Analog-to-digital
conversion (ADC)
Ideal ADC
•  Infinite precision of
quantization, fixed sampling
period 1/T
•  Real ADCs have finite
precision and clock jitter
Finite precision ADC
•
The width of each level in uniform bbit quantization
•
When the input signal is within the
dynamic range, the quantization error
is bounded to
Finite precision ADC
•  Assumptions of
quantization error
1)  Mean and variance of e[n]
are independent of n
2)  Probability density function
of e[n] is uniform
3)  E{e[n]e[m]} = 0, m ≠ n
4)  E{e[n]x[n]} = 0
Finite precision ADC
•  From 1) and 2)
•  From 3)
•  SNR in terms of peak-toaverage-ratio (PAR)
Quantization effect
•  Quantization effects for a
complex exponential using
6-bit and 10-bit quantizers
•  Xmax/Xpeak = 1, 10, 0.1
Finite precision ADC
•  Each quantization bit contributes ≈6 dB to SNRq
•  Effective number of bits (ENOB) is often used to measure the
performance of ADC
•  Typically one bit headroom is received for PAR
•  Typically quantization error floor is placed 6 dB below noise floor
•
ENOB = b-2
•  ADCs have improved significantly in sampling frequency but
only 1bit/decade in ENOB
•  Strong interference may saturate ADC
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λ Figure 9.1.5
The peak-to-average ratio for binary PAM with the SRRC pulse shape as a function of
λ excess bandwidth.
λ
Table 9.1.1
λ
Clock jitter effects
λ
λ Figure 9.1.7
The relationship between sampling clock jitter and amplitude error.
Clock jitter effects
λ Figure 9.1.8
The effect of sample clock jitter on the effective number of bits of conversion.
λ
Oversampling ADC
•  Noise power/unit bandwidth
•  In-band noise power, M
denotes oversampling ratio
•  Word-length β with
oversampling vs. Nyquist
sampling
Doubling the sampling ratio
gives half a bit more resolution
Sigma-delta ADC
•  SNRq can be further improved by sigma-delta ADC
•  1-bit samples in the output of ADC correspond to b-bit
samples after low-pass filtering and downsampling
•  Difference of input and delayed output is accumulated/
integrates and quantized by 1 bit
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Sigma-delta quantizer
•
•
•
•
•
Discrete-time model:
y[n] = w[n] + e[n]
w[n] = x[n] - y[n-1] + w[n-1]
y[n] = x[n] + (e[n]-e[n-1])
Transfer function of the
noise
G(z) = (1-z-1)
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Sigma-delta quantizer for constant
input
w0 = 0;
for k = 2:N+1;
w1 = x(k-1) - y(k-1) + w0;
y(k) = sign(w1);
w0 = w1;
end
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Oversampling factor
FT
Word-length F
with
oversampling
compared to Nyquist sampling 1
M =
T
M =
2F
m
= b + log2 M
2Fm
2
1
Word-length
withcompared
oversampling
compared
sampling
= bsampling
+ to
logNyquist
Word-length with
oversampling
to Nyquist
2M
2
Power
spectral density of the modulation noise
1
1
Power
density
the modulation
noise
= bof+
log2 M
= spectral
b + log
2⇡f T
2M
2
2
Py (f ) = |G(ej2⇡f T )|2 Pe (f ) = 4 sin2 (
)Pe (f )
2
2⇡f
T
T 2
Power spectralPower
densityspectral
of the modulation
noise
Pdensity
|G(e
)| Pe (f noise
) = 4 sin2 (
)Pe (f )
thej2⇡f
modulation
y (f ) =of
Substituting the noise
2 density
2 2⇡f T
j2⇡f T 2
j2⇡f
T density
Py (f ) = |G(eP
)|
P
(f
)
=
4
sin
) 2 ( 2⇡f T )P (f )
Substituting
the
noise
e
2 2
)|(2 Pe2(f ))P
=e (f
4 sin
y (f ) = |G(e
e P (f ) =
sin2 (⇡f T )
y
2
3 FT
2 2
2
Substituting the noise density
2
Py (f ) =
sin (⇡f T )
Substituting the noise density
When
f
<<
F
T
3 FT
2 2
2 2
2
2
PWhen
(⇡f
T
)
P
(f
)
⇡
(⇡f T )2
y (f ) =f << sin
2
y
FT
2
3 FT P
3
F
T
sin (⇡f
T)
y (f ) =
2 2
3
F
2
T
Py (f ) ⇡ In-band
(⇡fnoise
T ) power becomes
When f << FT
T2
3 FT
Z Fm
When fP<<
FT 2
2
2
(⇡f T )becomes
y (f ) ⇡noise power
In-band
3
2
2
Pi =
Py (f )df = ⇡ 2 2 T 3 Fm
3 FT
2
9
Py (f Z
)⇡
(⇡f T )
0
Fm 3 F
In-band noise power becomes
2 2 2 3 3
T
Pi =
Py (f )df
= ⇡
T sigma-delta
Fm
Improvement
of
compared to oversampling is 5.1718 +
In-bandZ noise
power becomes
9
Fm
2 2 20 3 3 20 log10 M
dB
Pi =
P+
⇡ M
T dB
Fm
y (f )df =
Z
•  -5.1718
20log
m
Improvement
of 9Fsigma-delta
compared
is 5.1718 +
0
10
2 2 2to 3oversampling
3
Py (f )df = ⇡
T Fm
20 log10 M PdB
i =
Improvement of sigma-delta
compared0 to oversampling is
9 5.1718 +
0 log10 M dB
Sigma-delta quantizer
•  Power spectral density of modulation noise
•  Substituting the pdf of quantization noise Δ /12
•  When f << F
•  In-band noise power becomes
•  Improvement of sigma-delta ADC vs. oversampling ADC
Improvement of sigma-delta compared to oversampling is
20 log10 M dB
5.1718 +
3
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3
Matlab exercise 2.2
sigma-delta
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Sigma-delta quantization with RTL
1.  Take the real (or imaginary) part of the received signal
2.  Implement sigma-delta quantization and low-pass filter
•  For low-pass filter design, see firpmord() and firpm()
3.  Compare the input and output
4.  Return the code and a pdf made by Matlab’s publish function
Example of sigma-delta quantization
sigma-delta quantization of sinusoid
sigma-delta quantization of sinusoid
input
'-"
output
1
0.5
Amplitude
0.5
Amplitude
input
'-"
output
1
0
0
-0.5
-0.5
-1
-1
1850
1900
1950
Time
2000
2050
2100
1940
1950
1960
1970
1980
1990
2000
2010
Time
•  No oversampling/downsampling (M=1)
•  Output depends on the cut-off of the low-pass filter
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