quantizational process and noise shaping
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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.
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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 rangeB
RQ
2
B
Q
R2
R is in the symmetrical range:
Quantization error:
e(nT) = xQ(nT) – x(nT)
In general case: e = xQ – xwhere, xQ is the quantized value
5
22
Qe
Q
2)(
2
RnTx
RQ
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: 01
2/
2/
Q
Q
edeQ
e
12
1 22/
2/
22 Qdee
Qe
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
Qe
Q
Qep
,0
22,
1
)(
Normalization 1/Q needed to guarantee
The statistical expectation
BQ
RSNR 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
22
es
s
e ff
12
Over Sampling ratio
Noise power within a given interval
Power spectral density2
22
)()()()( fHf
fSfHfS NS
s
eeeNS
b
a
b
a
f
f
NS
s
e
f
f
dffHf
dffS2
2
)()(
s
s
f
fL
'
Quantization noise powers12
22 Qe
To maintain the same quality required the power spectral density remain the same '
'22
s
e
s
e
ff
13
12
''
22 Qe
Lff e
s
ese
222 '
'
'
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
2B2- 1
122
p
p
Lp
15
p
s
NSf
ffH
2
2
'
2)(
2/'sff
12
22
122
2
2/
2/
122
2
22
2
1
12'
'12'
'12'
'
2
'
'
p
p
e
p
s
sp
e
f
f
p
s
sp
e
p
ss
ee
Lpf
f
p
f
f
f
f
f
s
s
22 '/ ee
= 2 -2(B-B’) = 2 -2B
12
22
122
2
2/
2/
122
2
222
1
12'
'12'
'12'
'
2
'
'
p
p
e
p
s
sp
e
f
f
p
s
sp
e
p
ss
ee
Lpf
f
p
f
f
f
f
f
s
s
12
22
122
2
2/
2/
122
2
22
2
1
12'
'12'
'12'
'
2
'
'
p
p
e
p
s
sp
e
f
f
p
s
sp
e
p
ss
ee
Lpf
f
p
f
f
f
f
f
s
s
22 '/ ee
12log5.0log)5.0(
2
22p
LpBp
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.
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Unipolar natural binary
xQ = R(b12-1 + b22-2 + … + bB2
-B)
xQ = R2-B(b12B-1 + b22
B-2 + … + bB-121 + 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)
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19
Converter code for B=4bits, R=10volts
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4. Analog to Digital Converter (ADC)
Example: A sampled sinusoid x(n)=Acos(2pfn), A=3voltsAnd f=0.04 cycles/sample. The sinusoid is evaluated at the tenSampling times n=0,1,2…9 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 DitherDither 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
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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
1vve Q
The two common Rectangular and triangular dither probability densities
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10log2 = 3 dB10log3 = 4.8 dB10log4 = 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)
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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
240Undilhered Quantization Undilhered Spectrum
Ma
gn
itu
de
(Un
its o
f Q
)
Quantized
Original
Dithered Quantization Dithered Spectrum
-1,5
-1,0
-0.5
0
0.5
1,0
1,5
(Un
its o
f Q
)
60
120
180
240
Ma
gn
itu
de
0 20 40 60 80 100 120 0 0,1 0,2 0,3 0,4
Quantized
Dithered original
0,5
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