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7.0 Sampling
7.1 The Sampling Theorem
A link between Continuous-time/Discrete-time Systems
x(t)
y(t)
Sampling
x[n]
y[n]
h[n]
h(t)
x[n]=x(nT), T : sampling period
h(t)
x(t)
x[n]
h[n]
y[n]
Recovery
y(t)
Motivation: handling continuous-time signals/systems
digitally using computing environment
– accurate, programmable, flexible,
reproducible, powerful
– compatible to digital networks and relevant
technologies
– all signals look the same when digitized,
except at different rates, thus can be
supported by a single network
Question: under what kind of conditions can a
continuous-time signal be uniquely specified
by its discrete-time samples?
See Fig. 7.1, p.515 of text
– Sampling Theorem
Recovery from Samples ?
Impulse Train Sampling
pt  

  t  nT 
T : samplingperiod
n  
s  2 : samplingfrequency
–
x p t   xt  pt  

T
 xnT  t  nT 
n  
See Fig. 7.2, p.516 of text
X p  j   1 X  j   P j 
2
P j   2
T

    k 
k  
See Fig. 4.14, p.300 of text
s
Impulse Train Sampling
 s  2 : samplingfrequency
T
X p  j   1 X  j   P j 
2
P j   2
T
X p  j   1
T
–

    k 
k  
s

 X  j  k 
k  
s
periodic spectrum, superposition of scaled, shifted
replicas of X(jω)
See Fig. 7.3, p.517 of text
Impulse Train Sampling

Sampling Theorem (1/2)
X  j   0,   M
– x(t) uniquely specified by its samples x(nT), n=0, 1,
2……
if  s  2  2 M : Nyquist rate
T
– precisely reconstructed by an ideal lowpass filter
with Gain T and cutoff frequency ωM < ωc < ωs- ωM
applied on the impulse train of sample values
See Fig. 7.4, p.519 of text
Impulse Train Sampling

Sampling Theorem (2/2)
X  j   0,   M
– if ωs ≤ 2 ωM
spectrum overlapped, frequency components
confused --- aliasing effect
can’t be reconstructed by lowpass filtering
See Fig. 7.3, p.518 of text
Aliasing Effect
𝑥 𝑡 = 𝐴 cos 𝜔0 𝑡
2𝜋
𝑦 𝑡 = 𝑏 cos( 𝜔0 + 𝜔𝑠 )𝑡, 𝜔𝑠 =
𝑇
𝑥 𝑛𝑇 = 𝐴 cos 𝜔0 𝑛𝑇
2𝜋
𝑦 𝑛𝑇 = 𝑏 cos( 𝜔0 + 𝑘 )𝑛𝑇
𝑇
= 𝑏 cos 𝜔0 𝑛𝑇
𝐴
𝜔𝑠
𝜔0
𝑏
𝜔
𝜔0 + 𝜔𝑠
2𝜋
𝑇
000
𝐴
𝐴
𝐴
𝑏
𝑏
𝑏
𝜔𝑠 =
2𝜋
𝑇
𝜔0
After sampling with 𝜔𝑠 = , any two frequency components 𝜔1 , 𝜔2
become indistinguishable, or sharing identical samples, or should be
2𝜋
considered as identical frequency components if 𝜔1 − 𝜔2 = 𝑘
𝑇
𝑒
2𝜋
𝑗 𝜔0 +𝑘 𝑇 𝑛𝑇
𝜔𝑠
= 𝑒 𝑗𝜔0 𝑛𝑇 (𝑇 = 1 for discrete-time signals )
𝜔
Continuous/Discrete Sinusoidals (p.36 of 1.0)
Sampling
𝑥 𝑡
𝐹
sampling
𝑋(𝑗𝜔),
𝑥𝑝 (𝑡)
𝐹
𝑋𝑝 (𝑗𝜔)
𝑋𝑝 (𝑗𝜔)
𝑋(𝑗𝜔)
𝜔
0
𝑥 𝑡 =
sampling
𝑛
0 1 2 3
𝑥[𝑛]
0 1 2 3
𝜔
0
𝑥 𝑛 𝛿(𝑡 − 𝑛)
𝑡
𝐹(chap4)
𝑛
𝐹(chap5)
2𝜔𝑠
𝜔𝑠
𝜔𝑠
𝜔𝑠
2𝜋
(= 2𝜋, if 𝑇 = 1)
𝑇
2𝜋
𝜔0
𝑒 𝑗𝜔0 𝑛
𝜔
𝑒𝑗
𝜔0 +2𝜋 𝑛
𝑇 =1
Aliasing Effect
𝑧 𝑡 = 𝑥 𝑡 + 𝑦 𝑡 = 𝐴 cos 𝜔0 𝑡 + 𝑏 cos( 𝜔0 + 𝜔𝑠 )𝑡
𝑧 𝑛𝑇 = 𝑥 𝑛𝑇 + 𝑦 𝑛𝑇 = (𝐴 + 𝑏)cos𝜔0 𝑛𝑇
𝜔𝑠 < 2𝜔𝑀
Aliasing Effect
𝜔𝑠
0
−𝜔𝑀
𝜔𝑀
𝜔𝑠
𝜔𝑠
𝜔
Sampling Thm
𝜔𝑠 > 2𝜔𝑀
−𝜔𝑀
−𝜔𝑐
0
0
𝜔𝑠
𝜔𝑀
𝜔𝑐
2𝜔𝑠
𝜔
𝜔
Practical Sampling
𝑥𝑃 (t)
x(t)
𝑝(𝑡)
(any other pulse shape)
𝜏
𝑥𝑃 (t)
x(t)
t
T
𝜏
𝑃(𝑗𝜔)
T
1
T
t
𝜏
t
𝐹
𝐹
2𝜋
𝑇
𝜔
0
𝑋𝑃 (𝑗𝜔)
0
2𝜋
𝑇
𝜔
1
𝛼
𝜏
𝜔
−𝑇
0
2𝜋
𝑇
𝑇
t
𝐹
𝜔
Practical Sampling
𝑥(𝑡)
𝑥𝑃 𝑡 =
𝑥 𝑛𝑇 𝑝(𝑡 − 𝑛𝑇)
𝑥(𝑡)
𝑛
𝑇
𝑡
𝜏
𝑋(𝑗𝜔)
𝜏
𝐹
𝑡
𝑇
𝑡
2𝜋
𝑇
𝐹
𝜔
𝜔
𝑋𝑃 (𝑗𝜔)
𝜔
0
2𝜋
𝑇
0 𝜔𝑐
𝜔
(any other
pulse shape)
𝑡
1
𝛼
𝜏
𝐹
𝜔
Impulse Train Sampling

Practical Issues
– nonideal lowpass filters accurate enough for
practical purposes determined by acceptable level of
distortion
oversampling ωs = 2 ωM + ∆ ω
– sampled by pulse train with other pulse shapes
– signals practically not bandlimited : pre-filtering
Oversampling with Non-ideal
Lowpass Filters
𝐻(𝑗𝜔)
∆𝜔
∡𝐻(𝑗𝜔)
−𝜔𝑀 0
𝜔𝑀
𝜔𝑠
𝐹
ℎ(𝑡)
0
𝑡0
𝑡
𝜔
Signals not Bandlimited
𝑋(𝑗𝜔)
𝜔𝑠
0
0
𝜔𝑐
𝜔𝑠
𝜔
Sampling with A Zero-order Hold

Zero-order Hold:
– holding the sampled value until the next sample taken
– modeled by an impulse train sampler followed by a
system with rectangular impulse response

Reconstructed by a lowpass filter Hr(jω)
H  j 
H r  j  
H 0  j 
H 0  j   e
 j T 2
 2 sin T 2





H  j   ideal lowpass filt erin impulse t rainsampling
See Fig. 7.6, 7.7, 7.8, p.521, 522 of text
Interpolation

Impulse train sampling/ideal lowpass filtering
xt   x p t   ht  
ht  

 xnT  ht  nT 
n  
cT sin c t 
xr t  
c t

 xnT 
n  
cT sin c t  nT 

See Fig. 7.10, p.524 of text
c t  nT 
Ideal Interpolation
Interpolation

Zero-order hold can be viewed as a “coarse”
interpolation
See Fig. 7.11, p.524 of text

Sometimes additional lowpass filtering naturally
applied
e.g. viewed at a distance by human eyes, mosaic
smoothed naturally
See Fig. 7.12, p.525 of text
Interpolation

Higher order holds
– zero-order : output discontinuous
– first-order : output continuous, discontinuous
derivatives
H  j   1
T
 sin T 2


  2 
2
See Fig. 7.13, p.526, 527 of text
– second-order : continuous up to first derivative
discontinuous second derivative
Aliasing

Consider a signal x(t)=cos ω0t
2


– sampled at sampling frequency s
T
reconstructed by an ideal lowpass filter
s
with c 
2
xr(t) : reconstructed signal
fixed ωs, varying ω0
Aliasing

Consider a signal x(t)=cos ω0t
s
xr t   cos0t  xt 
– (a) (b) 0 
2
(c) (d)
s
2
 0  s xr t   coss  0  t  xt 
when aliasing occurs, the original frequency ω0 takes
on the identity of a lower frequency, ωs – ω0
– w0 confused with not only ωs + ω0, but ωs – ω0
See Fig. 7.15, 7.16, p.529-531 of text
Aliasing

Consider a signal x(t)=cos ω0t
– many xr(t) exist such that
xr nT   xnT , n  0,  1,  2, ...
the question is to choose the right one
– if x(t) = cos(ω0t + ϕ)
the impulses have extra phases ejϕ, e-jϕ


1 e jx  e  jx
cos
x

–
2
cos0t     1 e j 0t    e  j 0t  
2


Sinusoidals (p.68 of 4.0)
cos 𝜔0 𝑡
sin 𝜔0 𝑡
𝐹
𝜋 𝛿 𝜔 − 𝜔0 + 𝛿 𝜔 + 𝜔0 ,
𝐹 𝜋
𝑗
𝛿 𝜔 − 𝜔0 − 𝛿 𝜔 + 𝜔0 ,
[𝑒 𝑗𝜔0 𝑡 + 𝑒 −𝑗𝜔0 𝑡 ]
1
2
1
2𝑗
[𝑒 𝑗𝜔0 𝑡 − 𝑒 −𝑗𝜔0 𝑡 ]
cos 𝜔0 (𝑡 − 𝑡0 )
𝐼𝑚
−𝜔0
sin 𝜔0 𝑡
𝑅𝑒
𝜔0
cos 𝜔0 𝑡
𝜔
𝑥 𝑡 − 𝑡0
𝑒 −𝑗𝜔0 𝑡 ⋅ 𝑋(𝑗𝜔)
Aliasing

Consider a signal x(t)=cos ω0t
s
– (a) (b) 0 
2
xr t   cos0t     xt 
(c) (d)
s
2
 0   s
xr t   coss  0  t     xt 
phase also changed
Example 7.1 of Text
𝑥 𝑡 = cos(𝜔0 𝑡 + 𝜙), 𝜔𝑠 = 2𝜔0
𝜔𝑠
𝑥𝑟 𝑡 = (cos 𝜙) cos(𝜔0 𝑡), 𝑡 = 𝑛𝑇
Sampling is “time-varying”
0
0
0
−𝜔0
𝑡
(a) 𝜙 = 0
𝑡
(b) 𝜙 ≠ 0
𝑡
𝜋
2
(c) 𝜙 =
𝜔0
𝜔
Example 7.1 of Text
(a) 𝜙 = 0
𝑅𝑒
𝐼𝑚
(c) 𝜙 =
(b) 𝜙 ≠ 0 𝑅𝑒
𝜙
−𝜔0
𝐼𝑚
−𝜔0
0
0
𝐼𝑚
𝐼𝑚
𝑅𝑒
𝜔0
𝜙
𝑅𝑒
−𝜔0
𝑅𝑒
𝜙
0
𝜔0
𝐼𝑚
𝐼𝑚
𝜔
𝜋
2
𝜔
𝑅𝑒
𝜔0
𝜔
Example 7.1 of Text
𝜔𝑠
1 𝑗
𝑥 𝑡 = cos( 𝑡 + 𝜙) = [𝑒
2
2
𝜔𝑠
2
𝑛𝑇
𝜔𝑠
2 𝑡+𝜙
𝜔𝑠
2
𝑛𝑇
+
𝜔
−𝑗 2𝑠 𝑡+𝜙
𝑒
]
2𝜋
𝜔𝑠
𝑡 = 𝑛𝑡,
⋅ 𝑛𝑡 =
⋅𝑛
= 𝑛𝜋
1 𝑗𝑛𝜋 𝑗𝜙
𝑥(𝑛𝑡) = [𝑒
⋅ 𝑒 + 𝑒 −𝑗𝑛𝜋 ⋅ 𝑒 −𝑗𝜙 ]
2
𝑒 𝑗𝑛𝜋 = 𝑒 −𝑗𝑛𝜋 = ±1 = (𝑒 𝑗𝜋 )𝑛 = (𝑒 −𝑗𝜋 )𝑛 = (−1)𝑛
1 𝑗𝑛𝜋 𝑗𝜙
𝑥(𝑛𝑡) = 𝑒
𝑒 + 𝑒 −𝑗𝜙 = (𝑒 𝑗𝑛𝜋 ) ⋅ (cos 𝜙)
2
(−1)𝑛
𝜙=0
𝜙≠0
𝑛 = 0,2,4, ⋯
𝑛=
1,3,5, ⋯
cos 𝜙
𝑛 = 1,3,5, ⋯
𝑛=
0,2,4, ⋯
𝑛=
1,3,5, ⋯
𝑛 = 0,2,4, ⋯
Examples
• Example 7.1, p.532 of text
(Problem 7.39, p.571 of text)

xt   cos
s
2
t 

 
 cos  cos 2s t  sin   sin
t  nT ,
s
2
(nT ) 
s
2
n
2
s
 t
s
2
 n
xnT   cos  cosn   (1) n cos 
sampledand
andlow-pass
low - pass
filt ered
sampled
filtered
 
xr t   cos  cos 2s t
7.2 Discrete-time Processing of
Continuous-time Signals

Processing continuous-time signals digitally
xd[n]=xc(nT)
xc(t)
C/D
Conversion
yd[n]=yc(nT)
Discrete-time
System
A/D Converter
D/A Converter
h(t)
x(t)
x[n]
h[n]
yc(t)
D/C
Conversion
y[n]
Recovery
y(t)
Formal Formulation/Analysis

C/D Conversion
(1) impulse train sampling with sampling period T
(2) mapping the impulse train to a sequence with unity
spacing
– normalization (or scaling) in time
Formal Formulation/Analysis

Frequency Domain Representation
ω for continuous-time, Ω for discrete-time, only in this
section
F
xc t  , yc t  

 X c  j  , Yc  j 
  , Y e 
xd n , yd n 
 X d e
F
j
j
d
Formal Formulation/Analysis

Frequency Domain Relationships
– continuous-time
x p t  

 x nT  t  nT 
k  
X p  j  
c

 jnT


x
nT
e
 c
(4.9)
k  
– discrete-time
xd n   xc nT 
   x nT  e
X d e j 

k  
c
 jn
(5.9)
Formal Formulation/Analysis

Frequency Domain Relationships
– relationship
 
X d e j  X p  j T ,   
T
X p  j   1
T
 
X d e j  1
T

 X  j   k 
k  
c

s
 X  j   2k  T 
k  
c
See Fig. 7.22, p.537 of text
C/D Conversion
Formal Formulation/Analysis

Frequency Domain Relationships
– Xd(ejΩ) is a frequency-scaled (by T) version of Xp(jω)
xd[n]
is a time-scaled (by 1/T) version of xp(t)
– Xd(ejΩ)
periodic with period 2π
xd[n] discrete in time
Xp(jω)
periodic with period 2π/T=ωs
xp(t) obtained by impulse train sampling
Formal Formulation/Analysis

D/C Conversion
(1) mapping a sequence to an impulse train
(2) lowpass filtering
Formal Formulation/Analysis

Complete System
See Fig. 7.24, 7.25, 7.26, p.538, 539, 540 of text

Yc  j   X c  j H d e
jT

equivalent to a continuous-time system


H c  j   H d e jT ,   s 2
0,   s 2
if the sampling theorem is satisfied
h(t)
x(t)
x[n]
y(t)
y[n]
h[n]
Recovery
Discrete-time Processing of
Continuous-time Signals

Note
– the complete system is linear and time-invariant if
the sampling theorem is satisfied
– sampling process itself is NOT time-invariant
Examples

Digital Differentiator
– band-limited differentiator
H c  j   j ,   c  s 2
0,   c
– discrete-time equivalent
Hd
e   j T ,
j
 
See Fig. 7.27, 7.28, p.541, 542 of text
Examples

Delay
– yc(t)=xc(t-∆)
H c  j   e  j ,   c  s 2
0 ,   c
– discrete-time equivalent
 
H d e j  j  j T ,   
See Fig. 7.29, p.543 of text
Examples

Delay
– ∆/T an integer
yd n  xd n   T 
– ∆/T not an integer
xd n   T 
undefined in principle
but makes sense in terms of sampling if the
sampling theorem is satisfied
e.g.
∆/T=1/2, half-sample delay

1 
yd n   yc nT   xc  nT  T 
2 

See Fig. 7.30, p.544 of text
Up/Down Sampling
𝐹
′
𝜔𝑀
0
0 𝜔𝑀
0
2𝜋
3𝑇
2𝜋
𝑇
𝜔𝑠 = 2𝜋
3𝑇
𝜔𝑠′ = 2𝜋
𝑇
𝜔𝑠′ = 2𝜋
𝑇
𝜔
𝑡
𝐹
𝜔
Up
Sampling
𝑇
3𝑇
𝐹
𝜔
3𝑇
Down
Sampling
(P.10 of 7.0)
7.3 Change of Sampling Frequency
Impulse Train Sampling of Discrete-time
Signals

Completely in parallel with impulse train sampling
of continuous-time signals
pn  

  n  kN ,
N : samplingperiod
k  
x p n   xn pn  

 xkN  n  kN 
k  
 xn  if n is an int eger mult ipleof N
0
else
See Fig. 7.31, p.546 of text
Impulse Train Sampling of Discrete-time
Signals

Completely in parallel with impulse train sampling
of continuous-time signals
 
X p e j  1
2
   
2

P e j X e j    d
   2     k 
N
Pe

j
s
k  
 s  2 : samplingfrequency
N
 
X p e j  1
N
N 1

j   k s 
X
e

k 0

See Fig. 7.32, p.547 of text
Aliasing Effect (P.14 of 7.0)
𝑥 𝑡 = 𝐴 cos 𝜔0 𝑡
2𝜋
𝑦 𝑡 = 𝑏 cos( 𝜔0 + 𝜔𝑠 )𝑡, 𝜔𝑠 =
𝑇
𝑥 𝑛𝑇 = 𝐴 cos 𝜔0 𝑛𝑇
2𝜋
𝑦 𝑛𝑇 = 𝑏 cos( 𝜔0 + 𝑘 )𝑛𝑇
𝑇
= 𝑏 cos 𝜔0 𝑛𝑇
𝐴
𝜔𝑠
𝜔0
𝑏
𝜔
𝜔0 + 𝜔𝑠
2𝜋
𝑇
000
𝐴
𝐴
𝐴
𝑏
𝑏
𝑏
𝜔𝑠 =
2𝜋
𝑇
𝜔0
After sampling with 𝜔𝑠 = , any two frequency components 𝜔1 , 𝜔2
become indistinguishable, or sharing identical samples, or should be
2𝜋
considered as identical frequency components if 𝜔1 − 𝜔2 = 𝑘
𝑇
𝑒
2𝜋
𝑗 𝜔0 +𝑘 𝑇 𝑛𝑇
𝜔𝑠
= 𝑒 𝑗𝜔0 𝑛𝑇 (𝑇 = 1 for discrete-time signals )
𝜔
Sampling
𝑥 𝑡
𝐹
(P.16 of 7.0)
sampling
𝑋(𝑗𝜔),
𝑥𝑝 (𝑡)
𝐹
𝑋𝑝 (𝑗𝜔)
𝑋𝑝 (𝑗𝜔)
𝑋(𝑗𝜔)
𝜔
0
𝑥 𝑡 =
sampling
𝑛
0 1 2 3
𝑥[𝑛]
0 1 2 3
𝜔
0
𝑥 𝑛 𝛿(𝑡 − 𝑛)
𝑡
𝐹(chap4)
𝑛
𝐹(chap5)
2𝜔𝑠
𝜔𝑠
𝜔𝑠
𝜔𝑠
2𝜋
(= 2𝜋, if 𝑇 = 1)
𝑇
2𝜋
𝜔0
𝑒 𝑗𝜔0 𝑛
𝜔
𝑒𝑗
𝜔0 +2𝜋 𝑛
𝑇 =1
Aliasing Effect (P.17 of 7.0)
𝑧 𝑡 = 𝑥 𝑡 + 𝑦 𝑡 = 𝐴 cos 𝜔0 𝑡 + 𝑏 cos( 𝜔0 + 𝜔𝑠 )𝑡
𝑧 𝑛𝑇 = 𝑥 𝑛𝑇 + 𝑦 𝑛𝑇 = (𝐴 + 𝐵)𝑏cos𝜔0 𝑛𝑇
𝜔𝑠 < 2𝜔𝑀
Aliasing Effect
𝜔𝑠
0
−𝜔𝑀
𝜔𝑀
𝜔𝑠
𝜔𝑠
𝜔
Aliasing for Discrete-time Signals
𝑘𝑁
𝑥 𝑛 = 𝐴𝑒 𝑗𝜔0 𝑛
𝑦 𝑛 = 𝑏𝑒
2𝜋
𝑗(𝜔0 + )𝑛
𝑁
𝑘𝑁
2𝜋
𝑗 𝜔0 +
𝑘𝑁
𝑁
𝑒
= 𝑒 𝑗𝜔0 𝑘𝑁
𝑧 𝑛 =𝑥 𝑛 +𝑦 𝑛
𝑧 𝑘𝑁 = (𝐴 + 𝑏)𝑒 𝑗𝜔0 𝑘𝑁
𝑇
𝑇/𝑁
Impulse Train Sampling of Discrete-time
Signals

Completely in parallel with impulse train sampling
of continuous-time signals
– ωs > 2ωM, no aliasing
x[n] can be exactly recovered from xp[n] by a
lowpass filter
With Gain N and cutoff frequency ωM < ωc < ωs- ωM
See Fig. 7.33, p.548 of text
– ωs > 2ωM, aliasing occurs
filter output
xr[n] ≠ x[n]
but xr[kN] = x[kN], k=0, ±1, ±2, ……
Impulse Train Sampling of Discrete-time
Signals

Interpolation
– h[n] : impulse response of the lowpass filter
hn 
Nc sin c n

c n
xr n  x p n  hn


 xkN 
k  
Nc sin c n  kN 
c n  kN 

– in general a practical filter hr[n] is used
xr n   x p n   hr n 


 xkN h n  kN 
k  
r
Decimation/Interpolation

Decimation: reducing the sampling frequency by a
factor of N, downsampling : two reversible steps
– taking every N-th sample, leaving zeros in between
x p n  

 xkN  n  kN 
k  
– deleting all zero’s between non-zero samples to produce
a new sequence (inverse of time expansion property of
discrete-time Fourier transform)
xb n   x p nN   xnN 
– both steps reversible in both time/frequency domains
See Fig. 7.34, p.550 of text
(p.38 of 5.0)
(p.39 of 5.0)
 
xn
 X e
F

j
(p.37 of 5.0)
Time Expansion
define xk  n  xn / k , If n/k is an integer,
k: positive integer
 0, else
See Fig. 5.13, p.377 of text
jk
F



xk  n 
 X e 
See Fig. 5.14, p.378 of text
Decimation/Interpolation

Decimation:
   x k  e
X b e j 

k  
 jk
b

 j k


x
kN
e
 p
k  
 j n


x
n
e
 p


N
(k  n N )
n  integer
multipleof N


 j n


x
n
e
 p
N
n  -
( x p n   0 if n not int eger mult ipleof N )

 X p e j
N

See Figs. 7.34, 7.35, p. 550, 551 of text
Decimation/Interpolation

Decimation
– decimation without introducing aliasing requires
oversampling situation
See an example in Fig. 7.36, p. 552 of text
Decimation/Interpolation

Interpolation: increasing the sampling frequency by
a factor of N, upsampling
– reverse the two-step process in decimation
from xb[n] construct xp[n] by inserting N-1 zero’s
from xp[n] construct x[n] by lowpass filtering
See Fig. 7.37, p. 553 of text

Change of sampling frequency by a factor of N/M:
first interpolating by N, then decimating by M
Decimation/Interpolation
𝑥𝑑 𝑛 = 𝑥𝑐 𝑛𝑇 = 𝑥[𝑛]
𝑥𝑝 [𝑛]
𝑇
𝑥𝑐 (𝑡)
0
𝑡
𝑇
012
Interpolation
𝑥𝑑′ 𝑛 = 𝑥𝑐 𝑛𝑇 ′ = 𝑥𝑏 𝑛
= 𝑥𝑐 (𝑛𝑁𝑇)
𝑡
𝑇 ′ = 𝑁𝑇
𝑡
𝑁𝑇
𝑛
012
Decimation
𝑛
Decimation/Interpolation
𝑋(𝑒 𝑗Ω )
𝑋(𝑗𝜔)
0
𝑇
0
0
𝜔
𝑇′
= 𝑁𝑇
𝑋𝑝
′
𝑗Ω
(𝑒 )
2𝜋
Ω
𝑋𝑏 (𝑒 𝑗Ω )
𝜔
0
2𝜋 2𝜋
=
𝑇 ′ 𝑁𝑇
for 𝑥𝑝 [𝑛]
2𝜋 for 𝑥 [𝑛]
𝑏
0
0
𝜔
2𝜋
𝑇
2𝜋
Ω′
Ω
Examples
• Example 7.4/7.5, p.548, p.554 of text
 
 
xn  X e j , X e j  0 for 2    
9
sampling x[n] without aliasing
 s  2  2 M  2 2  ,  N  9 2
N
 9 
N max  4,  s  2  2  
N max
4
2
Examples
• Example 7.4/7.5, p.548, p.554 of text
 
 
xn  X e j , X e j  0 for 2    
9
maximum possible downsampling: using full band [-π, π]
xn 
 xb n
4:1
xn 
 xu n 
 xub n
1:2
9:1
( N max  4)
(N/M  9 2)
Examples
• Example 7.4/7.5, p.548, p.554 of text
Problem 7.6, p.557 of text
X 1  j   0,   1
X 2  j   0,   2
wt   x1 t x2 t 
W  j   1 X 1  j   X 2  j 
2
W  j   0,   1  2 ,   s  2  21  2 
T
Problem 7.20, p.560 of text
S A : inserting one zero after each sample
S B : decimation 2:1, extracting every second sample
x1 n
x2 n
x3 n
x1 n
Which of (a)(b) corresponds to low-pass filtering
with c   4 ?
Problem 7.20, p.560 of text
(a) yes
Problem 7.20, p.560 of text
(b) no
Problem 7.23, p.562 of text
Problem 7.23, p.562 of text
p t   p1 t   p1 t   
p1 t  

  t  2k 
k  
P1  j  

     k  

k  
 
P  j   P1  j   e
 j

P1  j 
for   2m   , m : int eger

e  j  1, et c.
Problem 7.23, p.562 of text
yt 
Problem 7.24, p.562 of text
2
st   s1 t   1
S  j  

   k    2  
a


k


k  
Problem 7.41, p.572 of text
Problem 7.41, p.572 of text
sc t   xc t    xc t  T0 , T  T0
sn  sc nT   xn   xn  1
   1  e X e 
1
He  
1  e
Se
j
 j
j
j
 j
differenceequation : yn  yn  1  sn
Problem 7.51, p.580 of text
dual problem for frequency domain sampling