Abstract
A new collaborative blind multiuser detection scheme based on precoding and signal subspace estimation is proposed for binary phase-shift keying/direct-sequence code-division multiple-access signals. In this scheme, precoding the phase of the data enables separation and detection of a desired user’s signal from the received signal with the prior knowledge of only the precoding sequence, signature waveform and timing of the user of interest. In this approach, instead of assigning unique precoding sequence to each user, same precoding sequence is assigned to a group of users which enables group precoding and deprecoding operations. The proposed collaborative blind multiuser receiver consists of two stages: group deprecoding stage to minimize multiple-access interference due to multiple groups in the system and blind subspace multiuser detection stage to detect desired user data from the respective group. The simulation results confirm that the performance gain afforded by the proposed scheme can be substantial when compared to the conventional linear subspace blind detectors.
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Appendix
Appendix
Theorem 1
A linear zero-forcing detector \( \varvec{w}^{(g)} \) for detecting the data bits of user group g is derived such that, in the absence of noise, both the ISI and interference from other groups are completely eliminated at the detector output.
Proof
In the absence of noise, user group \( g \) is assumed to be desired user group. From (14) we get
Subscripts re and im are used to represent real and imaginary parts respectively. Considered that \( \varvec{b}^{(g)} \) and \( \varvec{s}^{(g)} \) are \( L \) users data vectors in the binary constellation and effective signature vectors respectively of user group \( g \), similarly \( \varvec{B}^{(g)} \) and \( \varvec{S}^{(g)} \) are \( (GL - L) \) users data vectors in the quaternary constellation and effective signature vectors respectively. Then without the loss of generality, we can write
where \( \varvec{b}^{(g)} = [\varvec{b}_{g1}^{T} \varvec{b}_{g2}^{T} \ldots \varvec{b}_{gL}^{T} ]^{T} \), \( \varvec{S}^{(g)} = [\varvec{s}^{(1)} \varvec{ s}^{(2)} \ldots \varvec{s}^{(g - 1)} \varvec{ s}^{(g + 1)} \ldots \varvec{s}^{(G)} ] \), \( \varvec{B}^{\left( g \right)} = \left[ {\varvec{b}_{p1}^{H} \varvec{ b}_{p2}^{H} \ldots \varvec{b}_{p(g - 1)}^{H} \varvec{b}_{p(g + 1)}^{H} \ldots \varvec{b}_{pG}^{H} } \right]^{H} \), and \( \varvec{s}^{(g)} = \left[ {\varvec{s}_{1q} \;\varvec{s}_{2q} \ldots \varvec{s}_{Lq} } \right], \)where \( \varvec{b}_{gl } \) and \( \varvec{s}_{lq} \) are the \( l \)th user’s row data vector and effective signature column vector respectively of user group \( g \). Also, \( \varvec{b}_{pu } \) are data vectors of user group \( u \) in the quarternary constellation. Thus the real and imaginary parts of \( \varvec{r}^{{\left( \varvec{g} \right)}} \) can be written as
substituting (25) in (23) we get,
rearranging the terms, we get
(27) is rewritten as
Suppose that a block of \( 2M + 1 \) received data samples are processed together. Then defined matrix \( \left[ {\varvec{b}^{\left( g \right)} \; \varvec{B}_{re}^{\left( g \right)} \; \varvec{B}_{im}^{(g)} } \right]^{T} \) is tall of size \( \left( {L + 2\left( {GL - L} \right)} \right) \times (2M + 1) \), which necessitates that \( \left( {2M + 1} \right) > (2GL - 1) \). The number of samples \( \left( {2M + 1} \right) \) should be chosen such that the above said matrix has full rank for linearly independent data vectors and precoding sequences. Thus, from (28) we can write
From (29) it is easy to see that \( imag\left( {\varvec{w}^{{\left( g \right)^{H} }} \;\varvec{s}^{\left( g \right)} } \right) = 0 \), \( imag\left( {\varvec{w}^{{\left( g \right)^{H} }} \;\varvec{S}^{\left( g \right)} } \right) = 0 \) and \( real(\varvec{w}^{{\left( g \right)^{H} }} \;\varvec{S}^{\left( g \right)} ) = 0 \), hence we can write
For the desired user group g, from (14), we can also write
substituting (25) in (31) we get,
Using (29) we can write (32) as
Taking norm on both sides after simplification, (33) implies
Using (31) we can write (34) as
It is easy to see from (35) that \( \varvec{w}_{re}^{{\left( g \right)^{T} }} \varvec{s}_{re}^{(g)} - \varvec{w}_{im}^{{\left( g \right)^{T} }} \varvec{s}_{im}^{(g)} \) is the matrix of size \( L \times L \). Thus (35) implies that \( \left\| {real\left( {\varvec{w}^{{\left( g \right)^{H} }} \varvec{s}^{(g)} } \right)} \right\| = 1 \)
Note: \( \left\|real\left( {\varvec{w}^{{\left( g \right)^{H} }} \varvec{s}^{(g)} } \right)\right\| \,= 1 \) and \( imag\left( {\varvec{w}^{{\left( g \right)^{H} }} \varvec{s}^{(g)} } \right) = 0 \), \( imag(\varvec{w}^{{\left( g \right)^{H} }} \;\varvec{S}^{\left( g \right)} ) = 0 \) and \( real(\varvec{w}^{{\left( g \right)^{H} }} \;\varvec{S}^{\left( g \right)} ) = 0 \) that implies \( \varvec{w}^{{\left( g \right)^{H} }} \;\varvec{S}^{\left( g \right)} = 0 \)
Thus, proof is complete.
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Lenty Stuwart, S., Tamil Selvi, S. Precoder Based Collaborative Blind Multiuser Detection of CDMA Signals. Wireless Pers Commun 83, 135–148 (2015). https://doi.org/10.1007/s11277-015-2384-4
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DOI: https://doi.org/10.1007/s11277-015-2384-4