max2sf f

1 2( ) ,sjn t

s sn ns s

t nT eT T

nss

nss nTtnTxnTttxtx

2s m m s mf f f f f

Ts is called the Nyquist interval: It is the longest time interval that can be used for sampling abandlimited signal and still allow reconstruction of the signal at the receiver without distortion

Sampling Theorem: A finite energy function x(t) can be completely

reconstructed from its sampled value x(nTs) with

provided that =>

2 ( )sin

2( ) ( )

( )

s

ss s

n s

f t nT

Tx t T x nT

t nT( ) sin (2 ( ))s s s s

n

T x nT c f t nT

1 1

2s

s m

Tf f

( ) sp

n

t nTx t

( ) ( ) ( )s px t x t x t

2( ) sj nf tn

n

x t c e

( ) [ ( ) ( )]s pX f x t x t

2[ ( ) ]sj nf tn

n

c x t e

[ ]n sn

c X f nf

'( ) ( ) ( )x t x t t

( ) '( )* ( )sx t x t p t

( )* ( ) ( ) ( )* ( ) ( )sn

p t x t t p t x t t nT

( ) [ ( )]s sX f x t

( ) ( ) ( )sn

P f x t t nT

1( ) ( )* ( )s

ns

P f X f f nfT

1( ) ( )s

ns

P f X f nfT

( ) 3cos(50 ) 100sin(300 ) cos(100 )x t t t t

( ) 3cos2000 5sin6000 cos12000ax t t t t