# Simulation of Symbol Error Rate Vs SNR performance curve for 16-QAM in AWGN

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This post is a part of the ebook : Simulation of digital communication systems using Matlab – available in both PDF and EPUB format.

## M-QAM Modulation:

In M-ASK modulation the information symbols (each k=log2(M) bit wide) are encoded into the amplitude of the sinusoidal carrier. In M-PSK modulation the information is encoded into the phase of the sinusoidal carrier. M-QAM is a generic modulation technique where the information is encoded into both the amplitude and phase of the sinusoidal carrier. It combines both M-ASK and M-PSK modulation techniques.M-QAM modulation technique is a two dimensional modulation technique and it requires two orthonormal basis functions

$latex \begin{matrix}\phi_I(t) = \sqrt{\frac{2}{T_s}} cos(2 \pi f_c t)& 0\leq t\leq T_s \\ \phi_Q(t) = \sqrt{\frac{2}{T_s}} sin(2 \pi f_c t) & 0\leq t\leq T_s \end{matrix} &s=2$

The M-QAM modulated signal is represented as

$latex \begin{matrix} S_i(t) = V_{I,i} \sqrt{\frac{2}{T_s}} cos(2 \pi f_c t) + V_{Q,i} \sqrt{\frac{2}{T_s}} sin(2 \pi f_c t) & 0\leq t\leq T_s\\ & i=1,2,…,M \end{matrix} &s=2$

Here $latex V_{I,i}$ and $latex V_{Q,i}$ are the amplitudes of the quadrature carriers amplitude modulated by the information symbols.

## Baseband Rectangular M-QAM 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 in-order 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 in-phase and quadrature bits and then mapped to M-QAM constellation. The rectangular configuration of QAM makes it easier to consolidate the previously mentioned steps into a simplified Look-Up-Table (LUT) approach.

Check here to know more on constructing a LUT for M-QAM modulation techniques.

## 16-QAM 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 M-QAM modulation can be generated as

The energy a single constellation point is calculated as $latex E = {V_I}^2 + {V_Q}^2$. Where $latex V_I$ and $latex V_Q$ are the I and Q components of the signaling points. For a set of n constellation points, the total energy is calculated as

$latex E = \sum_{i=1}^{n}\left ( {V_{I,i}}^2 + {V_{Q,i}}^2 \right ) &s=2$.

In 16 QAM there are 16 signal points in the constellation that are equally divided into four quadrants (each with four 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 top-right quadrant,

$latex E = \left ( 1^2+1^2\right )+\left ( 1^2+3^2\right )+\left ( 3^2+1^2\right )+\left ( 3^2+3^2\right )= 40 &s=2$

The average energy is $latex E_{avg} = E/4 = 10$ and the normalization factor will be $latex 1/\sqrt{E_{avg}}=1/\sqrt{10}$.

The values in the LUT (where the reference constellation is stored) are normalized by the above mentioned normalization factor and then the 16-QAM signal is generated.

## Simulation Model:

The simulation model for M-QAM 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 M-QAM modulation technique in AWGN when k=log2(M) is even,is given by

$latex P_s \leq 1-\left ( 1-2\left ( 1-\frac{1}{\sqrt{M}} \right )Q\left( \sqrt{\frac{3kE_b}{(M-1)N_0}}\right ) \right )^2 &s=2$

Or equivalently,
$latex P_s \leq 1-\left ( 1-\left ( 1-\frac{1}{\sqrt{M}} \right )erfc\left(\sqrt{ \frac{3kE_b}{2(M-1)N_0}}\right ) \right )^2 &s=2$

## Matlab Code:

Check this book for full Matlab code.
Simulation of Digital Communication Systems Using Matlab – by Mathuranathan Viswanathan