# Bit Error Rate of 64-QAM in AWGN

64-QAM is an important modulation scheme being used in WiMAX and LTE. It allows for transmission of 6 bits symbol which results in higher bit rate and spectral efficiency. The calculation of bit error rate of 64-QAM is a bit tricky as there are many different formulas available with varying degrees of accuracy. Here, we first calculate the bit error rate (BER) of 64-QAM using a simulation and then compare it to the theoretical curve for 64-QAM.

```%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% FUNCTION TO CALCULATE 64-QAM BER USING SIMULATION
% n_bits: Input, number of bits
% EbNodB: Input, energy per bit to noise PSD
% ber: Output, bit error rate
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function[ber]= M_QAM(n_bits,EbNodB);
M=64;
k=log2(M)
EbNo=10^(EbNodB/10);
x=transpose(round(rand(1,n_bits)));
h1=modem.qammod(M);
h1.inputtype='bit';
h1.symbolorder='gray';
y=modulate(h1,x);
n=randn(1,n_bits/k)+j*randn(1,n_bits/k);
y=y+sqrt(7/(2*EbNo))*n.';
h2=modem.qamdemod(M)
h2.outputtype='bit';
h2.symbolorder='gray';
h2.decisiontype='hard decision';
z=demodulate(h2,y);
ber=(n_bits-sum(x==z))/n_bits
return
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
```
```%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% CALCULATE 64-QAM BER USING FORMULA
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
EbNodB=0:2:16;
EbNo=10.^(EbNodB/10);
k=6;
M=64;
x=sqrt(3*k*EbNo/(M-1));
Pb=(4/k)*(1-1/sqrt(M))*(1/2)*erfc(x/sqrt(2));
semilogy(EbNodB,Pb)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
```

Using the above functions the BER of 64-QAM is calculated as shown below. Also shown is the constellation diagram of 64-QAM after addition of noise.

It is observed that the theoretical curve almost overlaps the simulation results. There is only a very small difference at very low signal to noise ratio. The BER of 64-QAM at 16dB is approximately equal to the BER for QPSK at 8dB. Therefore the 64-QAM can only be used in scenarios where there is a very good signal to noise ratio.

In this post we have used built in MATLAB functions for modulation and demodulation. In future posts we try to build up the simulation without using these functions!

#### Author: Yasir Ahmed (aka John)

More than 20 years of experience in various organizations in Pakistan, the USA, and Europe. Worked as a Research Assistant within the Mobile and Portable Radio Group (MPRG) of Virginia Tech and was one of the first researchers to propose Space Time Block Codes for eight transmit antennas. The collaboration with MPRG continued even after graduating with an MSEE degree and has resulted in 12 research publications and a book on Wireless Communications. Worked for Qualcomm USA as an Engineer with the key role of performance and conformance testing of UMTS modems. Qualcomm is the inventor of CDMA technology and owns patents critical to the 4G and 5G standards.

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## 5 thoughts on “Bit Error Rate of 64-QAM in AWGN”

1. Ginni says:

Could you please give a reference to the theoretical formula ?
Thanks a lot !

2. LINA says:

Hi there. May i know why can’t i run the codes successfully?

3. John says:

The constellation size is controlled by the parameter ‘M’. Additionally the signal to noise ratio must be carefully adjusted.

4. ananta says:

please help me to find BER of 128 qam in wimax,and also 256 qam bit error rate