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 S_{QPSK}(t)= \sqrt{\frac{2E_{s}}{T_{s}}}cos(2\pi f_{c}t +(2i-1)\frac{\pi}{4}), 0\leq t\leq 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 S_{QPSK}(t) = \sqrt{\frac{2E_{s}}{T_{s}}}cos (2\pi f_{c}t+(2i+1)\frac{\pi }{4} ), 0\leq t\leq 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)= \sqrt{\frac{2E_{s}}{T_{s}}}cos (2i-1)\frac{\pi }{4}cos2\pi f_{c}t - \sqrt{\frac{2E_{s}}{T_{s}}}sin (2i-1)\frac{\pi }{4}sin2\pi f_{c}twhere 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 _{1}(t)=\sqrt{\frac{2}{T_{s}}}cos 2\pi f_{c}t, 0\leq t\leq T_{s}, \Phi _{2}(t)=\sqrt{\frac{2}{T_{s}}}sin 2\pi f_{c}t, 0\leq t\leq T_{s}

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

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

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