# QAM bit error probability calculator

Calculate the M-ary QAM bit error probability with this online calculator. Enter

• energy per bit and
• the constellation size M.

M $E b N 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaala aabaGaamyramaaBaaaleaacaWGIbaabeaaaOqaaiaad6eadaWgaaWc baGaaGimaaqabaaaaaaa@3AC9@$ $\frac{{E}_{b}}{{N}_{0}}\text{ }\left[dB\right]$ Bit error probability ${p}_{b}$ Start simulation
0.07864960352514244

The bit error probability $p b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadc hadaWgaaWcbaGaamOyaaqabaaaaa@3922@$ of M-QAM over an AWGN channel is given by

$p B = 4 n ( 1− 1 M )Q( 3n M−1 ⋅ E B N 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadc hadaWgaaWcbaGaamOqaaqabaGccqGH9aqpdaWcaaqaaiaaisdaaeaa caWGUbaaamaabmaabaGaaGymaiabgkHiTmaalaaabaGaaGymaaqaam aakaaabaGaamytaaWcbeaaaaaakiaawIcacaGLPaaacaWGrbWaaeWa aeaadaGcaaqaamaalaaabaGaaG4maiaad6gaaeaacaWGnbGaeyOeI0 IaaGymaaaacqGHflY1daWcaaqaaiaadweadaWgaaWcbaGaamOqaaqa baaakeaacaWGobWaaSbaaSqaaiaaicdaaeqaaaaaaeqaaaGccaGLOa Gaayzkaaaaaa@4D52@$

or

$p b = 2 ld( M ) ( 1− 1 M )erfc( 3ld( M ) 2( M−1 ) ⋅ E B N 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadc hadaWgaaWcbaGaamOyaaqabaGccqGH9aqpdaWcaaqaaiaaikdaaeaa caWGSbGaamizamaabmaabaGaamytaaGaayjkaiaawMcaaaaadaqada qaaiaaigdacqGHsisldaWcaaqaaiaaigdaaeaadaGcaaqaaiaad2ea aSqabaaaaaGccaGLOaGaayzkaaGaamyzaiaadkhacaWGMbGaam4yam aabmaabaWaaOaaaeaadaWcaaqaaiaaiodacaWGSbGaamizamaabmaa baGaamytaaGaayjkaiaawMcaaaqaaiaaikdadaqadaqaaiaad2eacq GHsislcaaIXaaacaGLOaGaayzkaaaaaiabgwSixpaalaaabaGaamyr amaaBaaaleaacaWGcbaabeaaaOqaaiaad6eadaWgaaWcbaGaaGimaa qabaaaaaqabaaakiaawIcacaGLPaaaaaa@5918@$ .