Spectrum sensing based on goodness of fit test with unilateral alternative hypothesis

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Devanshi Piprottar

Conversion of continuous goodness of fit test equation into sampled goodness of fit test equation

Engineering

∫ , ( ) ( )- ( ( )) ( )
∫, ( ) ( )- ( ( )) ( ) ∫
, ( ) ( )-
( )( ( ))
( )
∫
, ( ) ( )-
( )( ( ))
( )
[ ∫
, ( )-
( )( ( ))
( ) ∫
, ( )-
( )( ( ))
( )] ( )
∫
, ( )-
( )( ( ))
( ) ( )
∫
, ( )-
( )( ( ))
( ) ( )
Note:
( )
-------------------------------------------------------------------------------------------------------------------------------
∫
, ( )-
( )( ( ))
( )
∫
, ( )-
( )( ( ))
( ) ∫
, ( )-
( )( ( ))
( )
∫
, ( )-
( )( ( ))
( ) ∫
, ( )-
( )( ( ))
( )
( )
∫
, ( )-
( )( ( ))
( ) ∫
, ( )-
( )( ( ))
( )
∫
, ( )-
( )( ( ))
( ) ∫
, ( )-
( )( ( ))
( )

{ ( ( ))} { ( ( ))} { ( ( ))} { ( ( ))}
{ ( ( )) } { ( ( )) ( ( ))}
{ ( ( )) ( ( ))} { ( ( ))}
( )
∫
, ( )-
( )( ( ))
( )
∫
, ( )-
( )( ( ))
( ) ∫
, ( )-
( )( ( ))
( )
∫
, ( )-
( )( ( ))
( ) ∫
, ( )-
( )( ( ))
( )
( )
∫
( )( ( ))
( ) ∫
[ ⁄ ]
( )( ( ))
( )
∫
[ ⁄ ]
( )( ( ))
( ) ∫
, ⁄ -
( )( ( ))
( )
∫
[ ⁄ ]* ( ) ( ( ))+
( )( ( ))
( ) ∫
[ ⁄ ]* ( ) ( ( ))+
( )( ( ))
( )
∫
[ ⁄ ]* ( ) ( ( ))+
( )( ( ))
( ) ∫
, -* ( ) ( ( ))+
( )( ( ))
( )
[ ∫
[ ⁄ ]* ( ) ( ( ))+
( )( ( ))
( ) ∫
[ ⁄ ]* ( ) ( ( ))+
( )( ( ))
( )
∫
[ ⁄ ]* ( ) ( ( ))+
( )( ( ))
( ) ∫
, -* ( ) ( ( ))+
( )( ( ))
( )]

[[ ⁄ ]{ ( ( )) ( ( ))} [ ⁄ ]{ ( ( )) ( ( ))}
[ ⁄ ]{ ( ( )) ( ( ))} { ( ( )) ( ( ))} ]
[[ ⁄ ]{ ( ( )) ( ( )) ( ( )) ( ( ))}
[ ⁄ ]{ ( ( )) ( ( )) ( ( )) ( ( ))}
[ ⁄ ]{ ( ( )) ( ( )) ( ( )) ( ( ))}
{ ( ( )) ( ( ))}]
[ ⁄ ] [
( ( )) ( ( )) ( ( )) ( ( ))
( ( )) ( ( )) ( ( )) ( ( ))
]
[ ⁄ ] ∑ ( ( )) ( ( )) ( )
Now, from equation (1)
( )
Put values from equation (4) and (5)
[ [ ⁄ ] ∑ ( ( )) ( ( )) ]
[∑ ( ( )) ( ( ))]

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Spectrum sensing based on goodness of fit test with unilateral alternative hypothesis

1. ∫ , ( ) ( )- ( ( )) ( ) ∫, ( ) ( )- ( ( )) ( ) ∫ , ( ) ( )- ( )( ( )) ( ) ∫ , ( ) ( )- ( )( ( )) ( ) [ ∫ , ( )- ( )( ( )) ( ) ∫ , ( )- ( )( ( )) ( )] ( ) ∫ , ( )- ( )( ( )) ( ) ( ) ∫ , ( )- ( )( ( )) ( ) ( ) Note: ( ) ------------------------------------------------------------------------------------------------------------------------------- ∫ , ( )- ( )( ( )) ( ) ∫ , ( )- ( )( ( )) ( ) ∫ , ( )- ( )( ( )) ( ) ∫ , ( )- ( )( ( )) ( ) ∫ , ( )- ( )( ( )) ( ) ( ) ∫ , ( )- ( )( ( )) ( ) ∫ , ( )- ( )( ( )) ( ) ∫ , ( )- ( )( ( )) ( ) ∫ , ( )- ( )( ( )) ( )

2. { ( ( ))} { ( ( ))} { ( ( ))} { ( ( ))} { ( ( )) } { ( ( )) ( ( ))} { ( ( )) ( ( ))} { ( ( ))} ( ) ∫ , ( )- ( )( ( )) ( ) ∫ , ( )- ( )( ( )) ( ) ∫ , ( )- ( )( ( )) ( ) ∫ , ( )- ( )( ( )) ( ) ∫ , ( )- ( )( ( )) ( ) ( ) ∫ ( )( ( )) ( ) ∫ [ ⁄ ] ( )( ( )) ( ) ∫ [ ⁄ ] ( )( ( )) ( ) ∫ , ⁄ - ( )( ( )) ( ) ∫ [ ⁄ ]* ( ) ( ( ))+ ( )( ( )) ( ) ∫ [ ⁄ ]* ( ) ( ( ))+ ( )( ( )) ( ) ∫ [ ⁄ ]* ( ) ( ( ))+ ( )( ( )) ( ) ∫ , -* ( ) ( ( ))+ ( )( ( )) ( ) [ ∫ [ ⁄ ]* ( ) ( ( ))+ ( )( ( )) ( ) ∫ [ ⁄ ]* ( ) ( ( ))+ ( )( ( )) ( ) ∫ [ ⁄ ]* ( ) ( ( ))+ ( )( ( )) ( ) ∫ , -* ( ) ( ( ))+ ( )( ( )) ( )]

3. [[ ⁄ ]{ ( ( )) ( ( ))} [ ⁄ ]{ ( ( )) ( ( ))} [ ⁄ ]{ ( ( )) ( ( ))} { ( ( )) ( ( ))} ] [[ ⁄ ]{ ( ( )) ( ( )) ( ( )) ( ( ))} [ ⁄ ]{ ( ( )) ( ( )) ( ( )) ( ( ))} [ ⁄ ]{ ( ( )) ( ( )) ( ( )) ( ( ))} { ( ( )) ( ( ))}] [ ⁄ ] [ ( ( )) ( ( )) ( ( )) ( ( )) ( ( )) ( ( )) ( ( )) ( ( )) ] [ ⁄ ] ∑ ( ( )) ( ( )) ( ) Now, from equation (1) ( ) Put values from equation (4) and (5) [ [ ⁄ ] ∑ ( ( )) ( ( )) ] [∑ ( ( )) ( ( ))]

Spectrum sensing based on goodness of fit test with unilateral alternative hypothesis

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