Simulation of Symbol Error Rate Vs SNR performance curve for 64QAM in AWGN
This post is a part of the ebook : Simulation of digital communication systems using Matlab – available in both PDF and EPUB format.
MQAM Modulation:
In MASK modulation the information symbols (each k=log2(M) bit wide) are encoded into the amplitude of the sinusoidal carrier. In MPSK modulation the information is encoded into the phase of the sinusoidal carrier. MQAM is a generic modulation technique where the information is encoded into both the amplitude and phase of the sinusoidal carrier. It combines both MASK and MPSK modulation techniques.MQAM modulation technique is a two dimensional modulation technique and it requires two orthonormal basis functions
$$ \begin{matrix}\phi_I(t) = \sqrt{\frac{2}{T_s}} cos(2 \pi f_c t)& 0\leq t\leq T_s \\ \phi_Q(t) = \sqrt{\frac{2}{T_s}} sin(2 \pi f_c t) & 0\leq t\leq T_s \end{matrix} $$
The MQAM modulated signal is represented as
$$ \begin{matrix} S_i(t) = V_{I,i} \sqrt{\frac{2}{T_s}} cos(2 \pi f_c t) + V_{Q,i} \sqrt{\frac{2}{T_s}} sin(2 \pi f_c t) & 0\leq t\leq T_s\\ & i=1,2,…,M \end{matrix} $$
Here \( V_{I,i} \) and \( V_{Q,i} \) are the amplitudes of the quadrature carriers amplitude modulated by the information symbols.
Baseband Rectangular MQAM modulator:
There exist other constellations that are more efficient (in terms of energy required to achieve same error probability) than the standard rectangular constellation. But due to its simplicity in modulation and demodulation rectangular constellations are preferred.
In practice, the information symbols are gray coded inorder to restrict the erroneous symbol decisions to single bit error, the adjacent symbols in the transmitter constellation should not differ more than one bit. Usually the gray coded symbols are separated into inphase and quadrature bits and then mapped to MQAM constellation. The rectangular configuration of QAM makes it easier to consolidate the previously mentioned steps into a simplified LookUpTable (LUT) approach.
Check here to know more on constructing a LUT for MQAM modulation techniques.
64QAM Modulation Scaling Factor:
In order to get a fair comparison across all other modulations, the energy transmitted signal has to be normalized. In general the constellation points for a MQAM modulation can be generated as
The energy a single constellation point is calculated as \( E = {V_I}^2 + {V_Q}^2 \). Where \( V_I \) and \(V_Q\) are the \(I\) and \(Q\) components of the signaling points. For a set of n constellation points, the total energy is calculated as
$$ E = \sum_{i=1}^{n}\left ( {V_{I,i}}^2 + {V_{Q,i}}^2 \right ) $$.
In 64 QAM there are 64 signal points in the constellation that are equally divided into four quadrants (each with sixteen constellation points). Since the constellation is divided equally into four quadrants, normalizing the energy in a single quadrant will simplify things.
Calculating the total energy in any one of the quadrant, say for example the topright quadrant,
The average energy is \( E_{avg} = E/16 = 42\) and the normalization factor will be \( 1/\sqrt{E_{avg}}=1/\sqrt{42} \).
The values in the LUT (where the reference constellation is stored) are normalized by the above mentioned normalization factor and then the 64QAM signal is generated.
Simulation Model:
The simulation model for MQAM modulation is given in the next figure. The receiver uses Euclidean distance as a metric to decide on the received symbols.
Theoretical Symbol Error Rate:
The theoretical symbol error rate for MQAM modulation technique in AWGN when k=log2(M) is even,is given by
$$ P_s \leq 1\left ( 12\left ( 1\frac{1}{\sqrt{M}} \right )Q\left( \sqrt{\frac{3kE_b}{(M1)N_0}}\right ) \right )^2 $$
Or equivalently,
$$ P_s \leq 1\left ( 1\left ( 1\frac{1}{\sqrt{M}} \right )erfc\left(\sqrt{ \frac{3kE_b}{2(M1)N_0}}\right ) \right )^2 $$
Matlab Code:
Check this book for full Matlab code.
Simulation of Digital Communication Systems Using Matlab – by Mathuranathan Viswanathan
Simulation Results:
See Also
[1] Simulation of Symbol Error Rate Vs SNR performance curve for 16QAM in AWGN
[2] Constructing a rectangular constellation for 16QAM
[3] BER Vs Eb/N0 for 8PSK modulation over AWGN
[4] BER Vs Eb/N0 for QPSK modulation over AWGN
[5]QPSK modulation and Demodulation
[6] Simulation of BER Vs Eb/N0 for BPSK modulation over AWGN in Matlab
[7] Intuitive derivation of Performance of an optimum BPSK receiver in AWGN channel
[8] Simulation of MPSK modulation techniques in AWGN channel
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