3.1. Channel and received signal model
After the transmitted signal is passed through the multipath channel, the circular convolution between signal and channel is induced by the use of a CP. Thus, after removing the CP for the multiuser scenario, the i th received time-domain data block at base station can be expressed by
(10)
where Hk 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 . The NPK × NPK channel matrix Hk is a circulant matrix formed by cyclically shifting the zero-padded length-NPK vector of the k th user CIR , where L is the channel delay spread length. Under slow fading, we assume that hk is invariant within a packet, but may vary from packet-to-packet. Since Hk is a circulant matrix, it has the eigendecomposition Hk = QHΛkQ, where Q is the orthogonal FFT matrix. Further, Λk is the diagonal element given by the NPK-point FFT of hk , i.e., Λk = diag(Qhk ) with diag{·} being the diagonal matrix.
3.2. Development of parallel FSOK MC-CDMA receiver
The FSOK MC-CDMA receiver is developed based on the overall block diagram depicted in Figure 1b. The receiver is designed to detect the P parallel data substreams for the K decoupled users simultaneously. Its operation involves the following steps. First, taking the FFT of r
i
, the post-FFT received signal block is given by
(11)
where . Next, because of the interleaved subcarrier assignment, the post-FFT received signal y
i
with NPK × 1 vector can be divided into K length-NP vectors. For the k th user, the received vector is
(12)
where , such that Λ k (k, k) is the (k, k)th element of Λk and y
i,k
is the k th element of y
i
. Assuming that the channel response vector hk in (11) is perfectly estimated, a linear receiver for the k th user simply combines to obtain
(13)
where wk is the combiner weight vector. For the zero-forcing weight vector, , while in the high SNR scenario, . Thus, the normalized weight vector acts as the one-tap equalizer of the proposed MC-CDMA system without requiring a matrix inversion. Moreover, to combat the noise enhancement problem, we can apply the minimum mean square error (MMSE) weight vector for the linear equalizer, i.e.,
(14)
where SNR is the received signal-to-noise ratio. Following linear equalization, the equalized block data of the k th user can be despread by the m th repeated-modulated spreading sequence of the p th substream of the k th user, yielding the despread output as follows
(15)
for m = 0, 1,..., N - 1, where is the despread noise. In (15), for the high SNR scenario with 1/SNR approaching zero, the composite equalizer-channel matrix approximates the identity matrix. Therefore, in (15), the MSI can effectively be eliminated because of the orthogonality property in (5). Then, the despread output in (15) can be rewritten as
(16)
Moreover, the direct computation of the N correlation outputs in (15) requires O(N2) of complex multiplications. To alleviate this, an FFT/IFFT-based despreader is proposed, that is, employing the cyclic shift despreading property and some manipulation, we can express the correlation outputs as
(17)
where is the NP × NP FFT matrix and can be pre-calculated from the FFT of the base repeated-modulated spreading sequence . Obviously, (17) indicates that by pairwisely multiplying the two FFTs of and and then taking the IFFT, we obtain the desired N correlator outputs 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).
Next, the i th QPSK-FSOK symbol with (R + 2) bits of the p th substream of the k th user can be detected by the ML algorithm in [11]. Therefore, for the first R bits, the maximizing index of the despread data can be found by
(18)
Based on (18), we can detect the first R bits, i.e., , where the function dec2bin denotes the conversion of unsigned decimal numbers into binary digits. It is noted that for the high SNR scenario, if is a correct decision, i.e., is equal to m
t
in (2), the maximizing value of the despreader in (15) can be approximated as
(19)
Finally, the QPSK slicer is used for the maximal value of and to detect the other two bits 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 and different subcarrier extractions, respectively.
Through the above derivation, we have shown that the proposed transceiver can efficiently be realized and achieve MAI/MSI-free multiuser uplink transmission over multipath fading channel. Next, we verify its superior performance in terms of the matched filter bound (MFB). Assuming perfect MAI/MSI elimination, then each despread signal only contains its desired substream of the desired user and AWGN. Therefore, the matched filter's weight vector of the k th user is simply given by the composite signature of the frequency-domain channel response and spreading code sequence, i.e.,
(20)
Based on Equation 20, the maximized output SNR for the k th user can be obtained as
(21)
where , are the desired signal and noise power, respectively, and 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.