BPSK modulation and demodulation — constellation, decision rule, and BER intuition

Lesson 1 of 5. Start here: one bit on one carrier is the cleanest digital modem. Next: QPSK (two bits, same energy-efficiency story).

In one sentence: BPSK sends one bit as $latex \pm$ phase on a carrier; coherent detection in AWGN yields $latex P_b=Q(\sqrt{2E_b/N_0})$.

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 the modulation you reach for when the link is power-limited: deep-space probes, some satellite beacons, and the most robust rate of the older direct-sequence Wi-Fi modes. Bluetooth and current Wi-Fi use other constellations.

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:$latex \theta=0^{\circ}$ for binary 1 and$latex \theta=180^{\circ}$ 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$latex T_b$, 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,$latex A_c$ is the amplitude of the sinusoidal signal,$latex f_c$ is the carrier frequency$latex Hz$,$latex t$ being the instantaneous time in seconds,$latex T_b$ is the bit period in seconds. The signal$latex s_0(t)$ stands for the carrier signal when information bit$latex a_k=0$ was transmitted and the signal$latex s_1(t)$ denotes the carrier signal when information bit$latex a_k=1$ 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$latex 180^{\circ}$ 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 ($latex 1$ represented by positive voltage and$latex 0$ represented by negative voltage) and multiplying the output by a reference oscillator running at carrier frequency$latex f_c$.

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 ($latex L$) 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 ($latex f_s$) and the carrier frequency ($latex f_c$). If a carrier is present, the sampling rate has to clear the Nyquist rate of that carrier,$latex f_s > 2 f_c$. Taking the oversampling factor as$latex f_s/f_c$ only helps once that inequality is already true. For baseband waveform simulation, the oversampling factor can simply be chosen as the ratio of bit period ($latex T_b$) to the chosen sampling period ($latex T_s$), 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 -$latex cos( 2 \pi f_c t)$.

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$latex 0$. 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 the four panels of Figure 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 bit error rate of coherent BPSK in AWGN is exactly$latex P_b = Q\left(\sqrt{2 E_b/N_0}\right)$. The baseband model below is that receiver: symbols at$latex \pm 1$, noise of variance$latex N_0/2$ on the real line, and a threshold at zero.

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

import numpy as np
from scipy.special import erfc

def q_function(x):
    return 0.5 * erfc(np.asarray(x, dtype=float) / np.sqrt(2.0))

def bpsk_ber_sim(ebn0_db, n_bits=200000, seed=1):
    """Baseband coherent BPSK. Symbol energy is 1, so N0 = 1/(Eb/N0)."""
    rng = np.random.default_rng(seed)
    ebn0 = 10 ** (ebn0_db / 10.0)
    n0 = 1.0 / ebn0
    bits = rng.integers(0, 2, size=n_bits)
    symbols = 2 * bits - 1
    noise = np.sqrt(n0 / 2.0) * rng.normal(size=n_bits)
    detected = (symbols + noise) > 0
    return np.mean(detected != bits)

for db in (0, 2, 4, 6, 8):
    simulated = bpsk_ber_sim(db)
    theory = q_function(np.sqrt(2 * 10 ** (db / 10.0)))
    print("Eb/N0=%4.1f dB   simulated %.3e   theory %.3e" % (db, simulated, theory))

FAQ

What is the recommended reading order? Follow the five lessons at the top: BPSK → QPSK → M-PSK → QAM → EVM. Use the BER and EVM calculators to check numbers as you go.

Why learn BPSK before QPSK? Every later constellation reuses the same matched-filter / correlator idea. If BPSK BER vs $latex E_b/N_0$ is clear, QPSK and M-PSK are mostly bookkeeping.

Similar articles

• 1. BPSK
• 2. QPSK
• 3. M-PSK sim
• 4. QAM sim
• 5. EVM
• Tool: BER vs Eb/N0 calculator
• Eb/N0, Es/N0 and SNR conversions

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