# Phase Shift Keying (PSK)

It is the most efficient of the 3 digital modulation techniques ASK,PSK and FSK. It is used for high bit rates.

Definition:-

In BPSK, the Phase of the carrier is shifted/ Changed according to the incoming Binary data sequence.

BPSK was developed during the early days of the deep space programs. PSK is now widely used in both Military and commercial communication systems.

we represent data sequence $\left&space;\{&space;b_{k}&space;\right&space;\}$ in Bipolar-NRZ scheme.

Expressions of PSK:-

The carrier signal is a continuous wave (or) sinusoidal wave form

$S(t)=A&space;\cos&space;2\pi&space;f_{c}t$ .

The normalized power is $P=\frac{A^{2}}{2}$

$A=\sqrt{2P_{s}}$ .

The carrier signal can be expresses in terms of power as $S(t)=\sqrt{2P_{s}}&space;\cos&space;2\pi&space;f_{c}t$.

if energy per bit is $E_{b}$ and the bit interval as $T_{b}$ then the carrier signal is $S(t)=\sqrt{\frac{2E_{b}}{T_{b}}}&space;\cos&space;2\pi&space;f_{c}t$.

Now according to PSK Binary ‘1’ is represented with carrier phase $0^{o}$ and Binary ‘0’ is represented with  a phase  $180^{o}$.

$\left\{\begin{matrix}&space;S_{PSK}(t)=\sqrt{2P_{s}}&space;\cos&space;2\pi&space;f_{c}t\rightarrow&space;\&space;Binary\&space;'1'&space;\\&space;=\sqrt{2P_{s}}&space;\cos&space;(2\pi&space;f_{c}t&space;+\pi&space;)\&space;\rightarrow&space;\&space;Binary\&space;'0'&space;\end{matrix}\right.$

in terms of Energy and bit duration ASK signal can be written as

$\left\{\begin{matrix}&space;S_{PSK}(t)=\sqrt{\frac{2E_{b}}{T_{b}}}&space;\cos&space;2\pi&space;f_{c}t\rightarrow&space;\&space;Binary\&space;'1'&space;\\&space;\&space;\&space;=\sqrt{\frac{2E_{b}}{T_{b}}}&space;\cos&space;(2\pi&space;f_{c}t+\pi&space;)\&space;\rightarrow&space;\&space;Binary\&space;'0'&space;\end{matrix}\right.$

i.e,

$\left\{\begin{matrix}&space;S_{PSK}(t)=\sqrt{\frac{2E_{b}}{T_{b}}}&space;\cos&space;2\pi&space;f_{c}t\rightarrow&space;\&space;Binary\&space;'1'&space;\\&space;\&space;\&space;\&space;=-&space;\sqrt{\frac{2E_{b}}{T_{b}}}&space;\cos&space;2\pi&space;f_{c}t\&space;\rightarrow&space;\&space;Binary\&space;'0'&space;\end{matrix}\right.$.

The wave forms of PSK modulation scheme are shown in the figure , where b(t) represents the polar NRZ representation of binary sequence $\left&space;\{&space;b_{k}&space;\right&space;\}$

i.e, $b(t)&space;=&space;\left\{\begin{matrix}&space;\&space;+v&space;\&space;(or)&space;\&space;1\&space;volt&space;\&space;when&space;\left&space;\{&space;b_{k}&space;\right&space;\}=1\\&space;\&space;-v&space;\&space;(or)&space;\&space;-1\&space;volt&space;\&space;when&space;\left&space;\{&space;b_{k}&space;\right&space;\}=0&space;\end{matrix}\right.$ .

PSK Transmitter:-

The figure shows the PSK generator (or) PSK Transmitter

The incoming binary sequence $\left&space;\{&space;b_{k}&space;\right&space;\}$ (in the form of a signal)  into $\pm&space;1&space;\&space;volt$ by using NRZ level encoder.

$b(t)=&space;\left\{\begin{matrix}&space;+1V&space;\&space;for&space;\&space;symbol&space;\1\\&space;-1V&space;\&space;for&space;\&space;symbol&space;&space;\end{matrix}\right.$

Now b(t) and the carrier S(t) are applied to product modulator, to get the PSK modulated signal at the output.

i.e, $S_{PSK}(t)=b(t).S(t)$

finally,

$\left\{\begin{matrix}&space;S_{PSK}(t)=\sqrt{\frac{2E_{b}}{T_{b}}}&space;\cos&space;2\pi&space;f_{c}t\rightarrow&space;\&space;Binary\&space;'1'&space;\\&space;\&space;\&space;\&space;=-&space;\sqrt{\frac{2E_{b}}{T_{b}}}&space;\cos&space;2\pi&space;f_{c}t\&space;\rightarrow&space;\&space;Binary\&space;'0'&space;\end{matrix}\right.$

Coherent PSK Detector:-

The figure shows the Block Diagram of coherent PSK/BPSK Detector. The PSK signal $S_{PSK}(t)$ is applied to the correlator (The Block product Modulator followed up by the Integrator).

$S_{PSK}(t)$ is multiplied by local carrier C(t) this carrier C(t) is phase locked with that of the carrier used in the Transmitter. As this is coherent reception.

The product $S_{PSK}(t).C(t)$ is applied to the Integrator. The integrator eliminates the noise.

The Integrator integrates the input over one bit interval $T_{b}$ and the output is given to a threshold device. If the threshold voltage is set to 0 V.

the output of threshold device v(t) (or) v is either ‘1’ (or) ‘0’ based on the following condition.

$v>&space;0\rightarrow&space;\&space;a&space;\&space;symbol&space;\&space;'0'&space;\&space;is&space;\&space;detected.$

$v\leq&space;0\rightarrow&space;\&space;a&space;\&space;symbol&space;\&space;'1'&space;\&space;is&space;\&space;detected.$

Note:- The input to demodulator is not $S_{PSK}(t)$ always most of the times it is interfered with noise n(t) in the channel.

in coherent detection input to the demodulator is simply $S_{PSK}(t)$ signal where as in Non-coherent detection the input is noisy PSK signal.

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## Author: vikramarka

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