Bài giảng Digital Signal Processing - Chapter 2: Quantization - Võ Trung Dũng

1Digital Signal Processing
Quantization
Dr. Dung Trung Vo
Telecommunication Divisions
Department of Electrical and Electronics
August, 2013
Quantization Process
 Analog to digital conversion: Sampling and quantization are necessary for any 
digital signal processing operation on analog signals:
 Sample/hold and ADC may be separate modules or may reside on board 
the same chip
 Signal quantization:
 Full-scale range R: in practice are between 1–10 volts.
 Quantization width or the quantizer resolution:
 Number of quantization levels: 2B
B
RQ
2

A/D converter
 Bipolar ADC:
 Unipolar ADC:
 Rounding:
 Truncation:
ADC Classification
kQnTxQ )( 2)(2
QkQnTxQkQ 
kQnTxQ )( QkQnTxkQ  )(
2
)(
2
RnTxR Q 
RnTxQ  )(0
2 Quantization error:
 Error distribution:
 Mean:
 Variance:
 Root-mean-square (rms) error:
 Ratio of R and Q is a signal-to-noise ratio (SNR, dynamic range of the 
quantizer):
Quantization Process
)()()( nTxnTxnTe Q 
01
2/
2/
_  

Q
Q
ede
Q
e
12
1 22/
2/
2
_
2 Qdee
Q
e
Q
Q
 

12
_
2 Qeerms 
B
Q
RSNR 6log20 10 



Example: In a digital audio application, the signal is sampled at a rate of 44 kHz and each
sample quantized using an A/D converter having a full-scale range of 10 volts. Determine
the number of bits B if the rms quantization error must be kept below 50 microvolts. Then,
determine the actual rms error and the bit rate in bits per second
 Solve for B:
 Which is rounded to B = 16 bits, corresponding to 2B = 65536 
quantization levels
 Bitrate:
 Dynamic range of the quantizer:
Quantization Process
12/212/ Brms RQe

82.15
1210.50
10log
12
log
622




 
rmse
RB
VRe Brms 4412/2  
sec/70444.16 KbitsBfs 
dBB 9616.66 
Probabilistic interpretation of the quantization noise: quantized signal xQ(n) as 
a noisy version of the original unquantized signal x(n) to which a noise component e(n) has 
been added
 Statistical properties: are very complicated, may be assumed to be a stationary zero-
mean white noise sequence with uniform probability density over the range [−Q/2,Q/2]. 
Moreover, e(n) is assumed to be uncorrelated with the signal x(n)
 Variance:
 Autocorrelation:
 Cross-correlation:
Statistical properties - quantization noise
)()()( nenxnxQ 
 
12
)(
2
22 QneEe 
  )()()()( 2 knekneEkR eee 
  0)()()(  nxkneEkRex
 Purposes: Oversampling was mentioned earlier as a technique to alleviate the need for 
high quality prefilters and postfilters. It can also be used to trade off bits for samples: if we 
sample at a higher rate, we can use a coarser quantizer
 Power spectral density: quantized signal xQ(n) as a noisy version of the original 
unquantized signal x(n) to which a noise component e(n) has been added
 Analysis: Consider two cases, one with sampling rate fs and B bits per sample, and the 
other with higher sampling rate f s and B bits per sample.
 To maintain the same quality in the two cases, we require that the power spectral densities 
remain the same
Oversampling
s
e
ee f
fS
2
)( 
22
ss fff for
s
e
s
e
ff '
'22  
3 If sampling is done at the higher rate f’s , then the total power σ’e2 of the quantization noise is 
spread evenly over the f’s Nyquist interval.
 Number of bit difference:
 Analysis: a saving of half a bit per doubling of L. This is too small to be useful. For
example, in order to reduce a 16-bit quantizer for digital audio to a 1-bit quantizer, that
is, ∆B = 15, one would need the unreasonable oversampling ratio of L = 230.
Oversampling
BBB
e
eL   2)'(22
2
22'
 LB 2log5.0
 Number of bit difference: Noise shaping quantizers reshape the spectrum of the 
quantization white noise into a more convenient shape.
 Power spectral density:
 Analysis: A noise shaping quantizer operating at the higher rate fs can reshape the flat 
noise spectrum so that most of the power is squeezed out of the fs Nyquist interval and moved 
into the outside of that interval.
Noise shaping
2
2
2 )()()()( fH
f
fSfHfS NS
s
e
eeNS

 
 A typical pth order noise shaping:
 Small f range:
 Assuming: a large oversampling ratio L, we will have fs <<f’s .
Noise shaping
p
s
NS f
ffH
2
2
'
sin2)( 
22
ss fff for
p
s
NS f
ffH
2
2
'
2)( 


 
2
'sff for
12
1
12
'
'12
'
'
2
'
' 22
122
2
2/
2/
22
2










 ps
s
Lpf
f
p
df
f
f
f
p
e
p
s
s
p
e
f
f
p
ss
e
e

BBB
e
e   2)'(22
2
22'

12
1
12
2
 pLp
p
 Number of bit difference:
 Analysis: savings are (p+0.5) bits per doubling of L.
 Example: The first CD player built by Philips used a first-order noise shaper with 4-times 
oversampling, (p = 1, L = 4), achieves a savings of ∆B = 2.1 bits. The Philips CD player used a 
14-bit, instead of a 16-bit, D/A converter at the analog reconstructing stage.
 A general DSP system with oversampling/noise shaping technique
Noise shaping




 12log5,0log)5.0(
2
22 p
LpB
p
4 Number of bit difference:
 Unipolar natural binary:
 Bipolar offset binary:
 Two’s complement
D/A Converters
)2...22( 22
1
1
B
BQ bbbRx
 
B
BB bbbm   ...22 2211
QmxQ 
)5.02...22( 22
1
1   BBQ bbbRx
)5.02...22( 22
1
1   BBQ bbbRx
 B-bit A/D converter:
 Successive approximation A/D converter:
A/D Converters
D/A Converters
 Natural and offset binary cases, truncation:
A/D Converters
5D/A Converters
Example 2.4.1: Convert the analog values x = 3.5 and x = −1.5 volts to their offset binary
representation, assuming B = 4 bits and R = 10 volts, as in Table 2.3.2.
Solution: The following table shows the successive tests of the bits, the corresponding DAC
output xQ at each test, and the comparator output C = u(x − xQ).:
A/D Converters
 Natural and offset binary cases, rounding :
 Two’s complement case:
A/D Converters
Qxy
2
1
 Purpose: Dither is a low-level white noise signal added to the input before quantization for 
the purpose of eliminating granulation or quantization distortions and making the total 
quantization error behave like white noise
 Analog dither:
 Nonsubtractive dither process and quantization:
Analog and Digital Dither