Binary Phase Shift Keying (BPSK) is a type of digital modulation technique in which we are sending one bit per symbol i.e., ‘0’ or a ‘1’. Hence, the bit rate and symbol rate are the same. Depending upon the message bit, we can have a phase shift of 0^o or 180^o with respect to a reference carrier as shown in the figure above.

For example, we can have the following transmitted band-pass symbols:

or

Where ‘E’ is the symbol energy, ‘T’ is the symbol time period, f is the frequency of the carrier. Using Gram-schmidt orthogonalization, we get a single orthonormal basis function, given as:

Hence, the resulting constellation diagram can be given as:

 

There are only two in-phase components and no quadrature component.

Now, we can easily see that the two waveform of S_o and S₁ are inverted with respect to one another and we can use following scheme to design a BPSK modulator:

First the NRZ encoder converts these digital bits into impulses to add a notion of time into them. Then NRZ waveform is generated by up-sampling these impulses. Afterwards, multiplication with the carrier (orthonormal basis function) is carried out to generate the modulated BPSK waveform.

Demodulator Design:

We do coherent demodulation of the BPSK signal at the receiver. Coherent demodulation requires the received signal to be multiplied with the carrier having the same frequency and phase as at the transmitter. The phase synchronization is normally achieved using Phase Locked Loop (PLL) at the receiver. PLL implementation is not done here, rather we assume perfect phase synchronization. Block diagram of BPSK modulator is shown in the figure below.  After the multiplication with the carrier (orthonormal basis function), the signal is integrated over the symbol duration ‘T’ and sampled. Then thresholding is applied to determine if a ‘1’ was sent (+ve voltage) or a ‘0’ was sent (-ve voltage).

The Matlab simulation code is given below. Here for the sake of simplicity,  the bit rate is fixed to 1 bit/s (i.e., T=1 second). It is also assumed that Phased Locked Loop (PLL) has already achieved exact phase synchronization.

clear all;
close all;
%Nb is the number of bits to be transmitted
T=1;%Bit rate is assumed to be 1 bit/s;
%bits to be transmitted
b=[1 0 1 0 1]
%Rb is the bit rate in bits/second


NRZ_out=[];
 
%Vp is the peak voltage +v of the NRZ waveform
Vp=1;
%Here we encode input bitstream as Bipolar NRZ-L waveform
for index=1:size(b,2)
 if b(index)==1
 NRZ_out=[NRZ_out ones(1,200)*Vp];
 elseif b(index)==0
 NRZ_out=[NRZ_out ones(1,200)*(-Vp)];
 end
end

%Generated bit stream impulses
figure(1);
stem(b);
xlabel('Time (seconds)-->')
ylabel('Amplitude (volts)-->')
title('Impulses of bits to be transmitted');
figure(2);
plot(NRZ_out);
xlabel('Time (seconds)-->');
ylabel('Amplitude (volts)-->');
title('Generated NRZ signal');

t=0.005:0.005:5;
%Frequency of the carrier
f=5;
%Here we generate the modulated signal by multiplying it with 
%carrier (basis function)
Modulated=NRZ_out.*(sqrt(2/T)*cos(2*pi*f*t));
figure;
plot(Modulated);
xlabel('Time (seconds)-->');
ylabel('Amplitude (volts)-->');
title('BPSK Modulated signal');


y=[];
%We begin demodulation by multiplying the received signal again with 
%the carrier (basis function)
demodulated=Modulated.*(sqrt(2/T)*cos(2*pi*f*t));
%Here we perform the integration over time period T using trapz 
%Integrator is an important part of correlator receiver used here
for i=1:200:size(demodulated,2)
 y=[y trapz(t(i:i+199),demodulated(i:i+199))];
end
received=y>0;
figure;
stem(received)
title('Impulses of Received bits');
xlabel('Time (seconds)-->');
ylabel('Amplitude (volts)')

 

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