DSP_FOEHU - Lec 03 - Sampling of Continuous Time Signals

6. max2sf f

7. 1 2
( ) ,sjn t
s s
n ns s
t nT e
T T
n
ss
n
ss nTtnTxnTttxtx

8. 2s m m s mf f f f f

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

10. 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
s
s s
n s
f t nT
T
x t T x nT
t nT
( ) sin (2 ( ))s s s s
n
T x nT c f t nT
1 1
2
s
s m
T
f f
( ) s
p
n
t nT
x t

11. ( ) ( ) ( )s px t x t x t
2
( ) sj nf t
n
n
x t c e
( ) [ ( ) ( )]s pX f x t x t
2
[ ( ) ]sj nf t
n
n
c x t e
[ ]n s
n
c X f nf

12. '( ) ( ) ( )x t x t t
( ) '( )* ( )sx t x t p t
( )* ( ) ( ) ( )* ( ) ( )s
n
p t x t t p t x t t nT

13. ( ) [ ( )]s sX f x t
( ) ( ) ( )s
n
P f x t t nT
1
( ) ( )* ( )s
ns
P f X f f nf
T
1
( ) ( )s
ns
P f X f nf
T

14. ( ) 3cos(50 ) 100sin(300 ) cos(100 )x t t t t
( ) 3cos2000 5sin6000 cos12000ax t t t t