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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
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