Novel low-PAPR parallel FSOK transceiver design for MC-CDMA system over multipath fading channels

2.1. Single substream transmission of each user

As shown in Figure 1a, there are P substreams of the k th user being transmitted simultaneously. For the i th symbol block and the p th substream, the transmitted data block is denoted as ${\mathbf{\text{s}}}_{i,p}^k={\left[s_{i,p}^k\left(0\right)\cdot \cdot \cdot s_{i,p}^k\left(R-1\right)s_{i,p}^k\left(R\right)s_{i,p}^k\left(R+1\right)\right]}^T$, which has R + 2 bits. Hence, the overall transmitted data block over P substreams is ${\mathbf{\text{s}}}_i^k={\left[{\mathbf{\text{s}}}_{i,1}^{k^T}{\mathbf{\text{s}}}_{i,2}^{k^T}\cdot \cdot \cdot {\mathbf{\text{s}}}_{i,p}^{k^T}\cdot \cdot \cdot {\mathbf{\text{s}}}_{i,p}^{k^T}\right]}^T$ with a total of P(R + 2) bits. The first R bits $\left[s_{i,p}^k\left(0\right)s_{i,p}^k\left(1\right)\cdot \cdot \cdot s_{i,p}^k\left(R-1\right)\right]$ of the p th substream are mapped on to one of the N codes, where N = 2 ^R . Moreover, as shown in Figure 2, the FSOK code set forms an N × N orthogonal matrix C = [c₀...c _m ...c_N-1], where the m th code vector is expressed as

with c₀ being the Chu sequence [19], f _m being an N-point frequency-shift sequence, and Θ being the element-by-element multiplication. Thus, the n th element of c _m is c_m,n= f_m,nc_0,nwhere f _m,n = exp{-j2π(n-1)m/N} and c_0,n= exp{jπ(n-1)²q/N}, with q and N being relatively prime. The FSOK Chu sequences retain the mutual orthogonality property ${\mathbf{\text{c}}}_m^H{\mathbf{\text{c}}}_n=N{\delta }_{m-n}$, and preserve the low-PAPR property in both the frequency and time domains. It is noted that the same orthogonal code matrix C is used for all K users to map their transmitted data. The MAI problem will be addressed later in Section 2.3.

where ${\mathbf{\text{c}}}_{m_i,p}^k$ is the m _i th mapped FSOK Chu sequence in (1), with m _i ∈ {0, 1,..., N-1} being the index mapped from the first R bits of the i th symbol.

2.2. Parallel substream transmission of each user

2.3. Subcarrier assignment for multiuser uplink transmission

where Q ^H denotes the NPK × NPK IFFT matrix. It is noted that, because of the constant modulus feature of the composite MC-CDMA FSOK Chu sequence in both the time and frequency domains, the proposed SISO MC-CDMA uplink system has a low-PAPR property. Finally, to prevent any interblock interference, a CP is inserted into each transmitted data block ${\mathbf{\text{t}}}_i^k$, with the length of the CP set larger than the length of the multipath channel response.

3.1. Channel and received signal model

where H ^k is the channel impulse response (CIR) matrix of the k th user and n _i is the additive white Gaussian noise (AWGN) vector with zero-mean and variance ${\sigma }_n^2$. The NPK × NPK channel matrix H ^k is a circulant matrix formed by cyclically shifting the zero-padded length-NPK vector of the k th user CIR ${\mathbf{\text{h}}}^k={\left[h_0^k{h}_1^k\cdot \cdot \cdot h_{L-1}^k\right]}^T$, where L is the channel delay spread length. Under slow fading, we assume that h ^k is invariant within a packet, but may vary from packet-to-packet. Since H ^k is a circulant matrix, it has the eigendecomposition H ^k = Q ^H Λ ^k Q, where Q is the orthogonal FFT matrix. Further, Λ ^k is the diagonal element given by the NPK-point FFT of h ^k , i.e., Λ ^k = diag(Qh ^k ) with diag{·} being the diagonal matrix.

3.2. Development of parallel FSOK MC-CDMA receiver

where $\stackrel{⌢}{\mathbf{\text{Q}}}$ is the NP × NP FFT matrix and ${\stackrel{⌢}{\mathbf{\text{C}}}}_p^{k^H}=diag{\left({\stackrel{⌢}{\mathbf{\text{Q}}}\stackrel{⌢}{\mathbf{\text{c}}}}_{0,p}^k\right)}^H$ can be pre-calculated from the FFT of the base repeated-modulated spreading sequence ${\stackrel{⌢}{\mathbf{\text{c}}}}_{0,p}^k$. Obviously, (17) indicates that by pairwisely multiplying the two FFTs of ${\stackrel{⌢}{\mathbf{\text{c}}}}_{0,p}^k$ and ${\mathbf{\text{z}}}_i^k$ and then taking the IFFT, we obtain the desired N correlator outputs $x_{m,p}^k\left(i\right)$ for m = 0, 1, ..., N-1. Moreover, when N is large, the computational complexity using (17) will be much lower than that associated with the original N-correlator bank in (15). Hence, the complexity (in number of complex multiplications) of the proposed despreader in (17) is reduced to O(N log₂N).

Finally, the QPSK slicer is used for the maximal value of $\mathsf{\text{Re}}\left\{x_{{\widehat{m}}_i^k,p}^k\left(i\right)\right\}$ and $\mathsf{\text{Im}}\left\{x_{{\widehat{m}}_i^k,p}^k\left(i\right)\right\}$ to detect the other two bits $\left[{ŝ}_{i,p}^k\left(R\right){ŝ}_{i,p}^k\left(R+1\right)\right]$ of the k th user, respectively. From (19), it is clear that a full frequency diversity gain is obtained for the p th substream of the k th user. As shown in Figure 1b, the data detection scheme can be extended to all the parallel substreams and all the simultaneous users with full diversity gain by employing the different repeated-modulated spreading sequence ${\stackrel{⌢}{\mathbf{\text{c}}}}_{m,q}^k$ and different subcarrier extractions, respectively.

where ${\sigma }_s^2$, ${\sigma }_n^2$ are the desired signal and noise power, respectively, and ${\mathbf{\text{c}}}_{m,p}^{k^H}{\Lambda }^{k^H}{\Lambda }^k{\mathbf{\text{c}}}_{m,p}^k$ represents the processing gain because of frequency diversity combining and despreading. From the MFB in (21), an error performance bound of the k th user can be evaluated and used for the verification of the superior performance of the proposed QPSK-FSOK transceiver.

4.1. MISO STBC transmitter

where the superscripts 1 and 2 are used to denote the 1st and 2nd transmit antennas, respectively.

4.2. MISO FSOK MC-CDMA receiver

where the MRC weight vectors are given by ${\mathbf{\text{V}}}_1^k={\left[{\bar{\Lambda }}^{k,1^T}{\bar{\Lambda }}^{k,2^H}\right]}^T$ and ${\mathbf{\text{V}}}_2^k={\left[{\bar{\Lambda }}^{k,2^T}-{\bar{\Lambda }}^{k,1^H}\right]}^T$; ${\widetilde{\mathbf{\text{n}}}}_i^k$ and ${\widetilde{\mathbf{\text{n}}}}_{i+1}^k$ are the post-MRC noise vectors; and ${\tilde{\Lambda }}^k={\bar{\Lambda }}^{k,1^H}{\bar{\Lambda }}^{k,1}+{\bar{\Lambda }}^{k,2^H}{\bar{\Lambda }}^{k,2}$ is a diagonal matrix with the (n, n)th element being $\mid {\bar{\Lambda }}^{k,1}\left(n,n\right){\mid }^2+\mid {\bar{\Lambda }}^{k,2}\left(n,n\right){\mid }^2$.

Similarly, from (13) in the SISO system, the linear receiver w ^k for the k th user can be used to equalize the channel effect, yielding the two consecutive data blocks, i.e., ${\overline{\mathbf{\text{z}}}}_i^k=diag{\left({\mathbf{\text{w}}}^k\right)}^H{\mathbf{\text{u}}}_i^k$ and ${\overline{\mathbf{\text{z}}}}_{i+1}^k=diag{\left({\mathbf{\text{w}}}^k\right)}^H{\mathbf{\text{u}}}_{i+1}^k$ with w ^k being ZF or MMSE vectors shown in (14). Next, to suppress the MSI, the i th and (i + 1)th equalized data blocks can be despread by the two independent repeated-modulated spreading sequences ${\stackrel{⌢}{\mathbf{\text{c}}}}_{m_1,p}^k$ and ${\stackrel{⌢}{\mathbf{\text{c}}}}_{m_2,p}^k$ to get the two outputs $x_{m_1,p}^k\left(i\right)={\stackrel{⌢}{\mathbf{\text{c}}}}_{m_1,p}^{k^H}{\overline{\mathbf{\text{z}}}}_i^k$ and $x_{m_2,p}^k\left(i+1\right)={\stackrel{⌢}{\mathbf{\text{c}}}}_{m_2,p}^{k^H}{\overline{\mathbf{\text{z}}}}_i^k$ for m₁, m₂ = 0, 1, ..., N, which is similar to (15). Finally, the ML algorithm in (18) can be used to demap and detect the two consecutive MISO QPSK-FSOK symbols of the p th substream, i.e., ${\left[{ŝ}_{i,p}^k\left(0\right){ŝ}_{i,p}^k\left(1\right)\cdots {ŝ}_{i,p}^k\left(R+1\right)\right]}^T$ and ${\left[{ŝ}_{i+1,p}^k\left(0\right){ŝ}_{i+1,p}^k\left(1\right)\cdots {ŝ}_{i+1,p}^k\left(R+1\right)\right]}^T$. For the high SNR scenario, if correct decisions regarding the two frequency shifts have been made, i.e., ${\widehat{m}}_1^k=m_i$ and ${\widehat{m}}_2^k=m_{i+1}$, then the maximizing values of the despreader for the i th and (i + 1)th symbol blocks are given by $x_{{\widehat{m}}_1^k,p}^k\left(i\right)\stackrel{\begin{array}{c}\mathsf{\text{For}}\\ \mathsf{\text{High}}\mathsf{\text{SNR}}\end{array}}{\approx }NP{d}_{i,p}^k$ and $x_{{\widehat{m}}_2^k,p}^k\left(i+1\right)\stackrel{\begin{array}{c}\mathsf{\text{For}}\\ \mathsf{\text{High}}\mathsf{\text{SNR}}\end{array}}{\approx }NP{d}_{i+1,p}^k$ for p = q. In such a way, we can detect all substreams for all users with full spatial and frequency diversity gain. Moreover, for the downlink system, the mobile user can acquire the spatial and frequency diversity gain from the BS with a two-antenna transmission downlink.

Derivation of the mutual orthogonality property for the repeated-modulated spreading sequence ${\widehat{\mathbf{\text{c}}}}_{m_i,p}^k$

Proof of the constant envelope of transmit signal ${\mathbf{\text{t}}}_i^k$

which indicates that if $\mid d_{i,m}^1\mid$ is constant (MPSK), the transmit signal ${\mathbf{\text{t}}}_i^1$ has a constant envelope.

For the other users with k = 2, 3,...,K, it is noteworthy that the same constant envelope result can be derived by referring to the procedures listed in (32)-(41).