chapter6 sampling
TRANSCRIPT
Signals & Systems
Chapter 6
Sampling
INC212 Signals and Systems : 2 / 2554
Chapter 6 Sampling
Overview Sampling theorem Signal reconstruction Interpolation formula Aliasing
INC212 Signals and Systems : 2 / 2554
The sampling theorem
INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
The sampling theorem
INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
The sampling theorem
INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
The sampling theorem
INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
Hz2BFs
ssn
s
nT
FT
nFT
F
nTtnTfttftf
22
;)(1
)(
)()()()()(
bandlimited
BF 2for0)(
The sampling theorem
INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
Te
Tt
Tdtt
Tdtet
Tc
tect
sk
tjkT
T
T
T
T
T
tjkTk
k
tjkkT
s
s
s
2,
1)(
1)1)((
1)(
1
,)(
2
2
2
2
Trigonometric form
0
,3,2,1,2
2
10
k
kk kT
cA
Tc
sssssT FT
tttT
t 22
,cos2coscos211
)(
2
2
)(1
;,)(T
T
tjkk
k
tjkk dtetx
Tctectx ss
The sampling theorem
INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
)2()2(2cos)(2 sss FFttf F
n
snFT
F )(1
)(
)()(cos)(2 sss FFttf F
ttfttfttftfT
tf sss cos)(22cos)(2cos)(2)(1
)(
)()()( ttftf T
Effect of undersampling and oversampling
INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
)5(sinc)( 2 ttf
20tri2.0)(F
Effect of undersampling and oversampling
INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
)5(sinc)( 2 ttf
20tri2.0)(F
sec1.0Hz10 TFs
Effect of undersampling and oversampling
INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
)5(sinc)( 2 ttf
20tri2.0)(F
sec2.0Hz5 TFs
Effect of undersampling and oversampling
INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
)5(sinc)( 2 ttf
20tri2.0)(F
sec05.0Hz20 TFs
Effect of undersampling and oversampling
INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
)5(sinc)( 2 ttf
20tri2.0)(F
1.0;Hz10;20
10for0)(
TF
F
ss
Effect of undersampling and oversampling
INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
Hz2
Hz todBandlimite2for0)(
BF
BBF
s
The minimum sampling rate = 2B The Nyquist rate
The sampling interval = 1/2B The Nyquist interval
Signal Reconstruction Zero-order hold
INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
Signal Reconstruction The Interpolation Formula
INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
Signal Reconstruction The Interpolation Formula
INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
BTH
4rect)(
Btth
BT
BtBTth
2sinc)(
1 2 rate;Nyquist theAssuming
2sinc2)(
n
n
n
n
nBtnTftf
nTtBnTf
nTthnTf
nTtnTfth
tfthtf
)2sinc()()(
))(2sinc()(
)()(
)()(*)(
)(*)()(
Aliasing
INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
Aliasing Amplitude spectrum of time-limited signal
INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
Bs 2
not be bandlimited
Aliasing Anti-aliasing
The sampling frequency may be as large as 10 or 20 times B.
INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling
Lowpass filter
Sampling
x (t) x [n]
INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling