We propose enhanced spatial multiplexing codes (E-SMCs) to enable various encoding rates. The symbol error rate (SER) performance of the E-SMC is investigated when zero-forcing (ZF) and maximal-ratio combining (MRC) techniques are used at a receiver. The proposed E-SMC allows a transmitted symbol to be repeated over time to achieve further diversity gain at the cost of the encoding rate. With the spatial correlation between transmit antennas, SER equations for
Multiple-input multiple-output (MIMO) schemes are typical for the purposes of maximizing data rate [
In this study, we address enhanced SMCs, referred to as E-SMCs, which extend the conventional SMC in [
In decoding E-SMCs, we separate received signals using a zero-forcing (ZF) receiver [
Given a required symbol error rate (SER) for each data stream, we provide a method that constructs E-SMCs consisting of a minimal number of time slots. And the codes are further optimized to have a maximum encoding rate. We will compare the data rates between E-SMCs and vr-STBCs when required SER is given. E-SMCs perform better than vr-STBCs when the time correlation of radio channels is low since vr-STBCs are designed on the assumption that the channel is constant during a transmitting code block.
To investigate SER performance for various encoding rates of E-SMCs, we assume an equal number of transmit and receiving antennas, an equal power allocation between the transmitters, and perfect channel state information known at the receivers. We also assume that spatial correlation exists only at the transmitters, which is valid when the transmitters are sufficiently high above the local scattering environments [
With numerical investigation, analytical and simulated results are compared for spatially correlated and time-varying Rayleigh fading channels where the time variability is modelled as first-order Markovian channel [
This paper is organized as follows. Section
An E-SMC matrix for Each entry of The sum of the entries in each column of
In the three properties, it is noted that modulo-2 arithmetic is not assumed. Thus, the properties imply that the E-SMC is based on a simple repetition code over time slots and not able to transmit the same symbols over transmit antennas in each time slot. With the E-SMC transmission matrix in (
We consider a MIMO system with
We consider time-varying and flat Rayleigh fading channels. We also assume the perfect channel information at a receiver and the presence of only transmit correlation; that is, there is sufficiently rich scattering at a receiver so that the receiver antennas would be uncorrelated. Analogous to a first-order Markovian model used in [
When a ZF receiver is used for the received signals in (
Considering that
We consider a MRC technique which renders a received SNR maximum. The ZF output signals are combined linearly with the following MRC weights:
Sample correlations of ZF output SNRs for an E-SMC in Rayleigh MIMO channels:
Without loss of generality, we assume that
Using a MGF-based approach, we write the average SER of symbol
The average SER of the
where
We assume that a symbol block to be encoded with the E-SMC consists of
When we have a single
Conditions (a) and (c) for the proposed E-SMC in Section
From (
Initialize all
If
If
The above SSM always finds
For mathematical induction, suppose that
For example, the following code blocks
It is noted that
Once the unused position in the code matrix is removed, a maximum code rate is achieved. Thus, a problem of removing the unused optimally can be formulated as
Since (
After achieving
In the simulation, we consider spatially transmit-correlated MIMO channels when
To obtain the numerical results, we test
Average SER of E-SMCs with ZF and MRC for QPSK in spatially uncorrelated Rayleigh MIMO channels.
Average SER of E-SMCs with ZF and MRC for QPSK over a spatially correlated Rayleigh MIMO channel with
Average SER for E-SMCs with ZF and MRC with 16-ary QAM in spatially uncorrelated Rayleigh MIMO channels.
Average SER for E-SMCs with ZF and MRC with 16-ary QAM over a spatially correlated Rayleigh MIMO channel with
Figures
Comparison of SER performance for the conventional SMC and the E-SMC in spatially uncorrelated Rayleigh MIMO channels when
Comparison of SER performance for the conventional SMC and the E-SMC with
Comparison of SER performance for the conventional SMC and the E-SMC with
Figure
Comparison of maximum encoding rates achieved by the E-SMC and the VS-SIC in time-varying and spatially uncorrelated Rayleigh fading channels when
Figure
(a) Optimal block length and (b) its encoding rate for
In this paper, we have proposed E-SMCs, which incorporate the symbol repetition over time. The E-SMC leads to a tradeoff between encoding rate and time diversity. For MIMO systems using E-SMCs, the received signals are decoded by ZF and MRC in this paper. SER equations of
The author declares that there is no conflict of interests regarding the publication of this paper.