{"id":1310,"date":"2011-12-08T04:45:23","date_gmt":"2011-12-08T04:45:23","guid":{"rendered":"http:\/\/www.raymaps.com\/?p=1310"},"modified":"2014-06-04T11:00:00","modified_gmt":"2014-06-04T11:00:00","slug":"qam-theoretical-ber","status":"publish","type":"post","link":"https:\/\/www.raymaps.com\/index.php\/qam-theoretical-ber\/","title":{"rendered":"QAM Theoretical BER in AWGN"},"content":{"rendered":"<p style=\"text-align: justify;\">Quadrature Amplitude Modulation (QAM) is an important modulation scheme as it allows for higher data rates and spectral efficiencies. The bit error rate (BER) of QAM can be calculated through Monte Carlo simulations. However this becomes quite complex as the constellation size of the modulation schemes increases. Therefore a theoretical approach is sometimes preferred. The BER for Gray coded QAM, for even number of bits per symbol, is shown below.<\/p>\n<figure id=\"attachment_1313\" aria-describedby=\"caption-attachment-1313\" style=\"width: 516px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/www.raymaps.com\/index.php\/qam-theoretical-ber\/qam_theoretical_ber\/\" rel=\"attachment wp-att-1313\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1313 \" title=\"BER of 4-QAM, 16-QAM, 64-QAM, 256-QAM\" src=\"http:\/\/www.raymaps.com\/wp-content\/uploads\/2011\/12\/QAM_Theoretical_BER.jpg\" alt=\"BER of 4-QAM, 16-QAM, 64-QAM, 256-QAM in AWGN\" width=\"516\" height=\"378\" srcset=\"https:\/\/www.raymaps.com\/wp-content\/uploads\/2011\/12\/QAM_Theoretical_BER.jpg 645w, https:\/\/www.raymaps.com\/wp-content\/uploads\/2011\/12\/QAM_Theoretical_BER-300x220.jpg 300w\" sizes=\"auto, (max-width: 516px) 100vw, 516px\" \/><\/a><figcaption id=\"caption-attachment-1313\" class=\"wp-caption-text\">BER of 4-QAM, 16-QAM, 64-QAM, 256-QAM in AWGN<\/figcaption><\/figure>\n<p style=\"text-align: justify;\">Gray coding ensures that a symbol error results in a single bit error. The code for calculating the theoretical QAM BER for k even (even number of bits per symbol) is given below. The formula for calculating the BER for k odd is different, however, the formula given below can be used a first estimate.<\/p>\n<pre lang=\"matlab\">EbNodB=-6:2:24\r\nEbNo=10.^(EbNodB\/10);\r\nk=8;\r\nM=2^k;\r\nx=sqrt(3*k*EbNo\/(M-1));\r\nPb=(4\/k)*(1-1\/sqrt(M))*(1\/2)*erfc(x\/sqrt(2));\r\nsemilogy(EbNodB,Pb)<\/pre>\n<p>Note:<br \/>\n1. Each additional bit\/symbol requires about 2dB extra in SNR to achieve the same BER.<br \/>\n2. 4-QAM is essentially QPSK modulation.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Quadrature Amplitude Modulation (QAM) is an important modulation scheme as it allows for higher data rates and spectral efficiencies. The bit error rate (BER) of QAM can be calculated through Monte Carlo simulations. However this becomes quite complex as the constellation size of the modulation schemes increases. Therefore a theoretical approach is sometimes preferred. The BER for Gray coded QAM, for even number of bits per symbol, is shown below. Gray coding ensures that a symbol error results in a single bit error. The code for calculating the theoretical QAM BER for k even (even number of bits per symbol) [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11,18,15],"tags":[47,77,34,114,27,36,112],"class_list":["post-1310","post","type-post","status-publish","format-standard","hentry","category-berp","category-lte","category-wimax","tag-4g","tag-awgn","tag-ber","tag-lte","tag-qam","tag-qpsk","tag-wimax"],"_links":{"self":[{"href":"https:\/\/www.raymaps.com\/index.php\/wp-json\/wp\/v2\/posts\/1310","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.raymaps.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.raymaps.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.raymaps.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.raymaps.com\/index.php\/wp-json\/wp\/v2\/comments?post=1310"}],"version-history":[{"count":30,"href":"https:\/\/www.raymaps.com\/index.php\/wp-json\/wp\/v2\/posts\/1310\/revisions"}],"predecessor-version":[{"id":2183,"href":"https:\/\/www.raymaps.com\/index.php\/wp-json\/wp\/v2\/posts\/1310\/revisions\/2183"}],"wp:attachment":[{"href":"https:\/\/www.raymaps.com\/index.php\/wp-json\/wp\/v2\/media?parent=1310"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.raymaps.com\/index.php\/wp-json\/wp\/v2\/categories?post=1310"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.raymaps.com\/index.php\/wp-json\/wp\/v2\/tags?post=1310"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}