Course 10 example application of random signals - oversampling and noise shaping
Quantizational Process and Noise Shaping
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Transcript of Quantizational Process and Noise Shaping
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DIGITAL SIGNAL PROCESSING
Lectured by Assoc. Prof. Thuong Le-Tien
Tel: 0903 787 989
Email: [email protected]
September 2011
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Quantization process and noise shaping
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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.
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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
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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
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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
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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:
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Mean: 01
2/
2/
Q
Q
edeQ
e
12
1 22/
2/
22 Qdee
Qe
Q
Q
12
2 Qeerms
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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
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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)
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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
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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 :
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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
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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
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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
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The total noise power in the Nyquist interval:
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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
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f
f
f
s
s
12
22
122
2
2/
2/
122
2
22
2
1
12'
'12'
'12'
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2
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22 '/ ee
12log5.0log)5.0(
2
22p
LpBp
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Oversampling and noise shaping system
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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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Converter code for B=4bits, R=10volts
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4. Analog to Digital Converter (ADC)
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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
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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
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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