# QPSK equation, wave forms and Signal space diagram

### QPSK equation:-

The meaning of QPSK is  that the carrier signal takes on different phases Π/4, 3Π/4, 5Π/4 and 7Π/4 based on incoming di-bit combination  or symbol.

$\large&space;S_{QPSK}(t)=&space;\sqrt{\frac{2E_{s}}{T_{s}}}cos(2\pi&space;f_{c}t&space;+(2i-1)\frac{\pi}{4}),&space;0\leq&space;t\leq&space;T_{s}$

= 0, elsewhere, where  i =  1,2,3,4.

Eb and Tb are the bit energy and bit-interval , Es and Ts are the energy per symbol  and symbol duration. Ts = 2 Tb

The carrier frequency fc = nc /Ts. where nc is a fixed integer.

each possible value of phase corresponds to a unique di-bit. then the foregoing phase values to represent the gray encoded set of di-bits 11,01,10 and 00, where only a single bit is changed from one di-bit to the next.

QPSK equation can be represented in another format as follows

$\large&space;S_{QPSK}(t)&space;=&space;\sqrt{\frac{2E_{s}}{T_{s}}}cos&space;(2\pi&space;f_{c}t+(2i+1)\frac{\pi&space;}{4}&space;),&space;0\leq&space;t\leq&space;T_{s}$

= 0, elsewhere   ,where i=0,1,2,3.

The above two equations are same, there is a change in i values. alternately the equation can be represented as follows.

$S_{QPSK}(t)=&space;\sqrt{\frac{2E_{s}}{T_{s}}}cos&space;(2i-1)\frac{\pi&space;}{4}cos2\pi&space;f_{c}t&space;-&space;\sqrt{\frac{2E_{s}}{T_{s}}}sin&space;(2i-1)\frac{\pi&space;}{4}sin2\pi&space;f_{c}t$where i= 1,2,3,4.

The above equation can be expanded cos(A+B). There are two orthogonal functions Φ1(t) and Φ2(t) where

$\Phi&space;_{1}(t)=\sqrt{\frac{2}{T_{s}}}cos&space;2\pi&space;f_{c}t,&space;0\leq&space;t\leq&space;T_{s},&space;\Phi&space;_{2}(t)=\sqrt{\frac{2}{T_{s}}}sin&space;2\pi&space;f_{c}t,&space;0\leq&space;t\leq&space;T_{s}$

$S_{QPSK}(t)=\sqrt{E_{s}}cos(2i-1)\frac{\pi&space;}{4}&space;*&space;\Phi&space;_{1}(t)&space;-&space;\sqrt{E_{s}}sin(2i-1)\frac{\pi&space;}{4}&space;*&space;\Phi&space;_{2}(t)$

Let    $b_{o}(t)=&space;\sqrt{E_{s}}cos(2i-1)\frac{\pi&space;}{4}$     and   $b_{e}(t)=&space;-\sqrt{E_{s}}sin(2i-1)\frac{\pi&space;}{4}$

then the resultant equation is:     $S_{QPSK}(t)=&space;b_{o}(t)&space;*&space;\Phi&space;_{1}(t)&space;+&space;b_{e}(t)&space;*&space;\Phi&space;_{2}(t)$.

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## Author: Lakshmi Prasanna Ponnala

Completed M.Tech in Digital Electronics and Communication Systems and currently working as a faculty.