# Frequency Shift Keying(FSK)

### Frequency Shift keying:-

In a Binary FSK system , Symbols ‘1’ and ‘0’ are distinguished from each other by transmitting one of two sinusoidal waves that differ in frequency by a fixed amount. (or) the frequency of the carrier signal is shifted to two frequencies fc1 (or)fH and fc2/fL for symbols ‘1’ and ‘0’ transmission.

The equation for FSK signal  is

$S_{FSK}(t)=\sqrt{\frac{2E_{b}}{T_{b}}}cos(2\pi&space;f_{ci})(t),0\leq&space;t\leq&space;T_{b}$

$=&space;0&space;elsewhere$

and i= 1,2.

$S_{FSK}(t)=\sqrt{\frac{2E_{b}}{T_{b}}}cos(2\pi&space;f_{H})(t),&space;for&space;symbol'1'$

$S_{FSK}(t)=\sqrt{\frac{2E_{b}}{T_{b}}}cos(2\pi&space;f_{L})(t),&space;for&space;symbol'0'$

where $f_{c1}=f_{H}$ is generally a high-frequency, $f_{c2}=f_{L}$ is a low-frequency. Vice-versa is also true.

$S_{FSK}(t)=\sqrt{2P_{s}}cos(2\pi&space;\omega&space;_{H})(t)---->&space;binary'1'$

$S_{FSK}(t)=\sqrt{2P_{s}}cos(2\pi&space;\omega&space;_{L})(t)---->&space;binary'0'$

where                   $\omega&space;_{H}=&space;\omega&space;_{c}&space;+&space;\Omega$                                          and                $\omega&space;_{L}=&space;\omega&space;_{c}&space;-&space;\Omega$

$2\pi&space;f_{H}&space;=&space;2\pi&space;f_{c}&space;+&space;\Omega$                                                      $2\pi&space;f_{L}&space;=&space;2\pi&space;f_{c}&space;-&space;\Omega$

$f_{H}&space;=&space;f_{c}+\frac{\Omega&space;}{2\pi&space;}$                                                              $f_{L}&space;=&space;f_{c}-\frac{\Omega&space;}{2\pi&space;}$

### FSK/BFSK  Generator :-

The input to the FSK generator is a binary sequence 1  0  1  0 …..etc.

• FSK generator uses two product modulators upper modulator and lower modulator with carriers     $\phi&space;_{1}(t)=\sqrt{\frac{2}{T_{b}}}cos&space;2\pi&space;f_{H}t$  and  $\phi&space;_{2}(t)=\sqrt{\frac{2}{T_{b}}}cos&space;2\pi&space;f_{L}t$
• A level shifter is there in which the output of level-shifter is $\sqrt{P_{s}T_{b}}$ volts when the input is a binary ‘1’ and ‘0’ volts for the input ‘0’.
 Input to the level shifter output of the level shifter ‘1’ $\sqrt{P_{s}T_{b}}$ ‘0’ 0
• The working of FSK generator is as follows, when input binary sequence ‘1’ on the upper modulator 1 has been shifted to a voltage $\sqrt{P_{s}T_{b}}$ so that the output of product modulator 1 is

$S_{1}(t)&space;=&space;\sqrt{P_{s}T_{b}}&space;X&space;\phi&space;_{1}(t)$

$=\sqrt{P_{s}T_{b}}&space;X\sqrt{\frac{2}{T_{b}}}&space;cos&space;2\pi&space;f_{H}t$

$=&space;\sqrt{2P_{s}}&space;cos&space;2\pi&space;f_{H}t$

and on the lower modulator input is ‘1’ is passed through an inverter and the output of inverter is ‘0’ then the output of level shifter will not change it remains at ‘0’ volts itself, then the  output of the second product modulator is  $S_{2}(t)&space;=&space;0&space;X&space;\phi&space;_{2}(t)$

$=&space;0$

∴ the over all output      $S_{FSK}(t)&space;=&space;S_{1}(t)&space;+&space;S_{2}(t)$

$=S_{1}(t)$   since $S_{2}(t)=0$

$=&space;\sqrt{2P_{s}}&space;cos&space;2\pi&space;f_{H}t$.

similarly, when the input sequence is a binary ‘0’ , intermediate outputs at the product modulators $S_{1}(t)=0$ and $s_{2}(t)=&space;\sqrt{2P_{s}}&space;cos&space;2\pi&space;f_{L}t$

∴ the over all output $S_{FSK}(t)=&space;\sqrt{2P_{s}}&space;cos&space;2\pi&space;f_{L}t$ which is the required output for binary ‘0’ transmission.