Sampling of Continuous-Time Signals
Introduction
Periodic Sampling
Frequency-Domain Representation of Sampling
Reconstruction of a Bandlimited Signal from Its Samples
Discrete-Time Processing of Continuous-Time Signals
Continuous-Time Processing of Discrete-Time Signals
Practical Considerations
1
1. Introduction
A/D Converter
Converts an analog signal
into a sequence of digits.
D/A Converter
Converts a sequence of
digits into an analog signal.
Processing
Change the sampling rates.
A/D
Converter
D/A
Converter
DigitalSignal
Processing
AnalogInputSignal
AnalogOutputSignal
DigitalInputSignal
DigitalOutputSignal
{3, 5, 4, 6 ...}
0
t
2
0. Introduction (c.1)
Antialiasing
FilterSampler Quantizer
& Coder
Smoothing
FilterInterpolator
DigitalSignal
Processing
AnalogInputSignal
AnalogOutputSignal
{3, 5, 4, 6 ...}t
0
t
t
{3, 5, 4, 6 ...}
Sampling Frequency Fs
Cut-off Frequency Fc
The number of bits
Sampling Frequency Fs*
t
t
3
1. Periodic Sampling
Ideal Continuous-to-Discrete-Time Converter
x[n] = xc(nT), - < n <
T is the sampling period.
fs=1/T is the sampling frequency.
Two Stages
Impulse train modulation
Conversion from an impulse train to a sequence.
4
2. Frequency-Domain Representation of
Sampling
Impulse Modulation
Modulating Signals
Sampling Signals
Frequency Representation
Convolution in Frequency Domain
s t t nTn
( ) ( )
x t x t s t x t t nT
x nT t nT
s c c
n
c
n
( ) ( ) ( ) ( ) ( )
( ) ( )
S jT
k s
k
( ) ( )
2
X j X j S js c( ) ( ) ( ) 1
2
X jT
X j kjs c
k
s( ) ( )
1
s N 2
s N 2
5
http://ccrma.stanford.edu/~jos/sasp/Impulse_Trains.html
Fourier transform of impulse train
Some derivation lemma
6
2. Frequency-Domain Representation
of Sampling
Continuous-Time Fourier Transform
7
dejFtf tj)(2
1)(
dtetfjF tj
)()(Fourier Transform
Inverse
Fourier Transform
2. Frequency-Domain Representation of
Sampling (c.1)
Comments
Bandlimited signals
The sampling frequency must be
Reconstruction
Ideal lowpass filter with gain T and cutoff frequency
N
s N 2
X j H j X js r s( ) ( ) ( )
N c s N ( )
8
2. Frequency-Domain Representation of
Sampling (c.2)
Example
Cosine signals
Aliasing when
Nyquist Sampling Theorem
Let xc(t) be a bandlimited signal with
Then xc(t) is uniquely determined by its samples x[n]=xc[nT], n=0, +-1, +-2, ..., if
x t tc( ) cos 0
02
s
X j forc N( ) 0
s NT
2
2 Nyquist frequency
Nyquist rate
9
2. Frequency-Domain Representation of
Sampling (c.3)
Discrete-Time Fourier Transform
The Fourier transform for sampling signals
Since that and
It follows that
Consequently
X j x nT es cj Tn
n
( ) ( )
x n x nTc[ ] ( ) X e x n ej j n
n
( ) [ ]
X j X e X esj
Tj T( ) ( ) ( )
X eT
X j jkj Tc
k
s( ) ( )
1
X eT
X jT
jk
T
jc
k
( ) ( )
1 2
Discrete Frequency
Normalization Factor
10
n
cs nTtnTxtx )()()(
3. Reconstruction of a Bandlimited Signal
from Its Samples
Interpolator
Output of the LP Filter
Frequency Domain
h tt T
t Tr ( )sin /
/
x t x nt nT T
t nT Trn
( ) ( )sin[ ( )/ ]
( )/
x t x n t nTsn
( ) [ ] ( )
x t x n h t nTr rn
( ) [ ] ( )
t
Sampling Frequency
Fs’
Smoothing
FilterInterpolator
AnalogOutputSignal
Cut-off Frequency
Fc’
t
{3, 5, 4, 6 ...}
t
xs xr
h h nT for nr r( ) ; ( ) , ,...0 1 0 1 2
X j x n H j er rj Tn
n
( ) [ ] ( )
X j H j X er rj T( ) ( ) ( )
x t x n h t nTr rn
( ) [ ] ( )
11
Proof of the Sinc function12
h tt T
t Tr ( )sin /
/
3. Reconstruction of a Bandlimited
Signal from Its Samples13
4. Discrete-Time Processing of
Continuous-Time Signals
I/O Relation of A/D
I/O Relation of D/A
A/DDiscrete-Time
SystemD/A
xc(t)x (n) y (n) yr (t)
y t y nt nT T
t nT Trn
( ) ( )sin[ ( )/ ]
( )/
X eT
X jT
jk
T
jc
k
( ) ( )
1 2
x n x nTc( ) ( )
Y j H j Y e
TY e T
otherwise
r rj T
j T
( ) ( ) ( )
( ), /
,
0
F
F
14
4. Discrete-Time Processing of
Continuous-Time Signals (c.1)
Linear Time-Invariant Systems
Relation between the Input and the Output
Combining A/D, LTI Systems, and the D/A
The ideal lowpass reconstruction
If signal is bandlimited and the sampling rate is above the
Nyquist rate
Y e H e X ej j j( ) ( ) ( )
Y j H j Y e
H j H e X e H j H eT
X j jk
T
r rj T
rj T j T
rj T
ck
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
1 2
Y jH e X j T
Tr
j Tc( )
( ) ( ), /
, /
0
Y j H j Y jr eff( ) ( ) ( ) H jH e T
Teff
j T
( )( ), /
, /
0
15
5. Continuous-Time Processing of
Discrete-Time Signals
Overall Discrete-Time System
D/AContinuous-
Time SystemA/D
x (n) xc (t) yc (t) y (n)
y t y nt nT T
t nT Tcn
( ) ( )sin[ ( )/ ]
( )/
x t x nt nT T
t nT Tcn
( ) ( )sin[ ( )/ ]
( )/
X j TX e Tcj T( ) ( ), /
Y j H j X j Tc c c( ) ( ) ( ), /
Y eTY j Tjc( ) ( / ),
1
H e H j Tjc( ) ( / ),
or H j H ecj T( ) ( ),
16
6. Changing the Sampling Rate Using
Discrete-Time Processing
Changing Rate
Given x[n] = xc(nT)
How to obtain xd[n] = xc(nT’), where T’=MT
Rate Reduction by an Integer Factor
X eT
X jT
jk
T
jc
k
( ) ( )
1 2
X eMT
X jMT
jr
MT
Let r i kM
X eM T
X jMT
jk
Tj
i
MT
MX e
dj
c
r
dj
c
ki
M
j M i M
i
M
( ) ( )
( ) ( )
( )( / / )
1 2
1 1 2 2
1
0
1
2
0
1
x[n] xd [n]=x[nM]MA/D
xc(t)
Notes:
1. The relation between x[n]
and xd[n].
2. Aliasing Effects ?
17
6. Changing the Sampling Rate Using
Discrete-Time Processing (c.1)
Decimatorsx[n] xd [n]=x[nM]
MA/Dxc(t) x’[n]
LP
18
6. Changing the Sampling Rate Using
Discrete-Time Processing (c.2)
Rate Increase by an Integer Factor
Given x[n] = xc(nT)
How to obtain xI [n] = xc(nT’), where T’=T/L
Analysis
The relation with continuous time signals
x[n] xi [n]L
LP Filter
Gain = LCutoff = /L
xe[n]
x n x n L x nT L n L Li c[ ] [ / ] ( / ), , , ,.... 0 2
x n x k n kLx n L n L L
otherwisee
k
[ ] [ ] [ ][ / ], , , ,....
,
0 2
0
)(][][][][ Lj
k
Lkj
n
nj
k
j
e eXekxekLnkxeX
19
6. Changing the Sampling Rate Using
Discrete-Time Processing (c.3)
Interpolator
Ideal Interpolator
Linear Interpolator
h h n n L Li i[ ] ; [ ] , , ,...,0 1 0 2
x n x kn kL L
n kL Li
k
[ ] [ ]sin[ ( ) / ]
( ) /
h nn L
n Li[ ]
sin( / )
/
h nn L n L
otherwiselin[ ]
/ ,
,
1
0
x n x k h n k x k h n kLlin e lin
k
lin
k
[ ] [ ] [ ] [ ] [ ]
H eL
Llin
j( )sin( / )
sin( / )
1 2
2
2
20
6. Changing the Sampling Rate Using
Discrete-Time Processing (c.4)
Change Rate by a Noninteger Factor
Noninteger factor L/M
M > L ==>
M < L ==>
x[n] xi [n]L
LP Filter
Gain = LCutoff = /L
xe[n] xd [n]
MLP Filter
Gain = 1Cutoff = /M
xi[n]
x[n]
L
LP Filter
Gain = LCutoff =
Min( /L, /M)
xe[n] xd [n]
M
xi[n]
21
7. Practical Consideration--
Antialiasing Filters
Aliasing
F +- kFs are mapped into the same discrete frequency
0Fs/2 Fs-Fs -Fs/2
F
f
22
7. Practical Consideration-- Antialiasing
Filters
Purpose:
Delete the frequency components that will
be aliased to low frequency components.
Low-Pass Filters
Fc < Fs/2Fc
Low-Pass Filter
F
1
23
7. Practicle Consideration--
Quantization Quantization
Express each sample value as a
finite number of digits.
Quantization Error
The error introduced in
representing the continuous-value
signal by a discrete value levels.
Signal-to-quantization noise ratio,
SQNR(dB)
1.76 + 6.02b
16 bits CD audio data has a
quality of more than 96 dB
Output of Sampler
Output of Quantization
24
7. Practicle Consideration--
Quantization (c.1)
SQNR(dB)
The maximum root mean
square signal Srms is
The rms quantization error is
The power ratio is
SQ
rms
b
2
2
1
1 2/
E e p e deQ
e deQ Q
rms
2
1 22
1 2 21 2
1 2
1
12 12( )
( )
/ / /
/
S
EdB bb( ) log ( ) . .
10
3
22 6 02 1 762
25
7. Practicle Consideration--Interpolator
Optimal Interpolator:
• Sampling Theorems
• no distortion for the frequency components below Fs/2
• no frequency components above Fs/2 exist and smoothing filtering is not necessary
Suboptimal Interpolator
• Distortion exists for the frequency components below
Fs/2
• Result in passing frequencies above the folding
frequency and smoothing filtering is necessary
a a
n s
x t X n Fs g tn
F( ) ( / ) ( )
Fs 2Fs
F
Signal Mangitude Spectrum
Zero-orderInterpolator
First-orderInterpolator
Optimal Interpolator
26
7. Practicle Consideration-- Smoothing
Filters
Delete the frequency components above a threshold
frequency to avoid the image signal introduced by
suboptimal filters
Low-pass filteringSmoothing
Filter
tt
Cut-off Frequency Fc*
Fc'
Low-Pass Filter
F
1
0 Fs 2Fs2Fs
F
Signal Mangitude Spectrum
27
7. Practical Consideration-- Concluding
Remarks
Time/Frequency Illustration
Antialiasing filtering and Antiimaging filtering
28
8. Concluding Remarks
Introduction
Periodic Sampling
Frequency-Domain Representation of Sampling
Reconstruction of a Bandlimited Signal from Its Samples
Discrete-Time Processing of Continuous-Time Signals
Continuous-Time Processing of Discrete-Time Signals
Practical Considerations
29
Homeworks & References
Homeworks
(Deadline= Nov. 9): 4.22, 4.23, 4.24, 4.38, 4.39, 4.42,
4.43, 4.44
References
http://ccrma.stanford.edu/~jos/sasp/Impulse_Trains.htm
30
