##### Quadrature Phase Shift Keying (QPSK), Theory and Matlab Code
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Quadrature Phase Shift Keying (QPSK) is a type of phase shift keying (PSK) in which two bits of data are encoded in each symbol, allowing for higher data rates than simpler modulation schemes like binary phase shift keying (BPSK). In QPSK, the phase of the carrier signal can take on one of four possible values, corresponding to the four possible combinations of two bits of data.

The modulation process in QPSK involves mapping the input bits to one of the four possible phase states. The demodulation process involves determining the phase of the received signal and mapping it back to the original bits.

QPSK is widely used in wireless communications, such as in digital television and satellite communications. It is also less sensitive to noise and interference than other modulation schemes, making it a good choice for applications where the signal-to-noise ratio is low.

Like DPSK, QPSK implementation can be done in software or hardware, and can be extended to other forms of PSK like 8-PSK and 16-PSK, with the main difference being the number of possible phase states used for encoding.

## QPSK implementation

QPSK implementation typically involves the following steps:

1. Bit to symbol mapping: The input data is divided into two-bit symbols, and each symbol is mapped to one of the four possible phase states. This can be done using a look-up table or by using a mathematical function to calculate the phase value for each symbol.
2. Modulation: The carrier signal is modulated by the phase values obtained in the previous step. This can be done by multiplying the carrier signal with a complex exponential function, where the phase of the exponential function is equal to the phase of the symbol.
3. Transmission: The modulated signal is transmitted over a communication channel, such as a wired or wireless link.
4. Demodulation: The received signal is demodulated to recover the original data. This is typically done by multiplying the received signal with a complex exponential function of the same frequency as the carrier signal, but with a phase of zero. The result is a complex signal, which can be used to determine the phase of the received signal.
5. Symbol to bit mapping: The demodulated signal is mapped back to the original bits by determining the symbol to which the received phase corresponds. This can be done by using a look-up table or by using a mathematical function to calculate the symbol value for each phase.

It’s worth noting that the implementation of QPSK can be done in software or hardware, and it can be affected by the presence of noise, interference or fading in the channel. To mitigate these effects, different techniques such as channel coding, interleaving, and error correction can be used in conjunction with QPSK.

• Matlab

## Matlab Code

``````clc;
clear al;
close all;
Tb=1;
t=0:(Tb/100):Tb;
fc=1;
c1=sqrt(2/Tb)*cos(2*pi*fc*t);
c2=sqrt(2/Tb)*sin(2*pi*fc*t);

subplot(3,2,1);
plot(t,c1);
title('Carrier Signal-1');
xlabel('t ----->');
ylabel('c1(t)');
grid on;

subplot(3,2,2);
plot(t,c2);
title('Carrier Signal-2');
xlabel('t ----->');
ylabel('c2(t)');
grid on;

N=16;
m=rand(1,N);

t1=0;
t2=Tb;
for i=1:2:(N-1)
t=[t1:(Tb/100):t2];

if m(i)>0.5
m(i)=1;
m_s=ones(1,length(t));
else
m(i)=0;
m_s=-1*ones(1,length(t));
end
odd_sig(i,:)=c1.*m_s;

if m(i+1)>0.5
m(i+1)=1;
m_s=ones(1,length(t));
else
m(i+1)=0;
m_s=-1*ones(1,length(t));
end
even_sig(i,:)=c2.*m_s;

Qpsk =odd_sig + even_sig;

subplot(3,2,3);
stem(m);
title('Binary Data of Message Signal');
xlabel('n ----->');
ylabel('b(n)');
grid on;

subplot(3,2,4);
plot(t,Qpsk(i,:));
title('QPSK Modulated Signal');
xlabel('t ------>');
ylabel('s(t)');
grid on;
hold on;
t1=t1+(Tb+0.01);
t2=t2+(Tb+0.01);
end
hold off;

t1=0;
t2=Tb;
for i=1:N-1
t=[t1:(Tb/100):t2];

x1=sum(c1.*Qpsk(i,:));
x2=sum(c2.*Qpsk(i,:));

if (x1>0 && x2>0)
demod(i)=1;
demod(i+1)=1;
elseif (x1>0 && x2<0)
demod(i)=1;
demod(i+1)=0;
elseif (x1<0 && x2<0)
demod(i)=0;
demod(i+1)=0;
elseif (x1<0 && x2>0)
demod(i)=0;
demod(i+1)=1;
end

end

subplot(3,2,5);
stem(demod);
title('QPSK Demodulated Signal');
xlabel('n ------>');
ylabel('b(n)');
grid on;``````

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