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DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 1
Sampling of Continuous-time Signals Advantages of digital signal processing, e.g., audio/video CD.
Things to look at:
Continuous-to-discrete (C/D)
Discrete-to-continuous (D/C) – perfect reconstruction
Frequency-domain analysis of sampling process
Sampling rate conversion
Periodic Sampling Ideal continuous-to-discrete-time (C/D) converter
Continuous-time signal: )(txc
Discrete-time signal: ∞<<∞−= nnTxnx c ),(][ , T: sampling period
In theory, we break the C/D operation into two steps:
(1) Ideal sampling using “analog delta function (impulse)”
(2) Conversion from impulse train to discrete-time sequence
Step (1) can be modeled by mathematical equation.
Step (2) is a “concept”, no mathematical model.
In reality, the electronic analog-to-digital (A/D) circuits can approximate the ideal C/D
operation. This circuitry is one piece; it cannot be split into two steps.
)(txc ][nx C/D
)(txc
)(ts
)(txs
Conversion from impulse train
to discrete-time sequence ][nx
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 2
Ideal sampling
Ideal sampling signal: impulse train (an analog signal)
( ) ( )∑∞
−∞=−=
nnTtts δ , T: sampling period
Analog (continuous-time) signal: )(txc
Sampled (continuous-time) signal: )(txs
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )∑∑
∑∞
−∞=
∞
−∞=
∞
−∞=
−=−=
−==
nc
nc
nccs
nTtnTxnTttx
nTttxtstxtx
δδ
δ
Frequency-domain Representation of Sampling
( ) ( ) ( )∑∞
−∞=Ω−Ω=Ω↔
ksk
TjSts δπ2 , where Ts /2π=Ω
Remark: Ω : analog frequency (radians/s)
ω : discrete (normalized) frequency (radians/sample)
Tω=Ω ; πωπ ≤<− , TTππ
≤Ω<−
Step 1: Ideal Sampling (all in analog domain)
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )( )∑ ∑
∑∞
−∞=
∞
−∞=
∞
−∞=
Ω−Ω=Ω−Ω∗Ω=
Ω−Ω∗Ω=Ω∗Ω=Ω
k kscc
ksccs
kjXT
kjXT
kjXT
jSjXjX
11
121
δ
δπ
The sampled signal spectrum is the sum of shifted copies of the original.
Remark: In analog domain,
)(*)(
21
)(*)()()(
ΩΩ=
↔
jYjX
fYfXtytx
π
)(txc )(txsSampling
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 3
Step 2: Analog Impulses to Sequence (analog to discrete-time)
No mathematical model. The spectrum of )(txs , )( ΩjX s , is the same as the
spectrum of ][nx , )( TjeX Ω . (See the Appendix at the end.)
Now, ( )( )sk
cTj kjX
TeX Ω−Ω= ∑
∞
−∞=
Ω 1)(
Thus, ⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛ −= ∑
∞
−∞= Tk
TjX
TeX
kc
j πωω 21)(
Remark: In time domain, )(txs and ][nx are two very different signals but they
have the “same” spectra in frequency domain.
Two Cases:
(1) no aliasing: Ns Ω>Ω 2 , and
(2) aliasing: Ns Ω<Ω 2 , where NΩ is the highest nonzero frequency compo-
nent of )( ΩjX c .
After sampling, the replicas of )( ΩjX c overlap (in frequency domain). That is, the
higher frequency components of )( ΩjX c overlap with the lower frequency com-
ponents of ( ))( sc jX Ω−Ω .
⇒ t t
xc(t) xs(t)
T
⇓ ⇓Xc(jΩ)
Ω
ΩN
Xs(jΩ)
ΩΩS
FT FT
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 4
Nyquist Sampling Theorem:
Let x(t) be a bandlimited signal with 0)( =ΩjX c for NΩ≥Ω || . (i.e., no com-
ponents at frequencies greater than NΩ ) Then )(txc is uniquely determined by its
samples K,2,1,0),(][ ±±== nnTxnx c , if Ns TΩ≥=Ω 22π . (Nyquist,
Shannon)
-- Nyquist frequency = NΩ , the bandwidth of signal.
-- Nyquist rate = 2 NΩ , the minimum sampling rate without distortion. (In some books,
Nyquist frequency = Nyquist rate.)
-- Undersampling: Ns Ω<Ω 2
-- Overdampling: Ns Ω>Ω 2
Fourier Series, Fourier Transform, Discrete-Time
Fourier Series & Discrete-Time Fourier Transform Fourier Series x(t): periodic continuous-time signal with period T0
Power: ∑∫
∞
−∞=
==k
kTx XdttxT
P 22
0 0
)(1
T0
t
x(t)
k
Xk
complex valued
⇔
⎪⎪⎩
⎪⎪⎨
⎧
=Ω=
==
∫
∑∑Ω−
∞
−∞=
Ω∞
−∞=
00
0
2
2 ,)(1
)(
0
0
00
Tdtetx
TX
eXeXtx
T
tjkk
k
tjkk
k
Ttkj
k
π
π
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 5
Fourier Transform x(t): continuous-time signal
Energy: ∫∫∞
∞−
∞
∞−ΩΩ== djXdttxPx
22 )(21)(π
Remark: (1) Other Notations
(2) Relationships between F.S. & F.T.
t
x(t)
Ω
X(jΩ)
complex valued
⇔
⎪⎩
⎪⎨
⎧
=Ω
ΩΩ=
∫∫∞
∞−
Ω−
∞
∞−
Ω
)()(
)(21)(
dtetxjX
dejXtx
tj
tj
π
⎪⎩
⎪⎨
⎧
=
=
∫∫
∞
∞−
−
∞
∞−
)()(
)(21)(
dtetxwX
dwewXtx
jwt
jwt
π⎪⎩
⎪⎨⎧
=
=
∫∫
∞
∞−
−
∞
∞−
)()(
)()(2
2
dtetxfX
dfefXtxftj
ftj
π
π
k
Xk ⇓ F.S.
t T0
T0 → ∞
⇒ t
Ω
X(jΩ)⇓ F.T.
Energy SpectrumPower Spectrum
∫ Ω−=0
0)(1
0T
tjkk dtetx
TX ∫
∞
∞−
Ω−=Ω dtetxjX tj)()(
)(lim)( 000
00
kT
kT
k XTjXXT∞→
Ω→Ω∞→
=Ω⎯⎯⎯ →⎯
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 6
(3) Periodic Signal
Discrete-Time Fourier Series x[n]: periodic discrete-time signal with period N.
Power: ∑∑−
=
−
=
==1
0
21
0
2][1 N
kk
N
nx Xnx
NP
k
Xk
F.S.
t T0
Ω
X(jΩ)
F.T.
2π/T0
k
Xk
complex valued
D.F.S.
⇔ n
x[n]
N N
Ω
X(jΩ)
2π/T
⇑
t
x(t)
T
⇓
F.T.
⇒
⎪⎪⎩
⎪⎪⎨
⎧
=
=
∑
∑−
=
−
−
=
][
1][
1
0
2
1
0
2
N
n
Nnkj
k
N
k
Nnkj
k
enxX
eXN
nxπ
π
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 7
Discrete-Time Fourier Transform x[n]: discrete-time signal
Energy: ∫∑ −
∞
−∞=
==π
ππdweXnxE jw
nx
22 )(21][
Remark: X(ejw) v.s. X(jΩ)
X(ejw) is a frequency-scaled version of X(jΩ)
Tw
jweXjXΩ=
=Ω )()(
w
X(ejw)
DTFT
⇔ n
x[n]
2π
Ω
X(jΩ)
2π/T
⇑
t
x(t)
T
⇓
F.T.
⇒
⎪⎪⎩
⎪⎪⎨
⎧
=
=
∑∫
∞
−∞=
−
−
][)(
)(21][
n
jwnjw
jwnjw
enxeX
dweeXnxπ
ππ
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 8
Reconstruction of a Band-limited Signal from Its
Samples -- Perfect reconstruction: recover the original continuous-time signal without distortion,
e.g., ideal lowpass (bandpass) filter
Based on the frequency-domain analysis, if we can “clip” one copy of the original spectrum,
)( ΩjX c , without distortion, we can achieve the perfect reconstruction. For example, we
use the ideal low-pass filter as the reconstruction filter.
Remark: Note that )(txs is an analog signal (impulses).
∑ →−→−=→→ ][..)()()()( nxconvrseqnTtnTxtxsamplingtx sc δ
)(.)(][)(.][ txreconnTtnxtxconvrimpulsenx rs ∑ →→−=→−→ δ
{ } ∑∑ ∫
∫ ∑∞
−∞=
∞
−∞=
∞
∞−
∞
∞−
∞
−∞=
−=−−=
⎭⎬⎫
⎩⎨⎧ −−=∗=
nr
nr
nrrsr
nTthnxdthnTnx
dthnTnxthtxtx
)(][)()(][
)()(][)()()(
λλλδ
λλλδ
)()()()()()(
}][){()(][)(
ΩΩ=Ω=Ω=
Ω=Ω=Ω
Ω
Ω=
∞
−∞=
Ω−∞
−∞=
Ω− ∑∑jXjHeXjHeXjH
enxjHejHnxjX
rTj
rTw
jwr
n
Tnjr
n
Tnjrr
Ideal low-pass reconstruction filter:
⎩⎨⎧ ≤Ω<−
=Ωotherwise
TTTjH r 0
)(ππ
)()sin()(
TtTtthr π
π=
Reconstruction filter
)(txs )(txrConvert from Sequence to Impulse train
][nx
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 9
T 2T t
1
hr(t)
t
xs(t)
T
n
x[n]
t
xr(t)
T
Tπ− Tπ Ω
T
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 10
Discrete-time Processing of Continuous-time Sig-
nals
If this is an LTI system,
(1) ][][ nynx → : )()()( ωωω jjj eXeHeY =
(2) ][)( nxtxc → : ⎟⎠
⎞⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −= ∑
∞
−∞= Tk
TjX
TeX
kc
j πωω 21)(
(3) )(][ tyny r→ : )()()( Tjrr eYjHjY ΩΩ=Ω
Note that “T” is included in the expression of ),( ωjeY TΩ←ω . This means
“physical” frequency (not normalized).
(4) )()( tytx rc →→L :
∑∞
−∞=
Ω
ΩΩ
⎟⎠⎞
⎜⎝⎛ −ΩΩ=
Ω=Ω
kc
Tjr
TjTjrr
TkjjX
TeHjH
eXeHjHjY
π21)()(
)()()()(
If )( ΩjHr is an ideal low-pass reconstruction filter, then
⎩⎨⎧ <ΩΩ=Ω
Ω
otherwiseTjXeHjY c
Tj
r ,0/||),()()( π
In other words, if )(txc is band-limited and is ideally sampled at a rate above the
Nyquist rate, and the reconstruction filter is the ideal low-pass filter, then the
equivalent analog filter has the same spectrum shape of the discrete-time filter.
⎩⎨⎧ <Ω=Ω
Ω
otherwiseTeHjH
Tj
eff ,0/||),()( π
][nx ][ny)(txc )(tyr
C/D Discrete-time
system D/C
T T
)( ωjeH
)( ΩjHeff
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 11
Remark: In order to have the above equivalent relation between )( ωjeH and
)( ΩjHeff , we need
(i) The system is LTI;
(ii) The input is bandlimited;
(iii) The input is sampled without aliasing and the ideal impulse train is
used in sampling;
(iv) The ideal reconstruction filter is used to produce the analog output.
In practice, the above conditions are only approximately valid at best.
However, there are methods in designing the sampling and the reconstruction
processes to make the approximation better.
)()( Ω=Ω jHjH ceff
⇒ )(][ nTThnh c=
The impulse response of the discrete-time system is a scaled, sampled version of )(thc .
TjHT
TwjHeH
c
cjw
π
π
≥Ω=Ω
<=
for ,0)( s.t. chosen is
w )()(
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 12
Continuous-time Processing of Discrete-time Signals
⇒ π<== weXTwjH
TwjX
TwjH
TeY jw
cccjw )()()()(1)(
⇒ π<= wTwjHeH c
jw )()(
or, equivalently, T
jHeH cTj π
<ΩΩ=Ω )()(
Example: Noninteger Delay
π<= Δ− w )( jwjw eeH
][nx ][ny)(txc )(tycD/C
Continu.-time system C/D
T T
)( ωjeH
)( ΩjHc
π
π
π
<=
<ΩΩΩ=Ω
<Ω=Ω Ω
wTwjY
TeY
TjXjHjY
TeTXjX
cjw
ccc
Tjc
),(1)(
),()()(
),()(
n
x[n]
n
y[n]
yc(t) = xc(t-ΔT)
xc(t)
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 13
Change the Sampling Rate Using Discrete-time
Processing
⎩⎨⎧
=→→=→→
)'(]['')(][
)(nTxnxTnTxnxT
txc
cc
Original sampling period: T
New sampling period: 'T 'TT ≠
• Sampling rate reduction by an integer factor Sampling rate compressor:
MTT =' , where M is an integer
)(][][ nMTxnMxnx cd ==
Compressor
Aliasing: If the original signal BW is not small enough to meet the Nyquist rate requirement,
t
T1
⇓
Ω
1
2Tπ
Downsampling
Ω
)(1
1
ΩjXT c
2
2Tπ
)(1
2
ΩjXT c
⇒
⇒
FT
t
T2 = MT1
FT
M↓][nx )(][][ nMTxnMxnx cd ==
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 14
prefiltering is needed.
The Original
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛ −= ∑
∞
−∞= Tk
TjX
TeX
kc
j πωω 21)(
The Downsampled
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛ −= ∑
∞
−∞= '2
''1)(
Tr
TjX
TeX
rc
jd
πωω
LLLLLLLLL
1100001121)1(210)1()2(
)1()1(21012
−−−−−
+−−−
kMMMi
MMMr
Old and new index: kMir += ∞−−−∞= ,,2,1,0,1,2,,, LLkr , 1,,2,1,0 −= Mi L
∑ ∑
∑
∑
∞
−∞=
−
=
∞
−∞=
∞
−∞=
⎟⎠
⎞⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −−=
⎟⎠
⎞⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −=
⎟⎠
⎞⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −=
k
M
ic
rc
rc
jd
MTi
MTkM
MTjX
MT
MTr
MTjX
MT
Tr
TjX
TeX
1
0
221
21
'2
''1)(
ππω
πω
πωω
∑
∑ ∑
−
=
⎟⎠⎞
⎜⎝⎛ −
−
=
∞
−∞=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛=
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −
−=
1
0
2
1
0
1
2211)(
M
i
Mi
Mj
M
i kc
jd
eXM
Tkj
MTijX
TMeX
πω
ω ππω
The down-sampled spectrum = sum of shifted replica of the original
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 15
Downsampling with aliasing
To avoid aliasing
⇒ π<MwN
General System for Sampling Rate Reduction by M
Lowpass filter Cutoff= Mπ
M↓][nx ][~ nx ][~][ nMxnxd =
T T T’=MT
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 16
• Increasing sampling rate by an integer factor
Sampling rate expander
LTT /'= , where L is an integer
⎟⎠⎞
⎜⎝⎛=
LTnxnx ci ][
(1) shape is compressed; (2) replicas are removed
Lowpass filter Cutoff= Lπ
L↑ ][nx ][nxe ][nxi
T T’ T’=T/L
t
T1
⇓
1
2Tπ
Ω
Downsampling
Ω
2
2Tπ
⇒
⇒
FT
t
T2 = T1/L
FT
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 17
(1) Increase samples
<Time-domain>
∑∞
−∞=−=
⎩⎨⎧ ±±==
k
e
kLnkx
otherwiseLLnL
nxnx
][][
,0,2,,0],[][
δ
L
<Frequency-domain>
)(][][
][][)(
Lj
k n
nj
nj
n k
je
eXekLnkx
ekLnkxeX
ωω
ωω
δ
δ
=⎟⎠⎞
⎜⎝⎛ −=
⎟⎠
⎞⎜⎝
⎛ −=
∑ ∑
∑ ∑∞
−∞=
∞
−∞=
−
−∞
−∞=
∞
−∞=
Note that Lkj
n
nj eekLn ωωδ −∞
−∞=
− =−∑ ][
Remark: Essentially, the horizontal frequency axis is compressed.
The shape of the spectrum is not changed.
Told Ω=ω_ , LTTnew Ω=Ω= '_ω , Lnewold ⋅= ωω __
Remark: At this point, we only insert zeros into the original signal. In time domain,
this signal doesn’t look like the original.
(2) Ideal lowpass filtering
<Frequency-domain>
⎩⎨⎧ ≤Ω<−=Ω otherwise
TLTLjHi 0)()(1)( ππ
<Time-domain>
)()sin(][
LnLnnhi π
π= , an interpolator!
[ ]∑∞
−∞= −−
=k
i LkLnLkLnkxnx
/)(/)(sin][][
ππ
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 18
Linear interpolation
2
)2sin()2sin(1)( ⎥⎦
⎤⎢⎣
⎡=
ωωω L
LeH j
lin
∑∞
−∞=−=
klinlin kLnhkxnx ][][][
⎩⎨⎧ ≤−= otherwise
LnLnnhlin ,0||,/||1][
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 19
• Changing sampling rate by a rational factor
Idea: Sampling period TLM
LTT ⎯⎯⎯ →⎯⎯⎯⎯⎯ →⎯ decimationioninterpolat
Remark: In general, if the factor is not rational, go back to the continuous signals.
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 20
• In summary:
-- Sampling
Time-domain Frequency -domain
Prefiltering Limit bandwidth Ns Ω>Ω 2
Analog sampling (impulse train) Duplicate and shift )(Ω
Analog to discrete ][)( nt δδ → ω→Ω
-- Reconstruction
Time-domain Frequency -domain
Discrete to analog )(][ tn δδ → Ω→ω
Interpolation Remove extra copies )(Ω
-- Down-sampling
Time-domain Frequency -domain
Prefiltering Limit bandwidth
Drop samples (rearrange index) Expand (by a factor of M) and duplicate
(insert (M-1) copies)
-- Up-sampling
Time-domain Frequency -domain
Insert zeros Shrink (by a factor of L)
Interpolation Remove extra copies in a π2 period
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 21
Digital Processing of Analog Signals Ideal C/D converter (approximation) analog-to-digital (A/D) converter
Ideal D/C converter (approximation) digital-to-analog (D/A) converter
Prefiltering to Avoid Aliasing Ideal antialiasing filter: Ideal low-pass filter (difficult to implement sharp-cutoff analog
filters).
A solution: simple prefilter and oversampling followed by sharp antialiasing filters in
discrete-time domain.
Remark: Sharp cutoff analog filters are expensive and difficult to implement.
A/D conversion ⇒ the input continuous-time signal is sampled at a very high
sampling rate.
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 22
A/D Conversion Digital: discrete in time and discrete in amplitude
Ideal sample-and-hold: Sample the (input) analog signal and hold its value for T sec-
onds.
∑∞
−∞=−=
nnTthnxtx )(][)( 00
)(})()({)()()( 000 thnTtnTxnTthnTxtxn
an
a ∗−=−= ∑∑∞
−∞=
∞
−∞=
δ
⎩⎨⎧ <<= otherwise
Ttth ,00,1)(0
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 23
Quantization: Transform the input sample ][nx into one of a finite set of prescribed
values.
])[(][ˆ nxQnx = , ][ˆ nx is the quantized sample
Note: Quantization is a non-linear operation.
(i) Uniform quantizer – uniformly spaced quantization levels; very popular (also called
linear quantizer)
(ii) Nonuniform quantizer – may be more efficient for certain applicaitons
Parameters in a quantizer
(1) Decision levels – partition the dynamic range of input signal
(2) Quantization (representation) levels – the output values of a quantizer; a quanti-
zation level represents all samples between two nearby decision levels
(3) Full-scale level – the quantizer input dynamic range
Note: Typically, when the decision levels are first chosen, then the best quantization
levels are decided (for a given input probability distribution). On the other hand,
when the quantization levels are chosen, the best decision levels are decided.
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 24
Quantization error analysis
For a uniform quantizer, there are two key parameters:
(i) step size Δ , and (ii) full-scale level ( mX± )
Assume (B+1) bits are used to represent the quantized values.
Bm
Bm XX
222
1 ==Δ+
Quantization error: ][][ˆ][ nxnxne −= = quanitized value – true value
It is clear that 2
][2
Δ<<
Δ− ne .
Statistical characteristics of ][ne :
(1) ][ne is stationary (probability distribution unchanged)
(2) ][ne is uncorrelated with ][nx
(3) ][ne , ]1[ +ne , … are uncorrelated (white)
(4) ][ne has a uniform distribution
The preceding assumptions are (approximately) valid if the signal is sufficiently com-
plex and the quantization steps are sufficiently small, …
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 25
Mean square error (MSE) of ][ne (= variance if zero mean)
{ }12
1)()(22/
2/222 Δ
=Δ
=−= ∫Δ
Δ−deeeeEeσ
-- Expressed in terms of B2 and mX
12
2 222 m
B
eX−
=σ
-- SNR (signal-to-noise ratio) due to quantization
x
m
m
xB
e
x XB
XSNR
σσ
σσ
102
22
102
2
10 log2002.68.10212
log10log10 −+=⋅
==
Remarks: (1) One bit buys a 6dB SNR improvement.
(2) If the input is Gaussian, a small percentage of the input samples would have an
amplitude greater than xσ4 .
If we choose xmX σ4= , dBBSNR 25.16 −≈
For example, a 96dB SNR requires a 16-bit quantizer.
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 26
D/A Conversion The ideal lowpass filter is replaced by a “practical” filter.
Examples of practical filters: zero-order hold and first-order hold.
Mathematical model:
∑∞
−∞=−=
nDA nTthnxtx )(][ˆ)( 0
= quantized input * impulse response of “practical” interpolation filter
)()(
)(][)(][)(
00
00
tetx
nTthnenTthnxtxnn
DA
+=
−+−= ∑∑∞
−∞=
∞
−∞=
Purpose: Find a compensation filter )(~ thr to compensate for the distortion caused by the
non-ideal )(0 th so that its output )(ˆ txr is close to the analog original
)(txa .
In frequency domain:
)()()(][
)(][)(][)(
00
000
Ω=Ω⎟⎠
⎞⎜⎝
⎛=
Ω=⎭⎬⎫
⎩⎨⎧ −=Ω
Ω∞
−∞=
Ω−
∞
−∞=
Ω−∞
−∞=
∑
∑∑
jHeXjHenx
ejHnxnTthnxFjX
Tj
n
nTj
n
nTj
nt
D/A Converter
Compensated reconstruction filter )(~ ΩjH r
][ˆ nx )(txDA )(ˆ txr
T
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 27
Because ( ) ∑∞
−∞=⎟⎠
⎞⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −Ω=Ω
ka T
kkjXT
jX π21 ,
( ) ( )Ω⎟⎠
⎞⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −Ω=Ω ∑
∞
−∞=jH
TkkjX
TjX
ka 00
21 π
[The interpolation filter )(0 ΩjH is used to remove the replicas.]
If )(0 ΩjH is not an ideal lowpass filter, we design a compensated reconstruction filter,
)()(
)(~0 Ω
Ω=Ω
jHjH
jH rr
, where )( ΩjH r is the ideal lowpass filter.
(1) Zero-order hold
or 2/0
)2/sin(2)( TjeTjH Ω−
ΩΩ
=Ω
Thus, the compensated reconstruction filter is
⎪⎩
⎪⎨⎧
>Ω
<ΩΩ
Ω=Ω
Ω
T
TeT
TjH
Tj
r
/||,0
/||,)2/sin(
2/)(~ 2/
π
π
Remark: A “practical” filter cannot achieve this approximation.
⎩⎨⎧ <<= otherwise
Ttth ,00,1)(0
DSP (Spring, 2007) Sampling of Continuous-time Signals
NCTU EE 28
Overall system:
Anti-aliasing Processing Zero-order-hold Compensated reconstruction.
)()()()(~)( 0 Ω⋅⋅Ω⋅Ω=Ω Ω jHeHjHjHjH aaTj
reff
22
0 )()()(~)( eTj
re eHjHjHjPa
σΩ⋅Ω⋅Ω=Ω where 12
22 Δ=eσ
)( ΩjH aa )( TjeH Ω )(0 ΩjH )(~ ΩjH r)( ΩjX a )( ΩjYa