In this article, the relationship between SNR-per-bit (Eb/N0) and SNR-per-symbol (Es/N0) are defined with respect to M-ary signaling schemes. Then the complex baseband model for an AWGN channel is discussed, followed by the theoretical error rates of various modulations over the additive white Gaussian noise (AWGN) channel. Finally, the complex baseband models for digital modulators and detectors developed in previous chapter of this book, are incorporated to build a complete communication system model.
If you would like to know more about the simulation and analysis of white noise, I urge you to read this article: White noise: Simulation & Analysis using Matlab.
Signal to noise ratio (SNR) definitions
Assuming a channel of bandwidth B, received signal power Pr and the power spectral density (PSD) of noise N0/2, the signal to noise ratio (SNR) is given by
Let a signal’s energy-per-bit is denoted as Eb and the energy-per-symbol as Es, then γb=Eb/N0 and γs=Es/N0 are the SNR-per-bit and the SNR-per-symbol respectively.
For uncoded M-ary signaling scheme with k = log2(M) bits per symbol, the signal energy per modulated symbol is given by
The SNR per symbol is given by
AWGN channel model
In order to simulate a specific SNR point in performance simulations, the modulated signal from the transmitter needs to be added with random noise of specific strength. The strength of the generated noise depends on the desired SNR level which usually is an input in such simulations. In practice, SNRs are specified in dB. Given a specific SNR point for simulation, let’s see how we can simulate an AWGN channel that adds correct level of white noise to the transmitted symbols.
Consider the AWGN channel model given in Figure 1. Given a specific SNR point to simulate, we wish to generate a white Gaussian noise vector
(1) Assume, s is a vector that represents the transmitted signal. We wish to generate a vector r that represents the signal after passing through the AWGN channel. The amount of noise added by the AWGN channel is controlled by the given SNR – γ
(2) For waveform simulation model, let the given oversampling ratio is denoted as L. On the other hand, if you are using the complex baseband models, set L=1.
(3) Let N denotes the length of the vector s. The signal power for the vector s can be measured as,
(4) The required power spectral density of the noise vector n is computed as
(5) Assuming complex IQ plane for all the digital modulations, the required noise variance (noise power) for generating Gaussian random noise is given by
(6) Generate the noise vector n drawn from normal distribution with mean set to zero and the standard deviation computed from the equation given above
(7) Finally add the generated noise vector (n) to the signal (s)
Matlab code
The following custom function written in Matlab, can be used for adding AWGN noise to an incoming signal. It can be used in waveform simulation as well as complex baseband simulation models.
%author - Mathuranathan Viswanathan (gaussianwaves.com
%This code is part of the books: Wireless communication systems using Matlab & Digital modulations using Matlab.
function [r,n,N0] = add_awgn_noise(s,SNRdB,L)
%Function to add AWGN to the given signal
%[r,n,N0]= add_awgn_noise(s,SNRdB) adds AWGN noise vector to signal
%'s' to generate a %resulting signal vector 'r' of specified SNR
%in dB. It also returns the noise vector 'n' that is added to the
%signal 's' and the spectral density N0 of noise added
%
%[r,n,N0]= add_awgn_noise(s,SNRdB,L) adds AWGN noise vector to
%signal 's' to generate a resulting signal vector 'r' of specified
%SNR in dB. The parameter 'L' specifies the oversampling ratio used
%in the system (for waveform simulation). It also returns the noise
%vector 'n' that is added to the signal 's' and the spectral
%density N0 of noise added
s_temp=s;
if iscolumn(s), s=s.'; end; %to return the result in same dim as 's'
gamma = 10ˆ(SNRdB/10); %SNR to linear scale
if nargin==2, L=1; end %if third argument is not given, set it to 1
if isvector(s),
P=L*sum(abs(s).ˆ2)/length(s);%Actual power in the vector
else %for multi-dimensional signals like MFSK
P=L*sum(sum(abs(s).ˆ2))/length(s); %if s is a matrix [MxN]
end
N0=P/gamma; %Find the noise spectral density
if(isreal(s)),
n = sqrt(N0/2)*randn(size(s));%computed noise
else
n = sqrt(N0/2)*(randn(size(s))+1i*randn(size(s)));%computed noise
end
r = s + n; %received signal
if iscolumn(s_temp), r=r.'; end;%return r in original format as s
endPython code
The following custom function written in Python 3, can be used for adding AWGN noise to an incoming signal. It can be used in waveform simulation as well as complex baseband simulation models.
# author - Mathuranathan Viswanathan (gaussianwaves.com
# This code is part of the book Digital Modulations using Python
from numpy import sum,isrealobj,sqrt
from numpy.random import standard_normal
def awgn(s,SNRdB,L=1):
"""
AWGN channel
Add AWGN noise to input signal. The function adds AWGN noise vector to signal 's' to generate a resulting signal vector 'r' of specified SNR in dB. It also
returns the noise vector 'n' that is added to the signal 's' and the power spectral density N0 of noise added
Parameters:
s : input/transmitted signal vector
SNRdB : desired signal to noise ratio (expressed in dB) for the received signal
L : oversampling factor (applicable for waveform simulation) default L = 1.
Returns:
r : received signal vector (r=s+n)
"""
gamma = 10**(SNRdB/10) #SNR to linear scale
if s.ndim==1:# if s is single dimensional vector
P=L*sum(abs(s)**2)/len(s) #Actual power in the vector
else: # multi-dimensional signals like MFSK
P=L*sum(sum(abs(s)**2))/len(s) # if s is a matrix [MxN]
N0=P/gamma # Find the noise spectral density
if isrealobj(s):# check if input is real/complex object type
n = sqrt(N0/2)*standard_normal(s.shape) # computed noise
else:
n = sqrt(N0/2)*(standard_normal(s.shape)+1j*standard_normal(s.shape))
r = s + n # received signal
return rTheoretical symbol error rates for digital modulations in AWGN channel
Denoting the symbol error rate (SER) as
The theoretical symbol error rates are coded as a reusable function. In this implementation, erfc function is used instead of the Q function shown in the Table 4.1. The following equation describes the relationship between the erfc function and the Q function.
Unified simulation model for performance simulation
In the previous chapter of the books, the code implementation for complex baseband models for various digital modulators and demodulator are given. Using these models, we can create a unified simulation code for simulating the performance of various modulation techniques over AWGN channel.
The complete simulation model for performance simulation over AWGN channel is given in Figure 2. The figure is illustrated for a coherent communication system model (applicable for MPSK/MQAM/MPAM modulations)
The Matlab code implementing the aforementioned simulation model is given in the books. Here, an unified approach is employed to simulate the performance of any of the given modulation technique – MPSK, MQAM, MPAM or MFSK (MFSK simulation technique is available in the following books: Digital Modulations using Python and Digital Modulations using Matlab).
This article is part of the following books |
The simulation code will automatically choose the selected modulation type, performs Monte Carlo simulation, computes symbol error rates and plots them against the theoretical symbol error rates. The simulated performance results obtained for MQAM and MPSK modulations are shown in the Figure 3 and Figure 4.
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References
Books by the author
 Wireless Communication Systems in Matlab Second Edition(PDF) Note: There is a rating embedded within this post, please visit this post to rate it. |  Digital Modulations using Python (PDF ebook) Note: There is a rating embedded within this post, please visit this post to rate it. |  Digital Modulations using Matlab (PDF ebook) Note: There is a rating embedded within this post, please visit this post to rate it. |
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