BPSK bit error rate simulation in Python & Matlab

Key focus: Simulate bit error rate performance of Binary Phase Shift Keying (BPSK) modulation over AWGN channel using complex baseband equivalent model in Python & Matlab.

Why complex baseband equivalent model

The passband model and equivalent baseband model are fundamental models for simulating a communication system. In the passband model, also called as waveform simulation model, the transmitted signal, channel noise and the received signal are all represented by samples of waveforms. Since every detail of the RF carrier gets simulated, it consumes more memory and time.

In the case of discrete-time equivalent baseband model, only the value of a symbol at the symbol-sampling time instant is considered. Therefore, it consumes less memory and yields results in a very short span of time when compared to the passband models. Such models operate near zero frequency, suppressing the RF carrier and hence the number of samples required for simulation is greatly reduced. They are more suitable for performance analysis simulations. If the behavior of the system is well understood, the model can be simplified further.

Passband model and converting it to equivalent complex baseband model is discussed in this article.

Simulation of bit error rate performance of BPSK using passband simulation model is discussed in this article.

BPSK constellation

In binary phase shift keying, all the information gets encoded in the phase of the carrier signal. The BPSK modulator accepts a series of information symbols drawn from the set m ∈ {0,1}, modulates them and transmits the modulated symbols over a channel.

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

Here, M denotes the modulation order and it defines the number of constellation points in the reference constellation. The value of M depends on the parameter k – the number of bits we wish to squeeze in a single M-PSK symbol. For example if we wish to squeeze in 3 bits (k=3) in one transmit symbol, then M = 2^k = 2³ = 8 and this results in 8-PSK configuration. M=2 gives BPSK (Binary Phase Shift Keying) configuration. The parameter A is the amplitude scaling factor, f_c is the carrier frequency and g(t) is the pulse shape that satisfies orthonormal properties of basis functions.

Using trigonometric identity, equation (1) can be separated into cosine and sine basis functions as follows

Therefore, the signaling set {s_i,s_q} or the constellation points for M-PSK modulation is given by,

For BPSK (M=2), the constellation points on the I-Q plane (Figure 1) are given by

Simulation methodology

Note: If you are interested in knowing more about BPSK modulation and demodulation, kindly visit this article.

In this simulation methodology, there is no need to simulate each and every sample of the BPSK waveform as per equation (1). Only the value of a symbol at the symbol-sampling time instant is considered. The steps for simulation of performance of BPSK over AWGN channel is as follows (Figure 2)

1. Generate a sequence of random bits of ones and zeros of certain length (N_sym typically set in the order of 10000)
2. Using the constellation points, map the bits to modulated symbols (For example, bit ‘0’ is mapped to amplitude value A, and bit ‘1’ is mapped to amplitude value -A)
3. Compute the total power in the sequence of modulated symbols and add noise for the given E_bN₀ (SNR) value (read this article on how to do this). The noise added symbols are the received symbols at the receiver.
4. Use thresholding technique, to detect the bits in the receiver. Based on the constellation diagram above, the detector at the receiver has to decide whether the receiver bit is above or below the threshold 0.
5. Compare the detected bits against the transmitted bits and compute the bit error rate (BER).
6. Plot the simulated BER against the SNR values and compare it with the theoretical BER curve for BPSK over AWGN (expressions for theoretical BER is available in this article)

Let’s simulate the performance of BPSK over AWGN channel in Python & Matlab.

Simulation using Python

Following standalone code simulates the bit error rate performance of BPSK modulation over AWGN using Python version 3. The results are plotted in Figure 3.

For more such examples refer the book (available as PDF and paperback) – Digital Modulations using Python

#Eb/N0 Vs BER for BPSK over AWGN (complex baseband model)
# © Author: Mathuranathan Viswanathan (gaussianwaves.com)
import numpy as np #for numerical computing
import matplotlib.pyplot as plt #for plotting functions
from scipy.special import erfc #erfc/Q function

#---------Input Fields------------------------
nSym = 10**5 # Number of symbols to transmit
EbN0dBs = np.arange(start=-4,stop = 13, step = 2) # Eb/N0 range in dB for simulation
BER_sim = np.zeros(len(EbN0dBs)) # simulated Bit error rates

M=2 #Number of points in BPSK constellation
m = np.arange(0,M) #all possible input symbols
A = 1; #amplitude
constellation = A*np.cos(m/M*2*np.pi)  #reference constellation for BPSK

#------------ Transmitter---------------
inputSyms = np.random.randint(low=0, high = M, size=nSym) #Random 1's and 0's as input to BPSK modulator
s = constellation[inputSyms] #modulated symbols

fig, ax1 = plt.subplots(nrows=1,ncols = 1)
ax1.plot(np.real(constellation),np.imag(constellation),'*')

#----------- Channel --------------
#Compute power in modulatedSyms and add AWGN noise for given SNRs
for j,EbN0dB in enumerate(EbN0dBs):
    gamma = 10**(EbN0dB/10) #SNRs to linear scale
    P=sum(abs(s)**2)/len(s) #Actual power in the vector
    N0=P/gamma # Find the noise spectral density
    n = np.sqrt(N0/2)*np.random.standard_normal(s.shape) # computed noise vector
    r = s + n # received signal
    
    #-------------- Receiver ------------
    detectedSyms = (r <= 0).astype(int) #thresolding at value 0
    BER_sim[j] = np.sum(detectedSyms != inputSyms)/nSym #calculate BER

BER_theory = 0.5*erfc(np.sqrt(10**(EbN0dBs/10)))

fig, ax = plt.subplots(nrows=1,ncols = 1)
ax.semilogy(EbN0dBs,BER_sim,color='r',marker='o',linestyle='',label='BPSK Sim')
ax.semilogy(EbN0dBs,BER_theory,marker='',linestyle='-',label='BPSK Theory')
ax.set_xlabel('$E_b/N_0(dB)$');ax.set_ylabel('BER ($P_b$)')
ax.set_title('Probability of Bit Error for BPSK over AWGN channel')
ax.set_xlim(-5,13);ax.grid(True);
ax.legend();plt.show()

Simulation using Matlab

Following code simulates the bit error rate performance of BPSK modulation over AWGN using basic installation of Matlab. You will need the add_awgn_noise function that was discussed in this article. The results will be same as Figure 3.

For more such examples refer the book (available as PDF and paperback) – Digital Modulations using Matlab: build simulation models from scratch

%Eb/N0 Vs BER for BPSK over AWGN (complex baseband model)
% © Author: Mathuranathan Viswanathan (gaussianwaves.com)
clearvars; clc;
%---------Input Fields------------------------
nSym=10^6;%Number of symbols to transmit
EbN0dB = -4:2:14; % Eb/N0 range in dB for simulation

BER_sim = zeros(1,length(EbN0dB));%simulated Symbol error rates
    
M=2; %number of constellation points in BPSK
m = [0,1];%all possible input bits
A = 1; %amplitude
constellation = A*cos(m/M*2*pi);%constellation points

d=floor(M.*rand(1,nSym));%uniform random symbols from 1:M
s=constellation(d+1);%BPSK modulated symbols
    
for i=1:length(EbN0dB)
    r  = add_awgn_noise(s,EbN0dB(i));%add AWGN noise
    dCap = (r<=0);%threshold detector
    BER_sim(i) = sum((d~=dCap))/nSym;%SER computation
end

semilogy(EbN0dB,BER_sim,'-*');
xlabel('Eb/N0(dB)');ylabel('BER (Pb)');
title(['Probability of Bit Error for BPSK over AWGN']);

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Reference

[1] Andrea Goldsmith, “Wireless Communications”, ISBN: 978-0521837163, Cambridge University Press; 1 edition, August 8, 2005.↗

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