Category: Signal Processing

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.

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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 \(f_c\).

BPSK transmitter (binary phase shift keying modulation)
Figure 1: BPSK transmitter

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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Topics 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

Understanding Gibbs Phenomenon in signal processing

Introduction

Gibbs phenomenon is a phenomenon that occurs in signal processing and Fourier analysis when approximating a discontinuous function using a series of Fourier coefficients. Specifically, it is the observation that the overshoots near the discontinuities of the approximated function do not decrease with increasing numbers of Fourier coefficients used in the approximation.

In other words, when a discontinuous function is approximated by its Fourier series, the resulting series will exhibit oscillations near the discontinuities that do not diminish as more terms are added to the series. This can lead to a “ringing” effect in the signal, where there are spurious oscillations near the discontinuity that can persist even when the number of Fourier coefficients used in the approximation is increased.

The Gibbs phenomenon is named after American physicist Josiah Willard Gibbs, who first described it in 1899. It is a fundamental limitation of the Fourier series approximation and can occur in many other areas of signal processing and analysis as well.

Gibbs phenomenon

Fourier transform represents signals in frequency domain as summation of unique combination of sinusoidal waves. Fourier transforms of various signals are shown in the Figure 1. Some of these signals, square wave and impulse, have abrupt discontinuities (sudden changes) in time domain. They also have infinite frequency content in the frequency domain.

Figure 1: Frequency response of various test signals

Therefore, abrupt discontinuities in the signals require infinite frequency content in frequency domain. As we know, in order to represent these signals in computer memory, we cannot dispense infinite memory (or infinite bandwidth when capturing/measurement) to hold those infinite frequency terms. Somewhere, the number of frequency terms has to be truncated. This truncation in frequency domain manifests are ringing artifacts in time domain and vice-versa. This is called Gibbs phenomenon.

Figure 2: Ringing artifact (Gibbs phenomenon) on a square wave when the number of frequency terms is truncated

These ringing artifacts result from trying to describe the given signal with less number of frequency terms than the ideal. In practical applications, the ringing artifacts can result from

● Truncation of frequency terms – For example, to represent a perfect square wave, an infinite number of frequency terms are required. Since we cannot have an instrument with infinite bandwidth, the measurement truncates the number of frequency terms, resulting in the ringing artifact.

● Shape of filters – The ringing artifacts resulting from filtering operation is related to the sharp transitions present in the shape of the filter impulse response.

FIR filters and Gibbs phenomenon

Owing to their many favorable properties, digital Finite Impulse Response (FIR) filters are extremely popular in many signal processing applications. FIR filters can be designed to exhibit linear phase response in passband, so that the filter does not cause delay distortion (or dispersion) where different frequency components undergo different delays.

The simplest FIR design technique is the Impulse Response Truncation, where an ideal impulse response of infinite duration is truncated to finite length and the samples are delayed to make it causal. This method comes with an undesirable effect due to Gibbs Phenomenon.

Ideal brick wall characteristics in frequency domain is desired for most of the filters. For example, a typical ideal low pass filter necessitates sharp transition between passband and stopband. Any discontinuity (abrupt transitions) in one domain requires infinite number of components in the other domain.

For example, a rectangular function with abrupt transition in frequency domain translates to a \(sinc(x)=sin(x)/x\) function of infinite duration in time domain. In practical filter design, the FIR filters are of finite length. Therefore, it is not possible to represent an ideal filter with abrupt discontinuities using finite number of taps and hence the \(sinc\) function in time domain should be truncated appropriately. This truncation of an infinite duration signal in time domain leads to a phenomenon called Gibbs phenomenon in frequency domain. Since some of the samples in time domain (equivalently harmonics in frequency domain) are not used in the reconstruction, it leads to oscillations and ringing effect in the other domain. This effect is called Gibbs phenomenon.

Similar effect can also be observed in the time domain if truncation is done in the frequency domain.

This effect due to abrupt discontinuities will exists no matter how large the number of samples  is made. The situation can be improved by using a smoothly tapering windows like Blackman, Hamming , Hanning , Keiser windows etc.,

Demonstration of Gibbs Phenomenon using Matlab:

In this demonstration, a sinc pulse in time domain is considered. Sinc pulse with infinite duration in time domain, manifests as perfect rectangular shape in frequency domain. In this demo, we truncate the sinc pulse in the time domain at various length and use FFT (Fast Fourier Transform) to visualize it frequency domain. As the duration of time domain samples increases, the ringing artifact become less pronounced and the shape approaches ideal brick wall filter response.

%Gibbs Phenomenon
clearvars; % clear all stored variables

Nsyms = 5:10:60; %filter spans in symbol duration

Tsym=1; %Symbol duration
L=16; %oversampling rate, each symbol contains L samples
Fs=L/Tsym; %sampling frequency

for Nsym=Nsyms,
    [p,t]=sincFunction(L,Nsym); %Sinc Pulse

    subplot(1,2,1);
    plot(t*Tsym,p,'LineWidth',1.5);axis tight;
    ylim([-0.3,1.1]);
    title('Sinc pulse');xlabel('Time (s)');ylabel('Amplitude');
    
    [fftVals,freqVals]=freqDomainView(p,Fs,'double'); %See Chapter 1
    subplot(1,2,2);
    plot(freqVals,abs(fftVals)/abs(fftVals(length(fftVals)/2+1)),'LineWidth',1.5);
    xlim([-2 2]); ylim([0,1.1]);
    title('Frequency response of Sinc (FFT)');
    xlabel('Normalized Frequency (Hz)');ylabel('Magnitude');
    pause;% wait for user input to continue
end

Simulation Results:

Figure 3: Sinc pulse constructed with Nsym = 5 (filter span in symbols) and L =16 (samples/symbol)
Figure 4: Sinc pulse constructed with Nsym = 15 (filter span in symbols) and L =16 (samples/symbol)
Figure 5: Sinc pulse constructed with Nsym = 35 (filter span in symbols) and L =16 (samples/symbol)

I have also discussed with examples, Gibbs phenomenon applied to truncation of Fourier series coefficients. You can read about it here.

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[5] How to Interpret FFT results – complex DFT, frequency bins and FFTShift
[6] How to interpret FFT results – obtaining magnitude and phase information
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