Digital Signal Processing - Quantization process and noise shaping

DIGITAL SIGNAL PROCESSING
Lectured by Assoc. Prof. Thuong Le-Tien
Tel: 0903 787 989
Email: Thuongle@hcmut.edu.vn
September 2011
1
Quantization process 
and noise shaping
Quantization process and noise shaping
1. Quantization process.
2. Over sampling and Noise Shaping.
3. Digital to Analog conversion DAC.
4. Analog to Digital Conversion ADC. 
2
1. Quantization Process
Analog to digital converter - ADC.
3
Analog
signal 
x(t)
Sample & hole
Sampler & quantizer
x(nT)
Sampled 
signal 
A/D
converter
Quantized
signal x (nT)Q
To DSP
B bits/sample
Quantized sample xQ(nT) represented by B bits take 
only one of 2B possible value.
Quantization width or quantizer resolution Q
4
R is the full-scale range
B
R
Q
2

B
Q
R
2
R is in the symmetrical range:
Quantization error: 
e(nT) = xQ(nT) – x(nT)
In general case: e = xQ – x
where, xQ is the quantized value
5
22
Q
e
Q

2
)(
2
R
nTx
R
Q 
Root Mean Square error: 
Quantization error e can be assumed as a random 
variable which is distributed uniformly over the 
range [-Q/2, Q/2] then having probability density: 
6
Mean: 0
1
2/
2/
 

Q
Q
ede
Q
e
12
1 2
2/
2/
22 Qdee
Q
e
Q
Q
 

12
2 Qeerms 
7


2/
2/
)(][
Q
Q
deeepeE
SNR (Signal-to-noise ratio): 
20log10(R/Q) = 20log10(2B) = 20Blog10(2)
1)(
2/
2/


Q
Q
deep



2/
2/
22 )(][
Q
Q
deepeeE
-Q/2 Q/20
p(e)
e







other
Q
e
Q
Qep
,0
22
,
1
)(
Normalization 1/Q needed to guarantee
The statistical expectation
B
Q
R
SNR 6log20 10 






8)(2 ke
Assumed e(n) is white noise then the 
autocorrelation function is the delta function
 
12
)(
2
22 QneEe 
The average power or variance of e(n)
Example: in digital audio application, signal sampled at 
44kHz and each sample quantized using a ADC having 
full scale of 10volts. Determine number of bits B if the 
rms quantization error must be kept below 50 microvolts. 
Then determine the actual rms error and bit rate
Sol:
Which is rounded to B=16
Then bit rate:
The dynamic range of the quantizer is 6B = 96 dB
10
2. Oversampling and noise shaping
Power spectrum of white quantization noise
Power spectrum density of e(n)
The noise power within at Nyquist sub-interval [fa, fb] 
with f = fb – fa :
The noise power over the entire interval f = fs
Noise shaping quantizers reshape the spectrum of the 
quantization noise into more convenient shape. This 
accomplished by filtering the white noise sequence e(n) 
by a noise shaping filter HNS(f). 
xQ(n) = x(n) + (n)
11
2
2
es
s
e f
f



12
Over Sampling ratio
Noise power within a given interval
Power spectral density
2
2
2
)()()()( fH
f
fSfHfS NS
s
e
eeNS

 
 
b
a
b
a
f
f
NS
s
e
f
f
dffH
f
dffS
2
2
)()(


s
s
f
f
L
'

Quantization noise powers
12
2
2 Q
e 
To maintain the same quality 
required the power spectral 
density remain the same '
'22
s
e
s
e
ff


13
12
'
'
2
2 Q
e 
Lf
f e
s
e
se
22
2 '
'
' 
 
BBB
e
eL   2)'(2
2
2
22
'


B = B-B’, or B = 0.5 log2 L
14
The total noise power in the Nyquist interval:










12
2
B2- 1
12
2
p
p
Lp

15
p
s
NS
f
f
fH
2
2
'
2
)( 






 2/'sff 


































12
2
2
12
2
2
2/
2/
12
2
2
2
2
2
1
12
'
'12
'
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'
'
2
'
'
p
p
e
p
s
s
p
e
f
f
p
s
s
p
e
p
ss
e
e
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f
p
f
f
p
df
f
f
f
s
s








22 '/ ee 
= 2 - (B-B’) = 2 -2B










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
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2
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2
2
2/
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12
2
2
22
2
1
2
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p
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f
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ss
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f
p
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df
f
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f
s
s




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
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
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
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
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
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

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
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


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

12
2
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2
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12
2
2
2
2
2
1
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

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
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


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



12
log5.0log)5.0(
2
22
p
LpB
p
16
Oversampling and noise shaping system
3. Digital to Analog Converter DAC
B bit 0 and 1 at input, b = [b1, b2,, bB], 
(a) unipolar natural binary,
(b) bipolar offset binary, 
(c) bipolar 2’s complement.
17
Unipolar natural binary
xQ = R(b12-
1 + b22
-2 +  + bB2
-B) 
xQ = R2
-B(b12
B-1 + b22
B-2 +  + bB-12
1 + bB)
Bipolar offset binary
xQ = R(b12
-1 + b22
-2 +  + bB2
-B – 0.5)
Two’s complement
xQ = R(b12
-1 + b22
-2 +  + bB2
-B – 0.5)
18
19
Converter code for B=4bits, R=10volts
20
4. Analog to Digital Converter (ADC)
Example: A sampled sinusoid x(n)=Acos(2pfn), A=3volts
And f=0.04 cycles/sample. The sinusoid is evaluated at the ten
Sampling times n=0,1,29 and x(n) is quantized using a 4-bit 
ADC with R=10volts. The following table shows the sampled 
and quantized values and its codes
22
5. Analog and Digital Dither
Dither is a low-level white noise signal added to the 
input before quantization for eliminating granulation 
or quantization distortion and making the total 
quantization error behave like white noise
Analog dither
23
Digital dither can be added to a digital prior to a 
requantization operation that reduces the number 
of bits representing the signal.
Nonsubtractive dither process and quantization
(Analog and digital dithers)
v(n) is dither noise
24
y(n) = x(n) + v(n)
Quantization error: e(n) = yQ(n) – y(n)
Total error resulting from dithering and quantization: 
(n) = yQ(n) – x(n)
(n) = (y(n) + e(n)) – x(n) = x(n) + v(n) + e(n) – x(n)
or
(n) = yQ(n) – x(n) = e(n) + v(n) 
Total error noise power
22222
12
1
vve Q   
The two common Rectangular and triangular dither probability densities
25
10log2 = 3 dB
10log3 = 4.8 dB
10log4 = 6 dB
Total error variance (the noise penalty in using dither)
Subtractive dither
Total error 
(n) = yout(n) – x(n) = (yQ(n) – v(n)) – x(n) = yQ(n) – (x(n) + v(n))
(n) = yQ(n) – y(n) = e(n)
26
0 20 40 60 80 100 120
-1,5
-1,0
-0.5
0
0.5
1,0
1,5
0 0,1 0,2 0,3 0,4 0,5
60
120
180
240
Undilhered Quantization Undilhered Spectrum
M
a
g
n
it
u
d
e
(U
n
it
s
 o
f 
Q
)
Quantized
Original
Dithered Quantization Dithered Spectrum
-1,5
-1,0
-0.5
0
0.5
1,0
1,5
(U
n
its
 o
f 
Q
)
60
120
180
240
M
a
g
n
it
u
d
e
0 20 40 60 80 100 120 0 0,1 0,2 0,3 0,4
Quantized
Dithered original
0,5