Question

Starting with the Fourier transform pair

x ( t ) = u ( t + 1 )   -   u ( t - 1 )     ⇔   X ( Ω ) = 2 sin ( Ω ) / Ω

and using no integration indicate the properties of the Fourier transform that will allow you to compute the Fourier transform of the following signals

x 1 ( t )   =   - u ( t + 2 ) + 2 u ( t ) - u ( t - 2 )

x 2 ( t )   =   2 sin ( t )/ t

x 3 ( t )   =   2 [ u ( t + 0 . 5 ) - u ( t - 0 . 5 ) ]

x 4 ( t )   =   cos ( 0 . 5 πt ) [ u ( t + 1 ) - u ( t - 1 ) ].

Answer 1

Fourier transfermation:

the term fourier transform refers mathematical technique that decomposes a function into its costituentt frequencies. a function of time x(t), to a function of frequency x() .

starting fourier transformation pair is

x ( t ) = u ( t + 1 ) - u ( t - 1 ) <=> x ( ) = 2 sin ( ) /

then we have

x 1 ( t ) = - u ( t + 2 ) + 2 u ( t ) - u ( t - 2 )

= - u ( t ) - u ( 2 ) + 2 u ( t ) - u ( t ) + u ( 2 )

= - 2 u ( t ) + 2 u ( t )

x 1 ( t ) = 0

x 2 ( t ) = 2 sin( t ) / t

x 2 ( ) = 2 sin() /

x 3 ( t ) = 2 [ u ( t + 0 . 5 ) - u ( t - 0 . 5 ) ]

= 2 [ u( t ) + u ( 0 . 5 ) - u ( t ) + u ( 0 . 5 ) ]

= 2 [ 2 u ( 0 . 5 ) ]

= 4 u ( 0 . 5 )

x 3 ( ) = 8 sin( / 2 ) / ( / 2 )   

x 4 ( t ) = cos( 0 . 5 π t ) [ u ( t + 1 ) - u ( t - 1 ) ]

= cos( 0 . 5 t ) [ 2 u ( 1 ) ]

x ( t ) = u ( t + 1 ) - u ( t - 1 ) <=> x ( ) = 2 sin ( ) /   

=cos ( 0 . 5 t ) 2 sin ( ) /

x 4 ( ​​​​​​​) = cos ( 0 . 5 ) 2 sin ( ) / .