# Simulation of M-PSK modulation techniques in AWGN channel

(2 votes, average: 3.50 out of 5)

A generic simulation technique to simulate all M-PSK modulation techniques (for upto (M=32)) is given here. The given simulation code is very generic, and it plots both simulated and theoretical symbol error rates for all M-PSK modulation techniques (upto (M=32)).

### M-PSK Modulation and simulation methodology:

The general expression for a M-PSK signal set is given by

$$s_i \left( t \right)=V cos \left[ 2 \pi f_c t – \frac{ \left( i – 1 \right) 2 \pi }{M}\right] \;\;\; \text{ where i = 1,2,…,M} \;\; \rightarrow (1)$$

Here M defines the number of constellation points in the constellation diagram and essentially the M-PSK type. For example (M=4) implies 4-PSK or QPSK, (M=8) implies 8-PSK. The value of M depends on another parameter (k) – the number of bits we wish to squeeze into a single M-PSK symbol. For example if we wish to squeeze in 3 bits ((k=3)) in one transmit symbol, (M = 2^k = 2^3 = 8). This gives us 8-PSK configuration.

Equation (1) can be separated into cos and sin terms as follows

$$s_i(t) = V cos \left[ \frac{(i-1)2\pi}{M} \right] cos(2 \pi f_c t) \; + \; V sin \left[ \frac{(i-1)2\pi}{M} \right] sin(2 \pi f_c t) \rightarrow \;\;\;(2)$$
This can be written as

$$\begin{matrix} s_i(t) = s_{i1}\phi_{1}(t) + s_{i2}\phi_{2}(t) \;\; \rightarrow (3)\ where,\ \ s_{i1} = \sqrt{E_s}\; cos \left[ \frac{(i-1)2\pi}{M} \right] \;,\; s_{i2} = \sqrt{E_s}\; sin \left[ \frac{(i-1)2\pi}{M} \right]\ \ \phi_{1}(t) = \frac{V cos(2\pi f_c t)}{\sqrt{E_s}},\phi_{2}(t) = \frac{V sin(2\pi f_c t)}{\sqrt{E_s}}\end{matrix}$$

Here (\phi_{1}(t)) and (\phi_{2}(t)) are orthonormal basis functions that follows from Gram-Schmidt orthogonalization procedure and (s_{i1}) and (s_{i2}) are the coefficients of each signaling point in the M-PSK constellation. (E_s) is the symbol energy usually normalized to (1/\sqrt{2}) The constellation points on the M-PSK constellation lie (2 \pi /M \;) radians apart and are placed on a circle of radius (\sqrt{E_s}). The coefficients (s_{i1}) and (s_{i2}) are termed as inphase (I) and quadrature-phase (Q) components respectively.

The ideal constellation diagram for M-PSK contains M equally spaced signaling points that are located at the distance (\sqrt{E_s}) from the origin. Figure 1 illustrates the ideal constellation diagram for 8-PSK constellation.

The generated I and Q components are then added with AWGN noise of required variance depending on the required Es/N0. The received signal’s constellation is corrupted with noise and the detection is based on comparing the received symbols with the ideal signaling points and making a decision based on the minimum distance.

Finally the simulated and theoretical symbol error rates are computed.

The theoretical symbol error rate [1] for M-PSK modulation is given by

\dpi{120} {\color{DarkBlue} \begin{align*} P_s &= 2Q\left[\sqrt{2E_s} \; sin\left(\frac{\pi}{M}\right)\right] \\ &= erfc\left [ \sqrt{E_s} \; sin\left(\frac{\pi}{M}\right)\right ] \end{align*} \;\;\;\;\; \rightarrow(4) }

### Matlab Simulation:

Check this book for full Matlab code.
Digital Modulations using Matlab: Build Simulation Models from Scratch – by Mathuranathan Viswanathan

### Recommended Books

• Satheesh Monikandan

Dear Sir,
How is the code rate related with snr in lte system downlink?

• AWGN model is a fundamental and the simplest mathematical model. Other propagation models exist that are suitable for modeling the radio channels encountered in LTE downlinks. Examples include

Stanford University Interim (SUI) model
Okumura model
Hata COST 231 model
COST Walfisch-Ikegami
Ericsson 9999 model

• Waseem Raza

I think there is a typo mistake in equation (1), phase term is not being added, (+) is missing.

• updated the equation. Thanks.

• Waseem Raza