In this work, we observe the behavior of block space-time code in wireless channel dynamics. The block space-time code is optimally constructed in slow fading. The block code in quasistatic fading channels provides affordable complexity in design and construction. Our results show that the performance of the block space-time code may not be as good as conventionally convolutional coding with serial transmission for some channel features. As channel approaches fast fading, a coded single antenna scheme can collect as much diversity as desired by correctly choosing the free distance of code. The results also point to the need for robust space-time code in dynamic wireless fading channels. We expect that self-encoded spread spec-trum with block space-time code will provide a robust performance in dynamic wireless fading channels.

Space-time codes introduce temporal and spatial correlation into signals transmitted from different antennas in order to provide diversity at the receiver as well as coding gain without sacrificing bandwidth [1,2]. Most optimal space-time codes have been developed in block code in slow fading channels [1,3–7]. The block code in quasi-static fading channels provides affordable complexity in design and construction. It has been shown that in a system with t transmit and r receive antennas, and a slow fading channel, the average channel capacity with perfect channel state information (CSI) at the receiver is about min{t,r} times larger than that of a single antenna system [

In this work, we consider the performance of current space-time code that is optimally constructed for slow fading channels (hereafter we called it the block space-time code). We raise the question whether the performance of the block space-time code would be at least as good as conventional serial code when channel characteristics change dynamically. In this paper, we analyze and compare two different transmitter structures: the parallel transmitter that employs the block spacetime code and the serial signal transmitter. Conventional channel code, such as convolutional code with a single transmit antenna, is used in the serial signal transmitter. We compare the BER of the two systems under the same bandwidth, average transmit power, data rate and a similar encoder processing complexity. The results suggest that the performance of the block space-time code can be degraded below the conventional code for some channel features.

Self-encoded spread spectrum (SESS) introduced in [

We consider a base station to a mobile communication where the base-station equipped with n antennas and the mobile is equipped with r antennas. Data are encoded by the channel encoder, and the encoded data go through a serial-to-parallel converter and are split into n streams of data. Each stream of data is the input to a pulse shaper. Then, the output of each shaper is modulated. We consider the 8-state trellis code [^{1}, and data-2, d^{2}, are encoded, pulse-shaped, modulated and transmitted in parallel over the two transmit antennas. Alternatively, the two data streams can be considered divided from a common data source. The signal constellation employed here is 4-PSK and the signal points are labeled by the elements of Z_{4}, the ring of integers modulo 4. The edge level c^{1}c^{2} in ^{1} is transmitted over the first antenna and that signal c^{2} is transmitted over the second antenna. This code can be described in terms of a sequence (d^{1},d^{2}) of binary inputs. The output signal pair at time t is given by [

At each time slot t, the output of modulator-i is a signal that is transmitted using transmit antenna i for 1≤i≤n. The n signals are transmitted simultaneously, each from a different transmit antenna, and all signals have the same transmission period.

The signal at each receive antenna is the sum of the n transmitted signals contaminated by a noise and corrupted by Rayleigh fadings. We assume that the elements of the signal constellation are normalized by a factor of, where E_{b} is the bit energy, so that the average energy of the constellation is the unity. A decision is based on the received signals at each receive antenna 1≤j≤r. The signal received by antenna j at time t is given by

where the noise at time t is a complex Gaussian random variable with a zero-mean and variance N_{0}/2 per dimension, independent for all j and t. The coefficient is the path gain from transmit antenna i to receive antenna j at time t. We are interested in the behavior of the block space-time codes that are optimally constructed for slow fading as channel dynamics change to independent path gains for every i, j and t.

A maximum-likelihood sequence detector is applied for decoding. We assume ideal channel state information; thus, the path gains , i=1,2,…n j=1,2,…r are precisely known to the receiver. Since is the received signal at receive antenna j at time t, the branch metric for a transition labeled is given by

Viterbi decoding is then applied to obtain the path with the lowest accumulated metric.

Convolutional coding is applied to each data stream as shown in _{free}) equal to 5. The two 4-state diagram of this code is shown in

Modulo 2 addition is performed to obtain the encoder output pairs. Likewise, the output of the second encoder can be generated. The encoders’ outputs are serial-to-parallel converted and fed to the modulator. The signal constellation employed here is 16-QAM for a single transmit antenna and the signal points are labeled by the elements of Z_{16}. Considering that 16-QAM and 4-PSK display approximately 5 dB difference for E_{b}/N_{0}≥0 dB, these modulation schemes are more favorable for the block space-time code system. Nevertheless, our results show that the conventional serial code system can outperform the block space-time code in some channel characteristics.

The output of a 16-QAM modulator can be represented as a complex number,

where ζ=. The signal received by antenna j at time t is given by

where is the path gain from the single transmit antenna to receive antenna j at time t. Notice that we scaled the transmit bit energy to maintain the same average bit energy in both systems for fair comparison. For Viterbi decoding, we replace Equation (3) with

(7)

From

where l is the block length, N_{0}/2 is the noise variance per dimension,

and

For r =1

Applying c= (0, 0, 0, 0) and e= (0, 2, 2, 0),

Since is Rayleigh fading, the probability density function (pdf) of can be shown as [

to maintain the same average received power without fading. U(v) is the unit step function. With r receiving antennas and

the pdf of X can be represented as a Gamma distribution with the parameter, 2r, as [

Therefore,

In Equation (14), we applied the approximation of the symbol error probability of QPSK [

with M=4. Therefore, the probability of the bit error can be found as [

where

B_{d} and k are the number of nonzero information bits and the total number of information bits, respectively, on the d_{free} path. The error coefficient, N_{dfree}, is the total number, or multiplicity, of the free distance code word. For the chosen codewords, c=(0, 0, 0, 0) and e=(0, 2, 2, 0), N_{dfree}, B_{d} and k are 1, 1 and 4, respectively.

From ^{2}(c,e) as follows for r=1:

Using c =(0, 0, 0)and e=(3, 1, 3),

With r receiving antennas, the pdf of the above equation can be represented as the combination of two random variables,

and

Consequently [

, and (19)

Therefore, the probability of bit error is

In Equation (20), we employ the approximation of the symbol error probability, , for 16- QAM modulation [_{dfree }, B_{d} and k are 1, 1 and 3 respectively.

We assume that the perfect channel state information is available at receive antennas in the following simulations. We consider the dynamic channel characteristics of independent fading for every bit interval. In _{b}/N_{0 }, the performance of the block space-time code may not be as good as the conventional convolutional code for some channel features. As channel dynamics reach fast fading, a coded single antenna scheme can collect as much diversity as desired by suitably choosing the free distance of code. Block space-time codes are most useful in slow fading when temporal diversity is not available. The advantage of the block space-time code can be diminished significantly as wireless channel approach fast fading.

In this paper, we show that the block space-time coding gain can be degraded below the conventional channel coding with a single transmit antenna for some channel characteristics. Our results suggest that there is a need for a robust space-time code in rapidly changing wireless channels. Our future work is to develop SESS block space-time code. Due to the inherent time diversity in SESS, we expect SESS block space-time code to provide a robust performance in dynamic wireless channels.

This work was supported in part by the contract award FA9550-08-1-0393 from the U.S. Air Force Office of Scientific Research. Thanks are due to Dr. J. A. Sjogren whose support has allowed the authors to investigate the feasibility of self-encoded block space-time code in dynamic wireless fading channels.

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