{T}.
If CP is used as guard interval at the transmitter, the receiver must remove the guard interval from the received signal to eliminate IBI. If ZP is used at the transmitter the IBI is null, and then, the guard interval removal is not necessary. This operation is represented by the matrix R, where $R={R}_{cp}=\left[{0}_{M\times {L}_{gi}}{I}_{M}\right]$ for CP systems and R= R_{ zp }= I_{ P }for ZP systems.
Finally, the spacetime decoding is performed by first stacking the received signals over two consecutive symbol periods, as:
where Q = M for CP systems and Q = P for ZP case, P_{rx} ∈ ℝ^{Q×Q}is a permutation matrix whose design depends on the overall system, and in conjunction with P_{tx} it is used to perform symbol decoupling in the received STBC symbols, as will be detailed below. F_{ Q }is a Q × Q matrix that implements a Qpoint discrete Fourier transform, normalized such that, ${F}_{Q}^{H}{F}_{Q}={F}_{Q}{F}_{Q}^{H}={I}_{Q}.$
Using (2) and (3), we can rewrite (4) as
where
Note that the term η(i) in (2), which accounts for IBI in nonZP guard interval systems, is removed by the joint operation of matrices T and R. It should be stressed that the guard interval length must be at least the channel order in order to avoid IBI, i.e., L_{ gi } ≥ L  1 [23].
The choice of ZP or CP as the guard interval is convenient, since it allows us the use of the following well known properties:
(p1) CP case: R_{ cp }H_{ j }(2i)T_{ cp }reduces to a circulant matrix, ${\mathbb{H}}_{j}\left(2i\right)$, of dimension M × M.
(p2) ZP case: R_{ zp }H_{ j }(2i)T_{ zp }is equivalent to ${\mathbb{H}}_{j}\left(2i\right){T}_{zp}$ where ${\mathbb{H}}_{j}\left(2i\right)$ is a circulant matrix of dimension P × P. The equivalence is due to R_{ zp }= I_{ P }and the structure of T_{ zp }.
(p3) A circulant matrix ${\mathbb{H}}_{j}\left(2i\right)$ of dimension Q × Q, as in (p1) and (p2), can be decomposed as ${\mathbb{H}}_{j}\left(2i\right)={F}_{Q}^{H}{\Lambda}_{j}\left(2i\right){F}_{Q}$ and ${\mathbb{H}}_{j}^{H}\left(2i\right)={F}_{Q}^{H}{\Lambda}_{j}^{*}\left(2i\right){F}_{Q}$, where Λ _{ j }(2i) is a diagonal matrix whose entries are the frequency response of the transmission channel h_{ j }(2i), i.e., ${\Lambda}_{j}\left(2i\right)=\mathtt{\text{diag}}\left({\stackrel{\u0303}{F}}_{Q\times L}{h}_{j}\left(2i\right)\right)$, where ${\stackrel{\u0303}{F}}_{Q\times L}$ is a Q × L matrix formed with the first L columns of the matrix that implements the (nonnormalized) Qpoint discrete Fourier transform.
In order to decouple the transmitted symbols and to provide FDE capabilities to the system, we choose P_{tx} and P_{rx} as in [21], where they are drawn from a set ${\left\{{P}_{J}^{\left(n\right)}\right\}}_{n=0}^{J1}$, where J is the dimension of P. Each ${P}_{J}^{\left(n\right)}$ performs a reverse cyclic shift that depends on n when applied to a J × 1 vector. This set of matrices has two useful properties:
(p4) Pre and postmultiplying a circulant matrix, ${\mathbb{H}}_{j}\left(2i\right)$, by ${P}_{J}^{\left(n\right)}$ yields ${H}_{j}^{T}\left(2i\right)$, i.e., ${P}_{J}^{\left(n\right)}{H}_{j}\left(2i\right){P}_{J}^{\left(n\right)}={\mathbb{H}}_{j}^{T}\left(2i\right)$ and ${P}_{J}^{\left(n\right)}{\mathbb{H}}_{j}^{*}\left(2i\right){P}_{J}^{\left(n\right)}={\mathbb{H}}_{j}^{H}\left(2i\right)$[21].
(p5) ${T}_{zp}{P}_{M}^{\left(0\right)}={P}_{P}^{\left(M\right)}{T}_{zp}$[22].
Then, if for CP systems we choose ${P}_{rx}={P}_{tx}={P}_{M}^{\left(0\right)}$ and using (p1), (p4) and (p3) we get:
where V= F_{ M }GCρ.
In the same way, for ZP systems we can choose ${P}_{rx}={P}_{P}^{\left(M\right)}$ and ${P}_{tx}={P}_{M}^{\left(0\right)}$, then using (p2), (p5), (p4) and (p3) we have:
where V= F_{ P }T_{ zp }GCρ.
Now, using the results (8)(11), the spacetime decoded vector in (5), can be expressed as
where ${\Lambda}_{j}\left(2i\right)=\mathtt{\text{diag}}\left({\stackrel{\u0303}{F}}_{Q\times L}{h}_{j}\left(2i\right)\right)\phantom{\rule{2.77695pt}{0ex}}\left(j=0,1\right)$ and V is the Q × K matrix defined in connection with (8)(11).
Note that the noise vector $\stackrel{\u0304}{n}\left(i\right)$ in (5) and (12) is still Gaussian with zero mean and covariance matrix N_{0}I_{2Q}.
3 Receiver Design
Assuming that perfect channel estimation is performed at the receiver, we can define an orthogonal matrix $U\left(2i\right)=\Lambda \left(2i\right)\left({I}_{2}\otimes {\Lambda}_{01}^{1}\left(2i\right)\right)$ of dimension 2Q × 2Q, where
Then, U^{H}(2i)Λ(2i) = I_{2} ⊗ Λ_{01}(2i), and
thus, U(2i) decouples the received vector allowing s(2i) and s(2i + 1) to be demodulated separately from:
where ${U}^{H}\left(2i\right)\stackrel{\u0304}{n}\left(i\right)={\left[{\stackrel{\u0304}{n}}_{0}^{T}\left(2i\right)\phantom{\rule{1em}{0ex}}{\stackrel{\u0304}{n}}_{1}^{T}\left(2i+1\right)\right]}^{T}$. As U(2i) is an orthogonal matrix, it does not alter the statistical properties of the noise vector $\stackrel{\u0304}{n}\left(i\right)$.
With z_{ j }(i) (j = 0, 1), as in (15), symbol detection can adopt different approaches as illustrated next.
3.1 Multiuser maximum likelihood (ML) detection
The optimal solution of the proposed system is to jointly detect the transmitted symbol of the different users within the transmitted vector, based on the received vector. Thus, the optimum receiver is a multiuser (MU) maximum likelihood (ML) receiver, implemented, in this case, by the minimum distance receiver, such that, for j = 0, 1:
As shown in the appendix, the maximum diversity gain G_{ d } = 2L, is attained for this receiver when Λ_{01}(2i)Ve^{2} ≠ 0, ∀e≠ 0. Then, a sufficient condition for maximum diversity gain, is to guarantee that, for any e≠ 0, at least L elements from Ve are different from zero.
Proof: In the following, we drop the time index to simplify the notation. Let λ_{01,l}be the real and nonnegative l th element of the diagonal of Λ_{01} and ${e}_{l}^{\prime}$ the l th element of Ve, then:
where ${\lambda}_{01,l}^{2}=\phantom{\rule{2.77695pt}{0ex}}{\lambda}_{0,l}{}^{2}+{\lambda}_{1,l}{}^{2}$, with λ_{j,l}(j = 0, 1) being the lth element of ${\stackrel{\u0303}{F}}_{Q\times L}{h}_{j}\left(j=0,1\right)$. Since the Qpoints discrete Fourier transform of a vector of size L can have at most L  1 zero elements, it then results that at most L  1 values of λ_{01,l}can be equal to zero. Then, if ${e}_{l}^{\prime}=V\phantom{\rule{2.77695pt}{0ex}}e$ has at least L elements different from zero, we have that ${\sum}_{l=0}^{Q1}{\lambda}_{01,l}^{2}{e}_{l}^{\prime}{}^{2}\phantom{\rule{2.77695pt}{0ex}}\ne 0$ and the maximum diversity gain is achieved. ■
3.1.1 ZP systems
For the ZP systems, V= F_{ P }T_{ zp }GCρ. Now using $V\phantom{\rule{2.77695pt}{0ex}}e={\stackrel{\u0303}{F}}_{P\times M}{e}_{0}$, where ${\stackrel{\u0303}{F}}_{P\times M}={F}_{P}{T}_{zp}$ is P × M matrix formed with the first M columns of F_{ P }and e_{0} = GCρe is a Mdimensional vector, if the user codes are linearly independent, and provided that G is nonsingular, then GC is full column rank and therefore e_{0} ≠ 0 for any e≠ 0. Again, since the Ppoints discrete Fourier transform of a sequence of M points can have at most M  1 zeros, then ${\stackrel{\u0303}{F}}_{P\times M}{e}_{0}$ has at least P  (M  1) = L + 1 elements different from zero, thus maximum diversity gain is achieved.
So, for ZP systems, the maximum diversity gain is achieved independent of the choice of the user codes, the precoding matrix and the number of users on the system, provided that GC is full column rank.
3.1.2 CP systems
For CP systems V= F_{ M }GCρ, then, in general, we cannot guarantee that the maximum diversity gain is achieved by the system. However, proper choices of the precoding matrix and spreading codes, can lead to full diversity gain. For example, if F_{ M }GC is a Vandermonde matrix (e.g., $G={F}_{M}^{H}$ and the codes c_{ k }are columns of a Vandermonde matrix^{1}), and GC is full column rank, then for any K dimensional vector e≠ 0, Ve has at most K  1 zeroes, or equivalently, it has at least M  K + 1 elements different from zero. Then, maximum diversity gain is achieved provided that K ≤ M  L + 1.
3.2 Minimum mean squared error single user detection
Using z_{ j }(i) (j = 0, 1), from (15), the k th user minimum mean squared error (MMSE) receiver for each symbol, w_{k,j}, is obtained by minimizing the meansquared error criterion
whose solution is given by [24]:
where ${R}_{{z}_{j}{z}_{j}}=\mathtt{\text{E}}\left[{z}_{j}\left(i\right){z}_{j}^{H}\left(i\right)\right]$ and ${p}_{{z}_{j}{s}_{k}}=\mathtt{\text{E}}\left[z\left(i\right){s}_{k}^{*}\left(2i+j\right)\right]$.
Symbol detection is performed for j = 0, 1 as:
where disc{n} returns the symbol constellation closer to n.
From (15) is easy to conclude that
and ${p}_{{z}_{0}{s}_{k}}={p}_{{z}_{1}{s}_{k}}={\u1ebc}_{s}{\Lambda}_{01}\left(2i\right){v}_{k}$, where v_{ k }is the k th column of V. It follows that w_{k,0}= w_{k,1}.
As we assume downlink transmission, in practical situations only the k th column of V is known to the k th user receiver. Then, an iterative procedure, like the recursive least squares (RLS) algorithm [24] or conjugated gradient algorithm [25], must be used to compute the desired solution (19).
4 Frequency domain equalization
From (15) one can see that FDE can be performed by applying an onetap chipequalizer to each component of the vectors z_{ j }(i) (j = 0, 1). Such operation is represented by the premultiplication of the vectors z_{ j }(i) (j = 0, 1) by a diagonal matrix Q(2i) ∈ ℂ^{Q×Q}, whose elements are the weights of the onetap chipequalizer.
Symbol detection is then performed over the frequency equalized and despread signal:
where disc{n} is the vector whose components are the symbols of the signal constellation closer to the components of vector n and (·)^{†} represent the MoorePenrose matrix inverse. In order to avoid high computational complexity algorithms, it is desired to have V^{†} = V^{H}, which is true, for example, if G is an orthogonal matrix and the spreading codes are orthogonal.
As in the downlink the receiver is only interested in the k th user, symbol detection is performed as
where v_{ k }is the k th column of V, and we assume that V^{†} = V^{H}.
Now, let us consider the noiseless part of the decision variable, ${v}_{k}^{H}Q\left(2i\right){z}_{j}\left(i\right)$. For the CP case we have
and for the ZP case
where $\stackrel{\u0303}{I}\left(2i\right)=Q\left(2i\right){\Lambda}_{01}\left(2i\right)$ is a diagonal matrix. Note that in general $\stackrel{\u0303}{I}\left(2i\right)\ne {I}_{Q}$, and thus ${\stackrel{\u0303}{I}}_{CP}\left(2i\right)={F}_{M}^{H}\stackrel{\u0303}{I}\left(2i\right){F}_{M}$ and ${\stackrel{\u0303}{I}}_{ZP}\left(2i\right)={F}_{P}^{H}\stackrel{\u0303}{I}\left(2i\right){F}_{P}$ are circulant matrices that introduce code distortion and interchip interference in the equalization process.
So, one can design the precoding matrix, G, in order to mitigate such undesired effects. Two simple approaches are commonly used. The first one is to choose the identity matrix as the precoding matrix, resulting into the wellknown singlecarrier block transmission systems. In both cases, CP and ZP systems, single carrier modulation presents code distortion and interchip interference, produced by the circulant matrix $\stackrel{\u0303}{I}\left(2i\right)$.
The second approach is to choose $G={F}_{M}^{H}$, leading the socalled multicarrier block transmission systems. In this case, for CP systems, the noiseless decision variable (24) reduces to ${c}_{k}^{H}\stackrel{\u0303}{I}\left(2i\right)C\phantom{\rule{2.77695pt}{0ex}}s\left(2i+j\right)$, that states the absence of interchip interference (as $\stackrel{\u0303}{I}\left(2i\right)$ is a diagonal matrix), however, code distortion is present.
We next consider the design of the onetap equalization matrix Q(2i) = diag(q_{0}(2i),..., q_{ Q } (2i)) following standard approaches.
4.1 MRC single user detection
This receiver weights each subchannel by its respective complex conjugate equivalent subchannel coefficient, leading in the present case to
where λ_{01,l}(2i) is the real and nonnegative l th element of the diagonal of Λ_{01}(2i) and λ_{j,l}(2i) (j = 0, 1) is the l th element of ${\stackrel{\u0303}{F}}_{Q\times L}{h}_{j}\left(2i\right)$.
Whit this choice, Q(2i) = Λ_{01}(2i) and then $\stackrel{\u0303}{I}\left(2i\right)={\Lambda}_{01}^{2}\left(2i\right)$. Thus, this equalize the spreading code distortion introduced by the transmission channel and may enhance the multiple access interference (MAI).
4.2 Zero forcing (ZF) singleuser receiver
Zero forcing applies channel inversion so that $Q\left(2i\right)={\Lambda}_{01}^{1}\left(2i\right)$ and $\stackrel{\u0303}{I}\left(2i\right)={I}_{Q}$. Thus, this equalizer eliminates code distortion, and for the case of multicarrier with CP transmissions systems also removes the interchip interference and can, therefore, eliminate MAI (if user codes are orthogonal). The equalizer coefficients are chosen as
The main drawback of this equalizer is that when λ_{01,l}(2i) ≈ 0, the noise effects are enhanced.
4.3 MMSE singleuser detection
The equalization coefficient based on the MMSE criterion for the proposed system results in [26]:
where σ^{2} is the variance of the noise. We have $Q\left(2i\right)={\left({\Lambda}_{01}^{2}\left(2i\right)+{\sigma}^{2}I\right)}^{1}{\Lambda}_{01}\left(2i\right)$ and $\stackrel{\u0303}{I}\left(2i\right)={\left({\Lambda}_{01}^{2}\left(2i\right)+{\sigma}^{2}I\right)}^{1}{\Lambda}_{01}^{2}\left(2i\right)$, thus, as with the MRC this equalizer does not eliminate code distortion. However, it offers a good trade off between code distortion reduction and noise enhancement.
5 Simulation results
We consider the downlink scenario of the proposed STBC CDMA transmission system with two transmit and one receive antenna. The data symbols are binary phaseshift keying (BPSK) modulated and spreading by a length M = 16 spreading code. The system is loaded with K = 4 users and two types of guard intervals are considered in combination with two different choices of precoding matrix, G= I_{ M }and $G={F}_{M}^{H}$, leading to four different systems, as shown in Table 1 where MC stands for multicarrier and SC for single carrier.
In all the experiments, we obtain the average biterror rate (BER) versus E_{ b } /N_{0} (E_{ b } is the energy per bit of the desired user) as the performance measure. In each experiment we run 30, 000 Monte Carlo realizations, with 2,000 symbols transmitted per run. The transmitted symbols are randomly generated and it is assumed that the first 500 symbols are used as the train sequence for the equalizers.
5.1 Performance of the transceiver
In the first experiment we compare two different transceivers, the one proposed in [4], adapted to the systems in Table 1 as reported in [16], denoted as STBC in the figures, and the transceiver proposed in this work. In order to allow a fair comparison between the transceivers, we employ a MMSE type of receiver (see Section 3.2), with a conjugated gradient [25] adaptive implementation. For comparison purposes, the performance of a single user MMSE type of receiver for a block CDMAbased system with only one antenna at the transmitter and one antenna at the receiver, denoted as SISO, is also presented.
The channel from each transmit antenna to the receiver is modeled here as a timevariant FIR filter, with coefficients given by h_{j,l}(i) = p_{j,l}α_{j,l}(i) (j = 1, 2 and l = 0, 1, 2,..., L  1) where α_{j,l}(i) is obtained with Clarke's model [27]. This procedure corresponds to the generation of independent sequences of correlated unit power complex Gaussian random variables $\left(\mathtt{\text{E}}\left[\left{\alpha}_{j,l}^{2}\left(i\right)\right\right]=1\right)$ with the path weights p_{j,l}normalized so that ${\sum}_{l=0}^{L1}{p}_{j,l}{}^{2}=1$. Here, the channel coefficients are kept constant during twosymbol period and each channel has L = 4 transmission paths of equal weight, i.e., p_{j,l}^{2} = 1/L, l = 0, 1, 2, 3. The guard interval length is L_{ gi } = 3. The results depend on the normalized Doppler frequency (f_{ d }T), where f_{ d } is the Doppler frequency and T is the duration of two symbols. A value f_{ d }T = 0.001 was assumed in all simulations. The systems use Hadamard codes of length M = 16. In each run, the user codes were randomly chosen, but we avoid the use of the first and second Hadamard code, which corresponds to the first and second column of the Hadamard matrix.
For the proposed transceiver, before the MMSE single user detection stage, decoupling of the received vector must be performed, as shown in (14), (15). For this purpose, two channel estimates were used. The first one is an ideally estimated channel, denoted as Proposed in the figures, while the second one, denoted 'Proposed w/Error' in the figures, is a noisy channel estimate, ${\widehat{h}}_{j}\left(2i\right)={h}_{j}\left(2i\right)+{\zeta}_{j}\left(2i\right)$, where ζ_{ j } (2i) is a complex white Gaussian noise vector with zero mean and covariance matrix $\mathtt{\text{E}}\left[{\zeta}_{j}\left(2i\right){\zeta}_{j}^{H}\left(2i\right)\right]={\sigma}_{\zeta}^{2}{I}_{L}$. In this experiment, we set the mean squared relative error of the channel estimate to 10 dB. Note, that the conjugated gradient algorithm that follows the decoupling does not need the channel knowledge to reach the MMSE solution.
Figures 2 and 3 show the BER results versus E_{ b } /N_{0} for CP and ZP systems, respectively. As can be noted, for multicarrier systems, the proposed receiver performs better than the STBC receiver, even in the presence of channel estimation errors. When comparing multicarrier systems versus single carrier systems, we observe the BER floor for single carrier systems. This is due, in part, to the better recovering of the spreading codes performed by the multicarrier systems.
In the case of SC CDMA ZP system of the Figure 3, it was observed that the receiver restored some of the orthogonality between user codes, resulting in good signal to interferenceplusnoise ratio, which in turn results in enhanced BER. For the case of SC CDMA CP of the Figure 2, the same receiver could not restore the orthogonality between user codes, resulting in poor signal to interferenceplusnoise ratio and then in worst BER.
5.2 Performance of FDE algorithms
In this experiment we compare different FDE algorithms for the proposed structure (see Section 4): MRC single user receiver (FDE MRC), zero forcing single user receiver (FDE ZF), and MMSE single user receiver (FDE MMSE). We use the same timevariant channel used in the first experiment and we assume that the channel was perfectly estimated. The system uses Hadamard codes of length M = 16 and were chosen as in the first experiment. Results for a matched filter single user algorithm (MF) are also shown in the figures. In this algorithm the receiver filter is matched to the user spreading code at the receiver.
Figures 4 and 5 show the BER results versus E_{ b } /N_{0} for CP and ZP systems, respectively. As expected, the FDE MMSE receiver outperforms the FDE ZF, the FDE MRC and the MF receivers. Furthermore, all these receivers have similar computational complexity.
Also note that for FDE ZF and FDE MMSE receivers, multicarrier systems perform better than for single carrier systems due to the small and even null interchip interference of multicarrier systems with FDE, as stated before in Section 4.
5.3 Performance with different channel covariance matrix
In the third experiment we compare the performance of the systems for three different transmission channels, all with L = 4 paths but different channel covariance matrices.
The process to generate the timevariant channel for each user is as in the first experiment. The first channel assumes uncorrelated transmission paths of equal weight, i.e., p_{j,l}^{2} = 1/L, j = 1, 2 and l = 0, 1, 2, 3, as in the first experiment. This type of channel is named Uniform in the performance curves. For the second channel we assume that the average power of each path decays exponentially, such that ${p}_{j,l}{}^{2}={\sigma}_{0}^{2}\mathtt{\text{exp}}\left(l\right)$, l = 0, 1, 2, 3, and ${\sigma}_{0}^{2}=1\mathrm{exp}(1)/(1{e}^{L})$[28]. This channel is named as Exponential in the figures. Finally, the third channel results from the multiplication of a matrix $\stackrel{\u0303}{K}$ by the channel vector generated as in the Exponential channel case. Matrix $\stackrel{\u0303}{K}$ was randomly generated and normalized such that the average power in each path is kept constant. This third channel is termed Correlated in the figures.
As in the two first experiments, we set a system with K = 4 users using Hadamard codes of length M = 16 and the guard interval length is L_{ gi } = 3. FDE with zero forcing single user receiver (FDE ZF) was employed.
The BER results for CP systems are shown in Figure 6, and ZP systems results are shown in Figure 7. As expected, the receiver performs worst for Correlated channels due to smaller coding gain that the systems exhibit for this type of channels (see Appendix).
5.4 Performance for different user codes
In this experiment we assess the BER for different codes. Four types of user codes are considered: Pseudonoise sequences (PN), WalshHadamard, Vandermonde and ZadoffChu (ZC) codes [29, 30], all of length 16. Vandermonde codes are taken from the columns of the Vandermonde matrix:
where $\frac{1}{\sqrt{\varpi}}$ normalizes the codes, such that ${c}_{k}^{H}{c}_{k}=1$. The parameters ε_{ j } can be chosen as equispaced points on the unit circle by setting ${\epsilon}_{j}=\mathrm{exp}(\sqrt{1}j(2\pi /M))$, j = 0, 1, ⋯, M  1 [31]. In this system ${\epsilon}_{j}=\mathrm{exp}(\sqrt{1}j(2\pi /M))$, j = 0, 1, ⋯, M  1.
We consider a scenario with K = 4 users and timevariant channels as in the first two experiments. In each run, the user codes were randomly chosen. BER results are shown in Figures 8 and 9 for the zero forcing single user FDE receiver (FDE ZF). We recall that for the multicarrier transmission system with CP as guard interval, the use of Vandermonde codes allows the maximum diversity gain when multiuser maximum likelihood detection is employed. Note, however, that the considered FDE ZF receiver (which is suboptimum) also exploits the diversity gain of the system in this case. For single carrier transmission system with CP as guard interval, it can be verified that the use of Vandermonde codes leads to a single carrier TDMA transmission system with ZP (in this case, $V={F}_{M}^{H}C={I}_{M\times K}$, where I_{M×K}is a truncated identity matrix), and the results in Figure 8 indicate that the FDE was not able to exploit the diversity gain.
The FDE ZF receiver for systems using Hadamard codes does not exploit the diversity gain of the system, but presents a better coding gain, as expressed by the offset on the BER curve. On the other hand, the use ZC or PN codes yields to better exploiting the coding gain of the systems, as shown by the bigger slope of the BER curve.
6 Conclusion
This work proposed a FDE STBC CDMAbased transmission system. The FDE algorithms used with single user detection resulted in a simple receiver design with reduced computational complexity. Simulations results have shown good performance in terms of BER when compared to previously proposed STBC CDMA systems. Diversity and coding gain analysis of the proposed structure was performed and conditions to achieve their maximum values, with multiuser maximum likelihood detection, were identified.
Note
^{1}See equation (29).
Appendix
Diversity and Coding Gain Analysis
Let us drop the time index and use only one equation from (15). If for a given detection system P(ε_{ k } h_{0}, h_{1}) is the conditional error probability of user k, then
where P_{ B } (·) is the block (block of symbols of K users) error probability and is given by
where χ represents the set of possible values for s and $P\left(\widehat{s}s,\phantom{\rule{2.77695pt}{0ex}}{h}_{0},\phantom{\rule{2.77695pt}{0ex}}{h}_{1}\right)$ is the conditional probability of the event that the detected block is $\widehat{s}$ when the transmitted block is $s\left(\widehat{s}\ne s\right)$.
Using standard procedures, we arrive, for equiprobable symbols at
where χ denotes the cardinality of χ, $e=\widehat{s}s$, ϑ(e) is the number of occurrences of a given vector e, when $\widehat{s}$ and s span χ, f(e) = E [exp(Λ_{01}Ve^{2}γ)] and $\gamma ={\u1ebc}_{s}\u2215S{N}_{0}$.
Let $K=\mathtt{\text{E}}\left[{h}_{j}{h}_{j}^{H}\right]$ be the covariance matrix of the channel vector h_{ j }(j = 0, 1), where it was assumed that the channels h_{ j }are identically distributed. Since K is square Hermitian, it always admits spectral decomposition, i.e., K= Ω D Ω^{H}, where D is a L × L diagonal matrix whose entries are the eigenvalues of K and Ω is a unitary matrix whose columns are the normalized eigenvectors of K.
If we assume that K is nonsingular and introduce the channel vector ${\stackrel{\u0303}{h}}_{j}={D}^{1\u22152}{\Omega}^{H}{h}_{j}$, which by construction has a identity covariance matrix, then we can write
where
If the channels h_{0} and h_{1} are modeled as statistically independent complex gaussian vectors, then after some algebraic manipulation, we obtain for f (e) in (32):
where λ_{ l } (e) are the eigenvalues of Γ (e) and κ(e) is the rank of Γ (e). Substituting (37) into (32) we arrive at
From (34) we have that if K is full rank, then κ(e) = rank(Γ(e)) = rank(Γ_{0}(e)), and its maximum value is L(Γ_{0}(e) is an L × L matrix). The system is said to achieve maximum diversity gain if min_{e≠0}κ(e) = L, and thus, κ(e) = L, ∀e≠ 0. That is, Γ(e) is a full rank matrix for any e≠ 0. We note from (33) that this is equivalent to have Λ_{01}Ve^{2} ≠ 0 for any vector e≠ 0. Under this condition we have
where det(·) denotes determinant.
We remark that since the diversity gain depends only on Γ_{0}(e) given by (35), it does not depend on the channel covariance matrix K(provided that is nonsingular). It depends, however, on the particular system employed and the relative power of the users.
From (34) we have that if K is full rank,
It then results from (39) that for a system that attains maximum diversity gain:
with G_{ c } = g_{0}[det(K)]^{1/L}, where g_{0} = [min_{e≠0}det(Γ_{0}(e))]^{1/L}, being the coding gain. As the trace of K= E [h^{2}] = 1 then det(K) < 1 and the maximum value of det(K) is obtained when all the eigenvalues of K are the same, i.e., λ_{K,i}= 1/L (i = 1, 2, ⋯, L). In this case det(K) = (1/L)^{L}, and therefore, for a system transmitting in a channel with L multipath components
equality (maximum coding gain) is achieved, for example, when channel has uncorrelated equal power coefficients (K= L^{1}I).
Abbreviations
 BER:

bit error rate
 CIBS:

chipinterleaved blockspread
 CP:

cyclic prefix
 DSCDMA:

directsequence code division multiple access
 FDE:

frequency domain equalization
 FIR:

finiteimpulse response
 IBI:

interblock interference
 ISI:

intersymbol interference
 MMSE:

minimum mean squared error
 MRC:

maximal ratio combiner
 OFDM:

orthogonal frequency division multiplexing
 RLS:

recursive least squares
 SCFDE:

singlecarrier frequencydomain equalization
 STBC:

spacetime blockcoded
 ZP:

zero padding.
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Acknowledgements
This study was supported by the Brazilian Council for Scientific and Technological Development (CNPq).
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Medina, C.A., SampaioNeto, R. Frequency domain equalization spacetime blockcoded CDMA transmission system. J Wireless Com Network 2011, 80 (2011). https://doi.org/10.1186/16871499201180
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Keywords
 frequency domain equalization (FDE)
 spacetime blockcodes (STBC)
 single carrier CDMA transmission systems
 multicarrier CDMA transmission systems