Quantization

1
Quantization

• Trac D. Tran
• ECE Department
• The Johns Hopkins University
• Baltimore, MD 21218

2
Outline

• Quantization
• Nonlinear mapping
• Forward and inverse quantization
• Quantization errors
• Clipping error
• Approximation error
• Signal-to-noise ratio (SNR)
• Examples

3
Quantization

• Sampling frequency controls the time resolution
  in the analog-to-digital conversion process
• Quantization
• Controls the resolution of the amplitude values
• Approximation of a signal source using a finite
  collection of discrete amplitudes
• Non-linear mapping a range of amplitude is
  mapped to a unique level
• Applications
• A/D conversion
• Compression

4
Typical Quantizer

Forward Quantizer
x
y
Q
input
output
clipping
y
quantization level or codeword
111
110
101
100
011
010
quantization cell
001
x
000
clipping
decision boundaries
5
Inverse Quantizer

• Typical reconstruction (cell centroid)
• Quantization error

y
111
110

x
x
-1
Q
Q
101
100

011
x
x

010
001
x
000
clipping
decision boundaries
6
Mid-rise versus Mid-tread
y
y
x
x
Uniform Midrise Quantizer
Uniform Midtread Quantizer

• Popular in ADC
• For a b-bit midrise

• Popular in compression
• For a b-bit midtread

• Quantization error bound

7
Quantization Errors

• Approximation error
• Lack of quantization resolution, too few
  quantization levels, too large quantization
  step-size
• Causes staircase effect
• Solution increases the number of quantization
  levels
• Clipping error
• Inadequate quantizer range limits, also known as
  overflow
• Solution
• Requires knowledge of the input signal
• Typical practical range for a zero-mean signal

8
Example
Design a 3-bit quantizer for a signal with range
0,128

• Maximum possible number of levels

• Quantization stepsize

• Quantization levels

• Reconstruction levels

• Maximum quantization error

9
Input-Output Relationship
y
7
6
5
4
3
2
1
x
80
96
112
0
32
16
48
128
64
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Example of Popular Quantization

• Round
• Floor
• Ceiling

y
x
Uniform midtread quantizer from Round and Floor
11
Quantization from Rounding
x
6
10
14
14
6
10
2
2
Uniform Quantizer, step-size4
12
Quantization Noise Power

• Definition power of a signal q(t)

y
x
13
Signal-to-Noise Ratio

• Definition of SNR in decibel (dB)

power of the signal
power of the noise

• For quantization noise

Suppose that we now add 1 more bit to our Q
resolution
14
SNR Tidbits

• Suppose the signal power is 1 million times the
  noise power
• SNR for CD music 16-bit uniform quantizer
• We can use the decibel scale to measure sound
  intensity as well
• Whisper 15 dB
• Typical conversation level 60 dB
• Car horn 110 dB
• Threshold-of-pain decibel level (jet engine) 120
  dB
• Gunshot 140 dB