F. Maloberti
DATA CONVERTERS
Springer
2007
Chapter 1
BACKGROUND ELEMENTS
Slide Set
Data Converters
—————————
Background Elements
0
F. Maloberti
DATA CONVERTERS
Springer
2007
Chapter 1
BACKGROUND ELEMENTS
Summary
• Introduction
• The Ideal Data Converter
• Sampling
• Amplitude Quantization
• Quantization Noise
• kT/C Noise
• Discrete and Fast Fourier Transforms
• The D/A Converter
• The z-Transform
1
F. Maloberti
DATA CONVERTERS
Springer
2007
Chapter 1
BACKGROUND ELEMENTS
2
The Ideal Data Converter
Transformation from continuous-amplitude, continuous-time into discreteamplitude discrete-time and vice-versa.
(a)
(b)
F. Maloberti
DATA CONVERTERS
Springer
2007
Chapter 1
BACKGROUND ELEMENTS
3
Sampling
A sampler transforms a continuous-time signal into its sampled-data equivalent.
x∗(t) = x∗(nT ) =
X
x(t)δ(t − nT )
(1)
x*(t)
x(t)
t
t
4T 5T
T 2T 3T
6T .....
Only the values at the sampling instant matter (independently on the real
representation).
Chapter 1
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DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
4
Sampling as a non-linear Process
The Laplace transform is:
T
X
X*
X
X*
∞
S d(t-nT)
-∞

L
∞
X

δ(t − nT ) =
−∞
∗
L x (nT ) =
∞
X
−∞
∞
X
e−nsT ;
(2)
−∞
(X(s − jnωs) =
∞
X
−∞
x(nT )e−nsT
(3)
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
Spectral Implications
f1 f2
fB
f
(a)
f1 f2
-2
-1
1
2
(b)
-fs
-2
-fs
-1
1
(c)
f
fs
f1 f2
-3
3
2
3
4
fs
is it possible to return back to the continuous-time?
5
f
5
F. Maloberti
DATA CONVERTERS
Springer
2007
Chapter 1
BACKGROUND ELEMENTS
6
Nyquist Theorem
A band limited signal, x(t), whose Fourier spectrum, X(jω), vanishes for
angular frequencies |ω| > ωs/2 is fully described by a uniform sampling
x(nT ), where T = 2π/ωs. The band limited signal x(t) is reconstructed
by
∞
X
sin(ωs(t − nT )/2)
x(t) =
x(nT )
(4)
ωs(t − nT )/2
−∞
Half the sampling frequency, fs /2 = 1/2T , is often named the Nyquist frequency. The
frequency interval 0 · · · fs /2 is referred to as the Nyquist band (or band-base) while frequency intervals, fs /2 · · · fs , fs · · · 3fs /2, · · · are named the second and third Nyquist
zones, and so forth.
F. Maloberti
DATA CONVERTERS
Springer
2007
Chapter 1
BACKGROUND ELEMENTS
Remember!
Something that you need to ...
The anti-aliasing filter protects the information content
of the signal. Use an antialiasing filter in front of every
quantizer to reject unwanted
interferences outside of the
band of the interest!
7
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
8
Antialiasing Filter
Pass
until here
Reject
from here
fB
fS/2
(a)
fS -fB
fS
H(s)
[dB]
Pass-Band
Transition-Band
Ripple in the Pass-Band
Stop-Band
ASB
Stop-band
Attenuation
(b)
The complexity of the antialiasing filter depends on the band-pass ripple
the transition region, and the attenuation in the stop-band
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
Undersampling
fL
Signal
Image
Signal
Signal
(a)
Image
(b)
fH
Image
Signal
Image
Image
fS
fL
Signal
Signal
Signal
(c)
Image
Image
(d)
fH
Image
Image
Signal
fS
Image
9
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DATA CONVERTERS
Springer
2007
Chapter 1
BACKGROUND ELEMENTS
10
Remember That
Aliasing is an important issue ...
Under-sampling requires an
anti-aliasing filter!
This removes unwanted spurs, which
can occur in the base-band
or be aliased back from any
other Nyquist zones.
The
anti-aliasing filter for undersampled systems is band-pass
around the signal band.
Chapter 1
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DATA CONVERTERS
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BACKGROUND ELEMENTS
11
Skip. Read as you wish.
Sampling-time Jitter
The sampling time depends on the used clock generator that can be affected by jiitter.
x(nT)
DX(2T)≈0
DX(3T)
DX(T)
d(T)
DX(0)
d(2T)
d(3T)
d(0)
0
T
The error is large with a large signal slopes.
2T
3T
t
F. Maloberti
DATA CONVERTERS
Springer
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BACKGROUND ELEMENTS
12
Error caused by Jitter
For a sine wave Xin(t) = A · sin(ωint), the error ∆X(nT ) is given by
∆X(nT ) = A · ωin · δ(nT ) cos(ωinnT )
(5)
Assume that δ(nT ) is the sampling of a random variable δji(t)
< xji(t)2 >=< [Aωin cos(ωinnT )]2 >< δij (t)2 >
A2ωin2
=
< δij (t)2 >
2
The SNR caused by jitter becomes
SN Rji,DB = −20 · log{< δij (t) > ωin}
(6)
(7)
(8)
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
Jitter Requirements
−10
10
−11
Clock jitter [s]
10
-66 dB
-78 dB
−12
10
-90 dB
-102 dB
−13
10
−14
10
6
10
7
10
Input frequency [Hz]
8
10
13
Chapter 1
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DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
14
Assume that the noise is a fixed part plus the one coming from jitter
2
f
2
−8
−8
vn = 0.4 · 10 + 0.1 · 10
20 · 106
81
80
79
SNR [dB]
78
77
76
75
74
73
72
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Normalized input frequency, f/fCK
0.8
0.9
1
(9)
F. Maloberti
DATA CONVERTERS
Springer
2007
Chapter 1
BACKGROUND ELEMENTS
15
Amplitude Quantization
Amplitude quantization changes a sampled-data signal from continuouslevel to discrete-level. The amplitude of each quantization interval or quantization step, ∆, is
XF S
∆=
(10)
M
An input level other than Xm,n the mid point of the n-th interval leads to the
quantization error, Q leading to a quantization output Y corresponding to
an Xin input
Y = Xin + Q = (n + 1/2)∆;
n∆ < Xin < (n + 1)∆
(11)
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
16
Quantization Error
The quantization error is an additive signal with amplitude limited by the
quantization step amplitude
X(nT)
Y(nT)
∑
eQ(nT)
eQ
(a)
D
D/2
Xin
D/2
Xmax
Xmin
(b)
F. Maloberti
DATA CONVERTERS
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Chapter 1
BACKGROUND ELEMENTS
17
Remark
Before going ahead it worths to ...
The quantization error is a
fundamental limit of the quantization process: Q cannot
be avoided: it becomes zero
only when the number of bits
goes to infinity, which is unfeasible in practice.
F. Maloberti
DATA CONVERTERS
Springer
2007
Chapter 1
BACKGROUND ELEMENTS
18
Quantization Noise
Remind the definition of the SNR
SN R|dB = 10 · log
Psign
Pnoise
Psign and Pnoise are the power of signal and noise in the band of interest.
It would be convenient to assimilate the quantization error to noise ...
That’s possible but only under certain conditions.
(12)
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
Quantization Noise: Conditions
? all the quantization levels are exercised with equal probability;
? a large number of quantization levels are used;
? the quantization steps are uniform;
? the quantization error is not correlated with the input.
−3
x 10
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
0
50
100
150
200
250
300
−1
19
Chapter 1
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Springer
2007
BACKGROUND ELEMENTS
20
Quantization Noise: Properties
Time average power
1
f or Q ∈ −∆/2 · · · ∆/2
∆
p(Q) = 0 otherwise
p(Q) =
(13)
The time average power of Q is given by
Z ∆/2 2
∆2
Q
2
PQ =
Q · p(Q)dQ =
dQ =
12
−∞
−∆/2 ∆
Z ∞
Memorize!
(14)
F. Maloberti
DATA CONVERTERS
Springer
2007
Chapter 1
BACKGROUND ELEMENTS
21
Quantization Noise: Calculation of the SNR
The knowledge of the Noise power enables calculating the SNR.
The power of a sine wave with maximum amplitude is
Z
XF2 S
1 T XF2 S
(∆ · 2n)2
2
Psin =
sin (2πf t)dt =
=
T 0
4
8
8
The power of a triangular wave with maximum amplitude is
(15)
XF2 S
(∆ · 2n)2
=
Psin =
12
12
(16)
SN Rsine|dB = (6.02 · n + 1.78) dB
(17)
SN Rtrian|dB = (6.02 · n) dB
(18)
leading to
Chapter 1
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Springer
2007
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22
Equivalent number of bits (ENB or ENoB)
The effective SNR is the real measure of the data converter resolution.
Measured in bits it is the ENB
EN Bsin =
SN Rtot|dB − 1.78
6.02
SN Rtot|dB
EN Btrinag =
6.02
(19)
(20)
If, for example there is quantization an jitter δji
EN B =
2 δ 2 + 2−2N /12) − 1.78
10 · log(π 2fin
ji
6.02
(21)
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
Equivalent number of bits with jitter
16
15
Equivalent number of bit
14
f= 40 MHz
13
f= 80 MHz
12
f= 120 MHz
11
10
f= 160 MHz
f= 200 MHz
9
8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Sampling uncertanty [psec]
0.8
0.9
1
23
Chapter 1
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DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
24
Quantization noise: Properties
Noise power spectrum
The power spectrum is the Laplace transform of the auto-correlation function. For sampled data signals
P(f ) =
Z ∞
−∞
R(τ )e−j2πf τ dτ =
∞
X
R(nT )e−j2πf nT
(22)
−∞
Assume that the auto correlation function, Re(nT ), goes rapidly to zero for
|n| > 0 or, for simplicity, use Re(0) only.
The auto correlation becomes a delta in the time domain and the Laplace
transform becomes frequency independent.
The power spectral density is white with power PQ = ∆2/12 spread uniformly over the unilateral Nyquist interval 0 · · · fs/2.
Chapter 1
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DATA CONVERTERS
Springer
2007
25
BACKGROUND ELEMENTS
Quantization noise: Properties
Noise power spectrum (ii)
The bilateral power spectrum is
∆2
p(f ) =
;
12 · fs
meeting the condition
Z ∞
−∞
p(f )df = ∆2/12
(23)
Signal
S
-fs/2
Quantized
Signal
fs/2
-fs/2
Quantization
Noise
area
D2/12
-fs/2
fs/2
fs/2
Chapter 1
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DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
26
kT/C Noise
An unavoidable limit is the kT/C noise due to the thermal noise associated
with the sampling switch.
Vout
+
SW
Vin
Rs
Cs
2
vn=4kTRs
(a)
Pn,Cs =
0
2
vn,C
(b)
2 (ω) =
vn,C
s
Z ∞
Cs
4kT Rs
1 + (ωRsCs)2
vn,out(f )df = 4kT Rs
Z ∞
0
df
kT
=
2
1 + (2πf RsCs)
Cs
(24)
(25)
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
kT/C Noise (ii)
−3
10
10-bit
11-bit
kT/C noise voltage
−4
10
12-bit
13-bit
14-bit
−5
15-bit
10
−6
10
−1
10
0
10
1
10
Sampling capacitance [pF]
2
10
kT /C noise voltage versus the capacitance value and quantization step for 1 VF S .
27
F. Maloberti
DATA CONVERTERS
Springer
2007
Chapter 1
BACKGROUND ELEMENTS
Remember
In addition to quantization ...
The kT/C noise is a fundamental limit caused by sampling. Sampling any signal
using 1 pF leads to 64.5 µV
noise voltage. If the sampling
capacitance increases by k
the noise voltage diminishes
√
by k.
28
Chapter 1
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DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
29
Example
A pipeline data-converter uses a cascade of two sample-and-hold circuits
in its first stage. The clock jitter is 1 psec. Determine the minimum sampling capacitance that enables 12 bit resolution. The full scale voltage is 1
V; the input frequency is 5 MHz.
Solution
The quantization noise power is ∆2/12. We assume that an extra 50%
noise is acceptable (the system would lose 1.76 dB, 0.29-bit). Thus the
noise budget for KT/C and jitter is
2
vn,budget
=
VF2S
−9 V 2
=
2.48
·
10
24 · 224
(26)
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
30
The noise jitter only affects the signal in the first stage: the second stage
samples the held signal from the first stage. The jitter noise is
VF S
2
· 2πf δji}2 = 2.47 · 10−10V 2
vn,ji = {
2
(27)
Thus, the total noise power that the sampler can generate is 2.23·10−9V 2.
Assuming equal capacitance in both S&H circuits the noise for each is
vn,C = 1.12 · 10−9V 2, leading to a sampling capacitance
kT
4.14 · 10−21
CS = 2 =
= 3.7pF
−9
1.12 · 10
vn,C
(28)
Observe that the noise jitter establishes a maximum achievable resolution
that can not be exceeded even with very large sampling capacitances. This
limit using the same 50% margin used above is 15.3 bit.
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
31
Discrete and Fast Fourier Transforms
The spectrum of a sampled data signal is estimated using
∗
L x (nT ) =
∞
X
x(nT )e−nsT
(29)
−∞
∗
F x (nT ) = X ∗(jω) =
∞
X
x(nT )e−jωnT
(30)
−∞
Unfortunately (30) requires an infinite number of samples that, of course,
are not available in practical cases.
A convenient approximation is the Discrete Fourier Transform (DFT ).
X(fk ) =
NX
−1
n=0
x(nT )e−j2πkn/(N −1)
(31)
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
32
Discrete and Fast Fourier Transforms (ii)
The DFT is a complex function; the real and the imaginary parts are
|X(fk )| =
q
Real [X(fk )]2 + Im [X(fk )]2
"
P h{X(fk )} = arctan
Im{X(fk )}
Real{X(fk )}
(32)
#
(33)
Equation (31) requires N 2 computations. For long series the Fast Fourier
Transform algorithm (FFT ) is more effective.
The FFT reduces the number of computations from N 2 down to N ·log2(N ).
Use power of 2 elements in the series
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
33
Windowing
The DFT and the FFT assume that the input is N-periodic.
Real signals are never periodic and the N-periodic assumption lead to discontinuity between the last and first samples of successive sequences.
1.5
1
0.5
0
−0.5
−1
−1.5
0
100
200
300
400
500
600
700
800
900
1000
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
34
Windowing (ii)
Windowing tackles the missed N-periodicity by tapering the endings of the series.
xw (kT ) = x(kT ) · W (k)
(34)
1
0. 9
Flattop
0. 8
Blackman-Harris
0. 7
Gaussian
0. 6
Hamming
0. 5
0. 4
0. 3
0. 2
0. 1
0
10
20
30
40
50
60
F. Maloberti
DATA CONVERTERS
Springer
2007
Chapter 1
BACKGROUND ELEMENTS
Useful Tips
It is suggested to follow some ...
When using FFT or DTF
make sure that the sequence
of samples is N-periodic. Otherwise use windowing.
With sine wave inputs avoid
repetitive patterns in the sequence: the ratio between
the sine wave period and the
sampling period should be a
prime number.
35
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
36
BACKGROUND ELEMENTS
Windowing Example
Input
1
0.5
0.5
0
0
−0.5
−0.5
−1
200
400
(a)
Windowed Input
1
600
800
1000
−1
200
400
(b)
600
800
1000
Time domain input of the example before (a) and after windowing (b).
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
Input Spectrum
350
120
250
100
200
80
150
60
100
40
50
20
10
20
30
40
50
60
Windowed Input Spectrum
140
300
0
37
70
80
90
100
0
10
20
30
40
50
Spectra with and without windowing.
60
70
80
90
100
F. Maloberti
DATA CONVERTERS
Springer
2007
Chapter 1
38
BACKGROUND ELEMENTS
Is windowing avoidable?
? Windowing is essentially an amplitude modulation of the input that produces undesired effects.
? A spike at the beginning or at the end of the sequence is completely masked being
windowed away almost completely
? Windowing gives rise to spectral leakage.
? The SNR measurement can be accurate but an input sine wave does not give a pure
tone.
With input sine waves use coherent sampling for which an integer
number of clock cycles,k, fits into the sampling window. In addition,
k must be a prime number.
k
fin = N
fs
2 −1
(35)
F. Maloberti
DATA CONVERTERS
Springer
2007
Chapter 1
BACKGROUND ELEMENTS
39
What the FFT Represents?
The FFT of an N sample sequence is made by N discrete lines equally spaced in the
frequency interval 0 − fs .
Each line gives the power falling within fs /N centered around the line itself.
The FFT operates like a spectrum analyzer with N channels whose bandwidth is fs /N .
If the number of points of the series increases then the channel
bandwidth of the equivalent spectrum analyzer decreases and each
channel will contain less noise power.
F. Maloberti
DATA CONVERTERS
Springer
2007
Chapter 1
BACKGROUND ELEMENTS
40
Processing Gain
? An M-bit quantization gives VF2S /12 · 22M noise power.
? The power of the full scale sine wave is VF2S /8.
? The FFT of the quantization noise is, on average, 3/2 · 22M /N below
the full scale
x2
noise |dB = Psign − 1.78 − 6.02 · M − 10 · log(N/2)
The term 10 · log(N/2) is called the processing gain of the F F T .
(36)
F. Maloberti
DATA CONVERTERS
Springer
2007
Chapter 1
BACKGROUND ELEMENTS
Notice
For distinguishing tones from noise ...
If the length of the input series increases by a factor 2
the floor of the FFT noise
spectrum diminish by 3 dB.
Tones caused by harmonic
distortion do not change.
Only long input series reveal
small tones.
41
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
42
BACKGROUND ELEMENTS
Example of FFT Spectra
Input Spectrum in DB
0
Input Spectrum in DB
0
4096 points
−20
32768 points
−20
SNR=62 dB
SNR=62 dB
−40
−40
−60
−60
Processing gain 33.12 dB
−80
−100
−120
Processing gain 42.14 dB
−80
−100
0
50
100
150
200
250
(a)
300
350
400
450
500
−120
0
50
100
150
200
250
(b)
300
350
400
450
500
F. Maloberti
DATA CONVERTERS
Springer
2007
Chapter 1
BACKGROUND ELEMENTS
43
Coding Schemes
• USB - Unipolar Straight Binary: is the simplest binary scheme. It is
used for unipolar signals. The U SB represents the first quantization
level, −Vref + 1/2VLSB with all zero’s (· · · 0000) . As the digital
code increases, the analog input increases by one LSB at a time, and
when the digital code is at the full scale (· · · 1111) the analog input is
above the last quantization level Vref − 1/2VLSB . The quantization
range is −Vref · · · + Vref .
• CSB - Complementary Straight Binary: the opposite of the USB.
CSB coding is also used for unipolar systems. The digital code (· · · 0000)
represents the full scale while the code (· · · 1111) corresponds to the
first quantization level.
F. Maloberti
DATA CONVERTERS
Springer
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Chapter 1
BACKGROUND ELEMENTS
44
• BOB - Bipolar Offset Binary: is a scheme suitable for bipolar systems
(where the quantized inputs can be positive and negative). The most
significant bit denotes the sign of the input: 1 for positive signals and 0
for negative signals. Therefore, (· · · 0000) represents the full negative
scale. The zero crossing occurs at (01 · · · 111) and the digital code
(1 · · · 1111) gives the full positive scale.
• COB - Complementary Offset Binary: this coding scheme is complementary to the BOB scheme. All the bits are complemented and the
meaning remains the same. Therefore, since (01 · · · 111) denotes
the zero crossing in the BOB scheme the zero crossing of COB is
becomes (10 · · · 000).
• BTC - Binary Two’s Complement: is one of the most used coding
schemes. The bit in the MSB position indicates the sign in a complemented way: it is 0 for positive inputs and 1 for negative inputs. The
F. Maloberti
DATA CONVERTERS
Springer
2007
Chapter 1
BACKGROUND ELEMENTS
45
zero crossing occurs at (· · · 0000)). For positive signals the digital
code increases normally for an increasing analog input. Thus, the
positive full scale is (0 · · · 1111). For negative signals, the digital
code is the two’s complement of the positive counterpart. This leads
(1 · · · 0000) to represent the negative full scale. The BTC coding
system is suitable for microprocessor based systems or for the implementation of mathematical algorithms. It is also the standard for digital
audio.
• CTC - Complementary Two’s Complement: is the complementary
code of BTC. All the bits are complemented and codes have the same
meaning. The negative full scale is (0 · · · 1111); the positive full scale
is (1 · · · 0000).
F. Maloberti
DATA CONVERTERS
Springer
2007
Chapter 1
BACKGROUND ELEMENTS
46
D/A Conversion
Ideal Reconstruction
fs
fs
HR,id(f ) = 1 f or −
<f <
2
2
HR,id(f ) = 0 otherwhise
(37)
sin(ωst/2)
r(t) =
(ωst/2)
(38)
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
47
Real Reconstruction
S&H followed by the reconstruction filter.
1 − e−sT
HS&H (s) =
sτ
(39)
sin(ωT /2)
T
HS&H (jω) = j e−jωT /2
τ
ωT /2
(40)
1
Ideal Reconstruction Filter
0.636
sin(wT/2)
fs=1/T
wT/2
f
fNyq
fs
2fs
3fs
4fs
5fs
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
48
Spectrum after the S&H
S&H response
1
Signal spectrum
Residual images
Rule of thumb
f
fB
fS -fB fS
2fS
3fS
The reconstruction must
use an in-band x/sin(x)
compensation if the band
of the signal occupies
about a quarter of the
Nyquist interval or more.
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
49
BACKGROUND ELEMENTS
Study of the S&H effect
Input Signal in the Time−domain
Spectrum of the Input
250
1
200
0.8
150
0.6
0.4
100
0.2
50
0
−0.2
100
200
300
400
500
600
700
800
900
1000
0
0
10
20
30
40
Input signal. Left: Time domain. Right: Spectrum.
50
60
70
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
50
BACKGROUND ELEMENTS
Study of the S&H effect (ii)
Spectrum of the Sampled Signal
Spectrum of the Sampled−and−Held Signal
250
15000
200
10000
150
100
5000
50
0
0
10
20
30
40
50
60
70
0
0
10
20
30
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60
Spectrum of the sampled signal and its sampled-and-held version.
70
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
51
The z-Transform
Is the time-discrete counterpart of the Laplace transform
Z{x(nT )} =
∞
X
x(nT )z −n
(41)
−∞
Z{a1 ·x1(nT )+a2 ·x2(nT )} = a1 ·Z{x1(nT )}+a2 ·Z{x2(nT )} (42)
Moreover, the Z-transform of a delayed signal is
Z{x1(nT − kT )} =
∞
X
−∞
x(nT − kT )z −(n−k)z −k = X(z)z −k
(43)
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
52
Mapping between s and z
z → esT
|z| → eσT ;
wT
p
p
p
s
p
p
ω→Ω
(44)
(45)
F. Maloberti
DATA CONVERTERS
Springer
2007
Chapter 1
BACKGROUND ELEMENTS
53
Features
Mapping is good because of the following
A sampled-data system is
stable if all the poles of its
transfer function are inside
the unity circle.
The frequency response of
the system is the z-transfer
function calculated on the
unity circle.
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
54
Other Mappings
Points of the s-plane mapping the z-plane points: z = −0.5; z = −0.5 +
√
j 3/2; z = −0.8j; z = 1.2
σ=
1
log |z|;
Ts
ω=
wT
p
p
s
-p
p
1
{phase(z) ± 2nπ}
Ts
(46)
F. Maloberti
DATA CONVERTERS
Springer
2007
Chapter 1
BACKGROUND ELEMENTS
55
Approximate integrator in the z-domain
1
.
sτ
the use of the following discrete-time equations
HI (s) =
(47)
y(nT + T ) = y(nT ) + x(nT + T )T,
(48)
y(nT + T ) = y(nT ) + x(nT )T,
(49)
y(nT + T ) = y(nT ) +
x(nT + T )x(nT )
T.
2
(50)
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
56
Approximate integrator in the z-domain (ii)
The use of the Z-transform obtains
zY = Y + zXT,
(51)
zY = Y + XT,
(52)
z+1
(53)
zY = Y + XT
2
giving rise to three different approximate expressions of the integral transfer function
HI,F = T
z
;
z−1
HI,B = T
1
;
z−1
HI,Bil = T
z+1
,
2(z − 1)
(54)
where the indexes ”F ”, ”B” and ”Bil” indicate forward, backward and bilinear.
Chapter 1
F. Maloberti
DATA CONVERTERS
Springer
2007
BACKGROUND ELEMENTS
57
Approximate integrator in the z-domain (iii)
The three above expression are equivalent to the continuous-time integration through approximate mappings given by
sT →
z−1
Tz
(F orward transf ormation),
(55)
sT →
z−1
T
(Backward transf ormation),
(56)
sT →
z−1
2T (z + 1)
(Bilinear transf ormation),
(57)
Notice that the above mappings moves the poles differently than the ideal
mapping s → ln(z)/T .
F. Maloberti
DATA CONVERTERS
Springer
2007
Chapter 1
BACKGROUND ELEMENTS
58
Wrap-up
The background knowledge studied in this first part is essential to properly
understand and design data converters.
We have seen how a data converter performs the transformation from
continuous-time and continuous-amplitude to discrete-time and quantizedamplitude (and vice-versa).
We have studies that data conversion affects the spectrum of the signal
and can sometimes modify its information content. It is therefore important
to know the theoretical implications and to be aware of the limits of the
approximations used for studying a data converter.
We also have studied the mathematical tools used for analysis and characterization of sampled-data systems.