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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_cis 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)

Generate a sequence of random bits of ones and zeros of certain length (N_sym typically set in the order of 10000)
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)
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.
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.
Compare the detected bits against the transmitted bits and compute the bit error rate (BER).
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)

*Figure 2: Simulation methodology for performance of BPSK modulation over AWGN channel*

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()

*Figure 3: Bit error rate performance of BPSK modulation over AWGN using Python*

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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Constellation diagram – investigate phase transitions

The phase transition properties of the different variants of QPSK schemes and MSK, are easily investigated using constellation diagram. Let’s demonstrate how to plot the signal space constellations, for the various modulations used in the transmitter.

Typically, in practical applications, the baseband modulated waveforms are passed through a pulse shaping filter for combating the phenomenon of intersymbol interference (ISI). The goal is to plot the constellation plots of various pulse-shaped baseband waveforms of the QPSK, O-QPSK and π/4-DQPSK schemes. A variety of pulse shaping filters are available and raised cosine filter is specifically chosen for this demo. The raised cosine (RC) pulse comes with an adjustable transition band roll-off parameter α, using which the decay of the transition band can be controlled.

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● Digital Modulations using Matlab : Build Simulation Models from Scratch, ISBN: 978-1521493885
● Digital Modulations using Python ISBN: 978-1712321638
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The RC pulse shaping function is expressed in frequency domain as

Equivalently, in time domain, the impulse response corresponds to

A simple evaluation of the equation (2) produces singularities (undefined points) at p(t = 0) and p(t = ±T_sym/(2α)). The value of the raised cosine pulse at these singularities can be obtained by applying L’Hospital’s rule [1] and the values are

Using the equations above, the raised cosine filter is implemented as a function (refer the books Digital Modulations using Python and Digital Modulations using Matlab for the code).

The function is then tested. It generates a raised cosine pulse for the given symbol duration T_sym = 1s and plots the time-domain view and the frequency response as shown in Figure 1. From the plot, it can be observed that the RC pulse falls off at the rate of 1/|t|³ as t→∞, which is a significant improvement when compared to the decay rate of a sinc pulse which is 1/|t|. It satisfies Nyquist criterion for zero ISI – the pulse hits zero crossings at desired sampling instants. The transition bands in the frequency domain can be made gradual (by controlling α) when compared to that of a sinc pulse.

***Figure 1: Raised-cosine pulse and its manifestation in frequency domain***

Plotting constellation diagram

Now that we have constructed a function for raised cosine pulse shaping filter, the next step is to generate modulated waveforms (using QPSK, O-QPSK and π/4-DQPSK schemes), pass them through a raised cosine filter having a roll-off factor, say α = 0.3 and finally plot the constellation. The constellation for MSK modulated waveform is also plotted.

**Figure 2: Constellations plots for: (a) a = 0.3 RC-filtered QPSK, (b) α = 0.3 RC-filtered O-QPSK, (c) α = 0.3 RC-filtered π/4-DQPSK and (d) MSK**

Conclusions

The resulting simulated plot is shown in the Figure 2. From the resulting constellation diagram, following conclusions can be reached.

Conventional QPSK has 180° phase transitions and hence it requires linear amplifiers with high Q factor
The phase transitions of Offset-QPSK are limited to 90° (the 180° phase transitions are eliminated)
The signaling points for π/4-DQPSK is toggled between two sets of QPSK constellations that are shifted by 45° with respect to each other. Both the 90° and 180° phase transitions are absent in this constellation. Therefore, this scheme produces the lower envelope variations than the rest of the two QPSK schemes.
MSK is a continuous phase modulation, therefore no abrupt phase transition occurs when a symbol changes. This is indicated by the smooth circle in the constellation plot. Hence, a band-limited MSK signal will not suffer any envelope variation, whereas, the rest of the QPSK schemes suffer varied levels of envelope variations, when they are band-limited.

References

[1] Clay S. Turner, Raised Cosine and Root Raised Cosine Formulae, Wireless Systems Engineering, Inc, (May 29, 2007) V1.2↗

In this chapter

Digital Modulators and Demodulators - Passband Simulation Models
● Introduction
● Binary Phase Shift Keying (BPSK)
□ BPSK transmitter
□ BPSK receiver
□ End-to-end simulation
● Coherent detection of Differentially Encoded BPSK (DEBPSK)
● Differential BPSK (D-BPSK)
□ Sub-optimum receiver for DBPSK
□ Optimum noncoherent receiver for DBPSK
● Quadrature Phase Shift Keying (QPSK)
□ QPSK transmitter
□ QPSK receiver
□ Performance simulation over AWGN
● Offset QPSK (O-QPSK)
● π/p=4-DQPSK
● Continuous Phase Modulation (CPM)
□ Motivation behind CPM
□ Continuous Phase Frequency Shift Keying (CPFSK) modulation
□ Minimum Shift Keying (MSK)
● Investigating phase transition properties
● Power Spectral Density (PSD) plots
● Gaussian Minimum Shift Keying (GMSK)
□ Pre-modulation Gaussian Low Pass Filter
□ Quadrature implementation of GMSK modulator
□ GMSK spectra
□ GMSK demodulator
□ Performance
● Frequency Shift Keying (FSK)
□ Binary-FSK (BFSK)
□ Orthogonality condition for non-coherent BFSK detection
□ Orthogonality condition for coherent BFSK
□ Modulator
□ Coherent Demodulator
□ Non-coherent Demodulator
□ Performance simulation
□ Power spectral density

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GMSK implementation and simulation – part 1

What’s the need for GMSK

Minimum shift keying (MSK) is a special case of binary CPFSK with modulation index . It has features such as constant envelope, compact spectrum and good error rate performance. The fundamental problem with MSK is that the spectrum is not compact enough to satisfy the stringent requirements with respect to out-of-band radiation for technologies like GSM and DECT standard. These technologies have very high data rates approaching the RF channel bandwidth. A plot of MSK spectrum (Figure 1) will reveal that the sidelobes with significant energy, extend well beyond the transmission data rate. This is problematic, since it causes severe out-of-band interference in systems with closely spaced adjacent channels.

Figure 1: PSD estimates for BPSK, QPSK and MSK signals

To satisfy such requirements, the MSK spectrum can be easily manipulated by using a pre-modulation low pass filter (LPF). The pre-modulation LPF should have the following properties and it is found that a Gaussian LPF will satisfy all of them [1]

Sharp cut-off and narrow bandwidth – needed to suppress high frequency components.
Lower overshoot in the impulse response – providing protection against excessive instantaneous frequency deviations.
Preservation of filter output pulse area – thereby coherent detection can be applicable.

Pre-modulation Gaussian low pass filter

Gaussian Minimum Shift Keying (GMSK) is a modified MSK modulation technique, where the spectrum of MSK is manipulated by passing the rectangular shaped information pulses through a Gaussian LPF prior to the frequency modulation of the carrier. A typical Gaussian LPF, used in GMSK modulation standards, is defined by the zero-mean Gaussian (bell-shaped) impulse response.

The parameter is the 3-dB bandwidth of the LPF, which is determined from a parameter called as discussed next. If the input to the filter is an isolated unit rectangular pulse (), the response of the filter will be [2]

where,

It is important to note the distinction between the two equations – (1) and (2). The equation for defines the impulse response of the LPF, whereas the equation for , also called as frequency pulse shaping function, defines the LPF’s output when the filter gets excited with a rectangular pulse. This distinction is captured in Figure 2.

Figure 2: Gaussian LPF: Relating h(t) and g(t)

The aim of using GMSK modulation is to have a controlled MSK spectrum. Effectively, a variable parameter called , the product of 3-dB bandwidth of the LPF and the desired data-rate , is often used by the designers to control the amount of spectrum efficiency required for the desired application. As a consequence, the 3-dB bandwidth of the aforementioned LPF is controlled by the design parameter. The range for the parameter is given as . When , the impulse response becomes a Dirac delta function , resulting in a transparent LPF and hence this configuration corresponds to MSK modulation.

The Matlab function to implement the Gaussian LPF’s impulse response (equation (1)), is given in the book (For Python implementation, refer this book). The Gaussian impulse response is of infinite duration and hence in digital implementations it has to be defined for a finite interval, as dictated by the function argument in the code shown next. For example, in GSM standard, is chosen as 0.3 and the time truncation is done to three bit-intervals .

It is also necessary to normalize the filter coefficients of the computed LPF as

Based on the gaussianLPF Matlab function, given in the book (For Python implementation, refer this book), we can compute and plot the impulse response and the response to an isolated unit rectangular pulse – . The resulting plot is shown in Figure 3.

Figure 3: Gaussian LPF:(a) impulse response and (b) response to isolated rectangular pulse

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References

[1] Murota, K. and Hirade, K., GMSK Modulation for Digital Mobile Radio Telephony, IEEE Transactions on Communications, vol COM-29, No. 7. pp. 1044-1050, July 1981.↗
[2] Marvin K. Simon, Bandwidth-Efficient Digital Modulation with Application to Deep Space Communications, JPL Deep Space Communications and Navigation Series,Wiley-Interscience, Hoboken, New Jersey, 2003, ISBN 0-471-44536-3,pp-57.↗

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Differentially encoded BPSK: coherent detection

In coherent detection, the receiver derives its demodulation frequency and phase references using a carrier synchronization loop. Such synchronization circuits may introduce phase ambiguity in the detected phase, which could lead to erroneous decisions in the demodulated bits. For example, Costas loop exhibits phase ambiguity of integral multiples of radians at the lock-in points. As a consequence, the carrier recovery may lock in radians out-of-phase thereby leading to a situation where all the detected bits are completely inverted when compared to the bits during perfect carrier synchronization. Phase ambiguity can be efficiently combated by applying differential encoding at the BPSK modulator input (Figure 1) and by performing differential decoding at the output of the coherent demodulator at the receiver side (Figure 2).

Figure 1: Differential encoded BPSK transmission

In ordinary BPSK transmission, the information is encoded as absolute phases: for binary 1 and for binary 0. With differential encoding, the information is encoded as the phase difference between two successive samples. Assuming is the message bit intended for transmission, the differential encoded output is given as

The differentially encoded bits are then BPSK modulated and transmitted. On the receiver side, the BPSK sequence is coherently detected and then decoded using a differential decoder. The differential decoding is mathematically represented as

This method can deal with the phase ambiguity introduced by synchronization circuits. However, it suffers from performance penalty due to the fact that the differential decoding produces wrong bits when: a) the preceding bit is in error and the present bit is not in error , or b) when the preceding bit is not in error and the present bit is in error. After differential decoding, the average bit error rate of coherently detected BPSK over AWGN channel is given by

Figure 2: Coherent detection of differentially encoded BPSK signal

Following is the Matlab implementation of the waveform simulation model for the method discussed above. Both the differential encoding and differential decoding blocks, illustrated in Figures 1 and 2, are linear time-invariant filters. The differential encoder is realized using IIR type digital filter and the differential decoder is realized as FIR filter.

File 1: dbpsk_coherent_detection.m: Coherent detection of D-BPSK over AWGN channel

Refer Digital Modulations using Matlab : Build Simulation Models from Scratch for full Matlab code. Refer Digital Modulations using Python for full Python code.

Figure 3 shows the simulated BER points together with the theoretical BER curves for differentially encoded BPSK and the conventional coherently detected BPSK system over AWGN channel.

Figure 3: Performance of differentially encoded BPSK and conventional BPSK over AWGN channel

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Complex Baseband Equivalent Models

Key focus: Complex Baseband Equivalent Models are behavioral models that simplify the simulation, saves computation memory requirements and run time.

This article is part of the book Digital Modulations using Matlab : Build Simulation Models from Scratch

Introduction

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.

Complex baseband representation of a modulated signal

By definition, a passband signal is a signal whose one-sided energy spectrum is centered on non-zero carrier frequency and does not extend to DC. A passband signal or any digitally modulated RF waveform is represented as

where,

Recognizing that the sine and cosine terms in the equation (1) are orthogonal components with respect to each other, the signal can be represented in complex form as

When represented in this form, the signal is called the complex envelope or the complex baseband equivalent representation of the real signal . The components and are called inphase and quadrature components respectively. Comparing equations (1) and (2), it is evident that in the complex baseband equivalent representation, the carrier frequency is suppressed. This greatly reduces both the sampling frequency requirements and the memory needed for simulating the model. Furthermore, equation (1), provides a practical way to convert a passband signal to its baseband equivalent and vice-versa [Proakis], as illustrated in Figure 1.

Figure 1: Conversion from baseband to passband and vice-versa

Complex baseband representation of channel response

In the typical communication system model shown in Figure 2(a), the signals represented are real passband signals. The digitally modulated signal occupies a band-limited spectrum around the carrier frequency (). The channel is modeled as a linear time invariant system which is also band-limited in nature. The effect of the channel on the modulated signal is represented as linear convolution (denoted by the operator). Then, the received signal is given by

The corresponding complex baseband equivalent (Figure 2(b)) is expressed as

Figure 2: Passband channel model and its baseband equivalent

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Reference:

[Proakis] Proakis J.G, Digital Communications, fifth edition, New York, McGraw–Hill, 2008

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Passband Simulation Models – Introduction

For a given modulation technique, there are two ways to implement the simulation model: passband model and equivalent baseband model. The passband model is also called waveform level simulation model. The waveform level simulation techniques, described in this chapter, are used to represent the physical interactions of the transmitted signal with the channel. In the waveform level simulations, the transmitted signal, the noise and received signal are all represented by samples of waveforms.

Typically, a waveform level simulation uses many samples per symbol. For the computation of error rate performance of various digital modulation techniques, the value of the symbol at the symbol-sampling time instant is all the more important than the look of the entire waveform. In such a case, the detailed waveform level simulation is not required, instead equivalent baseband discrete-time model, described in chapter 3 can be used. Discrete-time equivalent channel model requires only one sample per symbol, hence it consumes less memory and yields results in a very short span of time.

This article is part of the book Digital Modulations using Matlab : Build Simulation Models from Scratch, ISBN: 978-1521493885 available in ebook (PDF) format (click here) and Paperback (hardcopy) format (click here).

In any communication system, the transmitter operates by modulating the information bearing baseband waveform on to a sinusoidal RF carrier resulting in a passband signal. The carrier frequency, chosen for transmission, varies for different applications. For example, FM radio uses carrier frequency range, whereas for indoor wireless networks the center frequency of transmission is . Hence, the carrier frequency is not the component that contains the information, rather it is the baseband signal that contains the information that is being conveyed.

Actual RF transmission begins by converting the baseband signals to passband signals by the process of up-conversion. Similarly, the passband signals are down-converted to baseband at the receiver, before actual demodulation could begin. Based on this context, two basic types of behavioral models exist for simulation of communication systems – passband models and its baseband equivalent. In the passband model, every cycle of the RF carrier is simulated in detail and the power spectrum will be concentrated near the carrier frequency . Hence, passband models consume more memory, as every point in the RF carrier needs to be stored in computer memory for simulation.

On the other hand, the signals in baseband models are centered near zero frequency. In baseband equivalent models, the RF carrier is suppressed and therefore the number of samples required for simulation is greatly reduced. Furthermore, if the behavior of the system is well understood, the baseband model can be further simplified and the system can be implemented entirely based on the samples at symbol-sampling time instants.

Continue reading: Conversion of passband model to baseband equivalent model is discussed here.

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BPSK – Binary Phase Shift Keying

Key focus: BPSK, Binary Phase Shift Keying, bpsk modulation, bpsk demodulation, BPSK matlab, BPSK python implementation, BPSK constellation

BPSK – introduction

BPSK stands for Binary Phase Shift Keying. It is a type of modulation used in digital communication systems to transmit binary data over a communication channel.

In BPSK, the carrier signal is modulated by changing its phase by 180 degrees for each binary symbol. Specifically, a binary 0 is represented by a phase shift of 180 degrees, while a binary 1 is represented by no phase shift.

BPSK is a straightforward and effective modulation method and is frequently utilized in applications where the communication channel is susceptible to noise and interference. It is also utilized in different wireless communication systems like Wi-Fi, Bluetooth, and satellite communication.

Implementation details

Binary Phase Shift Keying (BPSK) is a two phase modulation scheme, where the 0’s and 1’s in a binary message are represented by two different phase states in the carrier signal: \(\theta=0^{\circ}\) for binary 1 and \(\theta=180^{\circ}\) for binary 0.

In digital modulation techniques, a set of basis functions are chosen for a particular modulation scheme. Generally, the basis functions are orthogonal to each other. Basis functions can be derived using Gram Schmidt orthogonalization procedure [1]. Once the basis functions are chosen, any vector in the signal space can be represented as a linear combination of them. In BPSK, only one sinusoid is taken as the basis function. Modulation is achieved by varying the phase of the sinusoid depending on the message bits. Therefore, within a bit duration \(T_b\), the two different phase states of the carrier signal are represented as,

\begin{align*} s_1(t) &= A_c\; cos\left(2 \pi f_c t \right), & 0 \leq t \leq T_b \quad \text{for binary 1}\\ s_0(t) &= A_c\; cos\left(2 \pi f_c t + \pi \right), & 0 \leq t \leq T_b \quad \text{for binary 0} \end{align*}

where, \(A_c\) is the amplitude of the sinusoidal signal, \(f_c\) is the carrier frequency \(Hz\), \(t\) being the instantaneous time in seconds, \(T_b\) is the bit period in seconds. The signal \(s_0(t)\) stands for the carrier signal when information bit \(a_k=0\) was transmitted and the signal \(s_1(t)\) denotes the carrier signal when information bit \(a_k=1\) was transmitted.

The constellation diagram for BPSK (Figure 3 below) will show two constellation points, lying entirely on the x axis (inphase). It has no projection on the y axis (quadrature). This means that the BPSK modulated signal will have an in-phase component but no quadrature component. This is because it has only one basis function. It can be noted that the carrier phases are \(180^{\circ}\) apart and it has constant envelope. The carrier’s phase contains all the information that is being transmitted.

BPSK transmitter

A BPSK transmitter, shown in Figure 1, is implemented by coding the message bits using NRZ coding (\(1\) represented by positive voltage and \(0\) represented by negative voltage) and multiplying the output by a reference oscillator running at carrier frequency \(f_c\).

The following function (bpsk_mod) implements a baseband BPSK transmitter according to Figure 1. The output of the function is in baseband and it can optionally be multiplied with the carrier frequency outside the function. In order to get nice continuous curves, the oversampling factor (\(L\)) in the simulation should be appropriately chosen. If a carrier signal is used, it is convenient to choose the oversampling factor as the ratio of sampling frequency (\(f_s\)) and the carrier frequency (\(f_c\)). The chosen sampling frequency must satisfy the Nyquist sampling theorem with respect to carrier frequency. For baseband waveform simulation, the oversampling factor can simply be chosen as the ratio of bit period (\(T_b\)) to the chosen sampling period (\(T_s\)), where the sampling period is sufficiently smaller than the bit period.

Refer Digital Modulations using Matlab : Build Simulation Models from Scratch for full Matlab code.
Refer Digital Modulations using Python for full Python code

File 1: bpsk_mod.m: Baseband BPSK modulator

function [s_bb,t] = bpsk_mod(ak,L)
%Function to modulate an incoming binary stream using BPSK(baseband)
%ak - input binary data stream (0's and 1's) to modulate
%L - oversampling factor (Tb/Ts)
%s_bb - BPSK modulated signal(baseband)
%t - generated time base for the modulated signal
N = length(ak); %number of symbols
a = 2*ak-1; %BPSK modulation
ai=repmat(a,1,L).'; %bit stream at Tb baud with rect pulse shape
ai = ai(:).';%serialize
t=0:N*L-1; %time base
s_bb = ai;%BPSK modulated baseband signal

BPSK receiver

A correlation type coherent detector, shown in Figure 2, is used for receiver implementation. In coherent detection technique, the knowledge of the carrier frequency and phase must be known to the receiver. This can be achieved by using a Costas loop or a Phase Lock Loop (PLL) at the receiver. For simulation purposes, we simply assume that the carrier phase recovery was done and therefore we directly use the generated reference frequency at the receiver – \(cos( 2 \pi f_c t)\).

*Figure 2: Coherent detection of BPSK (correlation type)*

In the coherent receiver, the received signal is multiplied by a reference frequency signal from the carrier recovery blocks like PLL or Costas loop. Here, it is assumed that the PLL/Costas loop is present and the output is completely synchronized. The multiplied output is integrated over one bit period using an integrator. A threshold detector makes a decision on each integrated bit based on a threshold. Since, NRZ signaling format was used in the transmitter, the threshold for the detector would be set to \(0\). The function bpsk_demod, implements a baseband BPSK receiver according to Figure 2. To use this function in waveform simulation, first, the received waveform has to be downconverted to baseband, and then the function may be called.

Refer Digital Modulations using Matlab : Build Simulation Models from Scratch for full Matlab code.
Refer Digital Modulations using Python for full Python code

File 2: bpsk_demod.m: Baseband BPSK detection (correlation receiver)

function [ak_cap] = bpsk_demod(r_bb,L)
%Function to demodulate an BPSK(baseband) signal
%r_bb - received signal at the receiver front end (baseband)
%N - number of symbols transmitted
%L - oversampling factor (Tsym/Ts)
%ak_cap - detected binary stream
x=real(r_bb); %I arm
x = conv(x,ones(1,L));%integrate for L (Tb) duration
x = x(L:L:end);%I arm - sample at every L
ak_cap = (x > 0).'; %threshold detector

End-to-end simulation

The complete waveform simulation for the end-to-end transmission of information using BPSK modulation is given next. The simulation involves: generating random message bits, modulating them using BPSK modulation, addition of AWGN noise according to the chosen signal-to-noise ratio and demodulating the noisy signal using a coherent receiver. The topic of adding AWGN noise according to the chosen signal-to-noise ratio is discussed in section 4.1 in chapter 4. The resulting waveform plots are shown in the Figure 2.3. The performance simulation for the BPSK transmitter/receiver combination is also coded in the program shown next (see chapter 4 for more details on theoretical error rates).

The resulting performance curves will be same as the ones obtained using the complex baseband equivalent simulation technique in Figure 4.4 of chapter 4.

Refer Digital Modulations using Matlab : Build Simulation Models from Scratch for full Matlab code.
Refer Digital Modulations using Python for full Python code

File 3: bpsk_wfm_sim.m: Waveform simulation for BPSK modulation and demodulation

*Figure 3: (a) Baseband BPSK signal,(b) transmitted BPSK signal – with carrier, (c) constellation at transmitter and (d) received signal with AWGN noise*

References:

[1] Lloyd N. Trefethen, David Bau III , Numerical linear algebra, Philadelphia: Society for Industrial and Applied Mathematics, ISBN 978-0-89871-361-9, pp.56

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