tag:blogger.com,1999:blog-49776078211787139842024-03-01T02:30:22.763-08:00Space Time Block CodesAragornhttp://www.blogger.com/profile/17936623185981002462noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-4977607821178713984.post-53415334554916426252011-07-06T05:22:00.001-07:002011-07-13T02:53:03.130-07:00SPACE TIME CODES<div dir="ltr" style="text-align: left;" trbidi="on"><br />
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: Arial,sans-serif; font-size: 12pt;">A space time code is a method employed to improve the reliability of data transmission in wireless communication systems using multiple transmit antennas. </span><span style="font-family: 'Times New Roman',serif; font-size: 12pt;"></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; mso-layout-grid-align: none; text-autospace: none;"><span style="font-family: Arial,sans-serif; font-size: 12pt;"> </span><span style="font-family: 'Times New Roman',serif; font-size: 12pt;"><br />
</span><span style="font-family: Arial,sans-serif; font-size: 12pt;"> Essentially, two different space-time coding methods, namely space-time trellis codes (STTCs) and Space time block codes (STBCs) have been proposed. STTC has been introduced as a coding technique that promises full diversity and substantial coding gain at the price of a quite high decoding complexity. To avoid this disadvantage, STBCs have been proposed by the pioneering work of Alamouti. </span><span style="font-family: 'Times New Roman',serif; font-size: 12pt;"></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: 5.0pt;"><span style="font-family: Arial,sans-serif; font-size: 12pt;"> </span><span style="font-family: 'Times New Roman',serif; font-size: 12pt;"><br />
</span><span style="font-family: Arial,sans-serif; font-size: 12pt;"> Space–time block coding is a technique used in wireless communications to transmit multiple copies of a data stream across a number of antennas and to exploit the various received versions of the data to improve the reliability of data-transfer. The fact that the transmitted signal must traverse a potentially difficult environment with scattering, reflection, refraction and so on and may then be further corrupted by thermal noise in the receiver means that some of the received copies of the data will be 'better' than others. This redundancy results in a higher chance of being able to use one or more of the received copies to correctly decode the received signal. In fact space time coding combines all the copies of the received signal in an optimal way to extract as much information from each of them as possible.</span><span style="font-family: 'Times New Roman',serif; font-size: 12pt;"></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: 5.0pt;"><span style="font-family: Arial,sans-serif; font-size: 12pt;"> </span><span style="font-family: 'Times New Roman',serif; font-size: 12pt;"><br />
</span><span style="font-family: Arial,sans-serif; font-size: 12pt;"> In our posts, we consider only </span><i><span style="font-family: Verdana,sans-serif; font-size: 12pt;">coherent STBCs</span></i><span style="font-family: Arial,sans-serif; font-size: 12pt;">; i.e. the receiver knows the channel impairments through training or some other form of estimation.Matlab simulations and the relevant results<b> </b>are given for each of the STBCs discussed. </span><span style="font-family: 'Times New Roman',serif; font-size: 12pt;"><br />
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</span><span style="font-family: Arial,sans-serif; font-size: 12pt;"> We start off with the description of the simplest STBC which is the </span><i><span style="font-family: Verdana,sans-serif; font-size: 12pt;">Alamouti STBC</span></i><span style="font-family: Arial,sans-serif; font-size: 12pt;"> for 2 transmit and 1 receive antenna.Then, we discuss the Alamouti code for 2 transmit and 2 receive antennas<b>. </b>The decoding is done by </span><i><span style="font-family: Verdana,sans-serif; font-size: 12pt;">Zero Forcing( ZF )</span></i><span style="font-family: Arial,sans-serif; font-size: 12pt;"> technique. The modulation used is 16-QAM. </span><span style="font-family: 'Times New Roman',serif; font-size: 12pt;"><br />
</span><b><span style="font-family: Arial,sans-serif; font-size: 12pt;"> </span></b><span style="font-family: Arial,sans-serif; font-size: 12pt;"> </span><span style="font-family: 'Times New Roman',serif; font-size: 12pt;"><br />
</span><span style="font-family: Arial,sans-serif; font-size: 12pt;"> We describe the </span><i><span style="font-family: Verdana,sans-serif; font-size: 12pt;">Golden code</span></i><span style="font-family: Arial,sans-serif; font-size: 12pt;">, a code for 2 transmit and 2 receive antennas which achieves full diversity and full rate.<b> </b>In the subsequent pages, we describe two decoding algorithms for Golden code. They are </span><i><span style="font-family: Verdana,sans-serif; font-size: 12pt;">Brute force ML</span></i><span style="font-family: Arial,sans-serif; font-size: 12pt;"> and </span><i><span style="font-family: Verdana,sans-serif; font-size: 12pt;">Sphere decoding</span></i><span style="font-family: Verdana,sans-serif; font-size: 12pt;">.</span><span style="font-family: 'Times New Roman',serif; font-size: 12pt;"><br />
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</span><span style="font-family: Arial,sans-serif; font-size: 12pt;"> We give a comparision of Alamouti code and the Golden code. For the comparision to be fair, we maintain the bit rates same for both the codes. So we use 4-QAM modulation for the </span><i><span style="font-family: Verdana,sans-serif; font-size: 12pt;">Golden code</span></i><span style="font-family: Arial,sans-serif; font-size: 12pt;">. </span><span style="font-family: 'Times New Roman',serif; font-size: 12pt;"><br />
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</span><b><span style="font-family: Arial,sans-serif; font-size: 12pt;"> </span></b><span style="font-family: Arial,sans-serif; font-size: 12pt;">We describe the </span><i><span style="font-family: Verdana,sans-serif; font-size: 12pt;">Silver code</span></i><b><span style="font-family: Arial,sans-serif; font-size: 12pt;">,</span></b><span style="font-family: Arial,sans-serif; font-size: 12pt;"> a code for 2 transmit and 2 receive antennas<b> </b>whose coding gain is slightly less than that of the Golden code but easier in implementation owing to its lesser decoding complexity. We decode it using</span><span style="font-family: Verdana,sans-serif; font-size: 12pt;"> </span><span style="font-family: Arial,sans-serif; font-size: 12pt;">the</span><i><span style="font-family: Verdana,sans-serif; font-size: 12pt;"> Sphere decoder</span></i><span style="font-family: Arial,sans-serif; font-size: 12pt;">.</span><span style="font-family: 'Times New Roman',serif; font-size: 12pt;"></span></div><span style="font-family: Arial,sans-serif; font-size: 12pt; line-height: 115%;"><br />
We then describe an </span><i><span style="font-family: Verdana,sans-serif; font-size: 12pt; line-height: 115%;">8X8 MIMO</span></i><b><span style="font-family: Verdana,sans-serif; font-size: 12pt; line-height: 115%;"> </span></b><span style="font-family: Arial,sans-serif; font-size: 12pt; line-height: 115%;">system that does not use any coding and decode it using ZF and V-BLAST techniques. Our aim here is to compare the performances of these two decoders.</span><br />
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<span style="font-family: Arial,sans-serif; font-size: 12pt; line-height: 115%;"><b>Related Posts</b></span><br />
<span style="font-family: Arial,sans-serif; font-size: 12pt; line-height: 115%;"><b><span class="Apple-style-span" style="font-weight: normal;"> <a href="http://spacetimecodes.blogspot.com/p/alamouti-code.html">Alamouti Code</a></span></b></span><br />
<span style="font-family: Arial,sans-serif; font-size: 12pt; line-height: 115%;"> <a href="http://spacetimecodes.blogspot.com/p/golden-code.html">Golden Code</a></span><br />
<a href="http://spacetimecodes.blogspot.com/p/silver-code.html">Silver Code</a><br />
<a href="http://spacetimecodes.blogspot.com/p/8-x-8-uncoded-mimo-system.html">8 X 8 Uncoded MIMO using VBlast and ZF </a> </div>Aragornhttp://www.blogger.com/profile/17936623185981002462noreply@blogger.com8tag:blogger.com,1999:blog-4977607821178713984.post-62601452327641489622011-07-06T05:16:00.000-07:002011-07-13T02:54:04.512-07:00A short note on Perfect STBCs<div dir="ltr" style="text-align: left;" trbidi="on"><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><b><i><span style="font-family: Arial,sans-serif; font-size: 12pt;">Perfect Codes</span></i></b><b><span style="font-family: Arial,sans-serif; font-size: 12pt;"> are Space-Time codes for the coherent MIMO channel.</span></b><span style="font-family: Arial,sans-serif; font-size: 12pt;"></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br />
</div><div class="MsoNormal" style="line-height: normal;"><span style="font-family: Arial,sans-serif; font-size: 12pt;">They are algebraic codes, built on non-commutative fields (or <i>division algebras </i>). <br />
The channel model considered is the following: if M is the number of transmit and receive antennas, </span></div><div align="center" class="MsoNormal" style="line-height: normal; text-align: center;"><span style="font-family: Arial,sans-serif; font-size: 12pt;">Y = H X + N </span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: Arial,sans-serif; font-size: 12pt;">where H ={hij} is the MxM channel matrix with complex fading coefficients and N the MxM complex Gaussian noise matrix.</span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br />
</div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: Arial,sans-serif; font-size: 12pt;">A square <i>n<sub>t</sub> × n<sub>t</sub> </i>STBC is called a <i>perfect </i>code if and only if:</span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br />
</div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><i><span style="font-family: Arial,sans-serif; font-size: 12pt;">• </span></i><span style="font-family: Arial,sans-serif; font-size: 12pt;">It is a full rate linear code using <i>n</i><sup>2</sup><i><sub>t</sub> </i>information symbols.</span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><i><span style="font-family: Arial,sans-serif; font-size: 12pt;">• </span></i><span style="font-family: Arial,sans-serif; font-size: 12pt;">The determinant of the difference of any two distinct codewords is different from 0 which ensures full-rank and in turn full-diversity.</span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><i><span style="font-family: Arial,sans-serif; font-size: 12pt;">• </span></i><span style="font-family: Arial,sans-serif; font-size: 12pt;">Non-vanishing determinant is the minimum determinant of a perfect code that is lower bounded away from zero by a constant. This constant is the measure of coding gain.( The coding advantage is an approximate measure of the gain over an uncoded system operating with the same diversity advantage).</span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br />
</div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><i><span style="font-family: Arial,sans-serif; font-size: 12pt;">• </span></i><span style="font-family: Arial,sans-serif; font-size: 12pt;">The energy required to send the linear combination of the information symbols on each layer is similar to the energy used for sending the symbols themselves (we do not increase the transmitted energy in encoding the information symbols).</span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br />
</div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><i><span style="font-family: Arial,sans-serif; font-size: 12pt;">• </span></i><span style="font-family: Arial,sans-serif; font-size: 12pt;">It induces uniform average transmitted energy per antenna in all <i>T </i>time slots, i.e., all the coded symbols in the code matrix have the same average energy.</span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: Arial,sans-serif; font-size: 12pt;"><br />
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Perfect codes only exist in dimension 2, 3, 4, and 6.</span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br />
</div><ul type="disc"><li class="MsoNormal" style="line-height: normal; mso-list: l0 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list .5in;"><span style="font-family: Arial,sans-serif; font-size: 12pt;">For M=2 antennas, QAM symbols are sent. There are infinitely many of them, but the most famous is the Golden code. The minimum determinant is 1/5. </span></li>
<li class="MsoNormal" style="line-height: normal; mso-list: l0 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list .5in;"><span style="font-family: Arial,sans-serif; font-size: 12pt;">For M=3 antennas, HEX symbols are sent. The minimum determinant is 1/49. </span></li>
<li class="MsoNormal" style="line-height: normal; mso-list: l0 level1 lfo1; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; tab-stops: list .5in;"><span style="font-family: Arial,sans-serif; font-size: 12pt;">For M=4 antennas, QAM symbols are sent. The minimum determinant is 1/1125. </span></li>
</ul><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: Arial,sans-serif; font-size: 12pt;">For M=6 antennas, HEX symbols are sent. The minimum determinant is between 1/(2<sup>6</sup> 7<sup>4</sup>) and 1/(2<sup>6</sup> 7<sup>5</sup>) .</span><span style="font-family: 'Times New Roman',serif; font-size: 12pt;"></span><br />
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<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span style="font-family: Arial,sans-serif; font-size: 12pt; line-height: 18px;"><b>Related Posts</b></span></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span style="font-family: Arial,sans-serif; font-size: 12pt; line-height: 18px;"><b><span class="Apple-style-span" style="font-weight: normal;"> <a href="http://spacetimecodes.blogspot.com/p/alamouti-code.html">Alamouti Code</a></span></b></span></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span style="font-family: Arial,sans-serif; font-size: 12pt; line-height: 18px;"> <a href="http://spacetimecodes.blogspot.com/p/golden-code.html">Golden Code</a></span></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"> <a href="http://spacetimecodes.blogspot.com/p/silver-code.html">Silver Code</a><br />
<a href="http://spacetimecodes.blogspot.com/p/8-x-8-uncoded-mimo-system.html">8 X 8 Uncoded MIMO using VBlast and ZF </a> </div></div></div>Aragornhttp://www.blogger.com/profile/17936623185981002462noreply@blogger.com2