Sampling & Quantization

Training material 2007 NHK-CTI

Digital Engineering Training material of NHK-CTI

2007.7.24 NHK Communications Training Institute Hideo Tsuji [email protected] 1

Training material of NHK-CTI

Table of contents Training material of NHK-CTI

• Overview of Digital System • Binary and Hexadecimal Notation • A/D Conversion – Sampling – Quantization

• Source Coding • D/A Conversion

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Training material 2007 NHK-CTI

Overview of digital system

Training material of NHK-CTI

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Training material of NHK-CTI

What is “Digital” ? Training material of NHK-CTI

• Definition of “analog” in a dictionary – of, relating to, or being a mechanism in which data is represented by continuously variable physical quantities

• Definition of “digital” in a dictionary – of, relating to, or using calculation by numerical methods or by discrete units – of or relating to data in the form of numerical digits

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Training material of NHK-CTI

“Analog” and “Digital” Training material of NHK-CTI

• Analog level

– Represents a level or a state with continuously variable physical quantities

time

• Digital level

– Represents a level or a state with discrete numerical units

0

1

0

1

1

0

time

5

Training material of NHK-CTI

Analog clock v.s. Digital clock Training material of NHK-CTI

• Analog Clock

• Digital Clock

– Hands turn continuously – Always shows actual time

– Digits change every minutes – Both ’12:34.00’ and ’12:34.59’ turn into ’12:34’ (Quantization error)

Represented Time

analog digital quantization error Actual Time

12:34.00

12:35.00

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Training material of NHK-CTI

Why “Digital”? ~ Advantage of digital system

Training material of NHK-CTI

• Easy processing with computers – Digital equipment is much smaller and energy-saving than analog equipment – Fusion of broadcast and communication

• Easy integration(multiplexing) of several media – Able to deal with audio/video/data in the same way

• Efficient data compression – Bandwidth requirement for ONE analog TV program and THREE digital TV programs are equivalent

• Error-proof performance – Never degrade with duplication or relaying 7

Training material of NHK-CTI

Disadvantage of digital system Training material of NHK-CTI

• Minor trouble may cause absolute failure – Analog system: noise increases gradually – Digital system: no influence with low error rate (owe to error correction), but suddenly fails absolutely at certain point

• Increase of delay – Example: delay of HDTV satellite broadcasting
Analog… 0.3 second (roundtrip time to the satellite)
Digital… 1.0 second (roundtrip + encode/decode delay)

• Easy to create pirated copy – How can we protect copyrighted material?
Copy management system: copyright holders can define the generation of duplication (none, child only, no limit…)
Watermark: implant invisible identification 8

Training material of NHK-CTI

Overview of digital system Training material of NHK-CTI

Sender (broadcaster) A/D Conversion

Source Coding

Modulation

Channel

Receiver (audience) D/A Conversion

Decoding

Demodulation

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Training material of NHK-CTI

Multi--media multiplexing Multi

A/D Conversion

Source Coding

A/D Conversion

Source Coding

Data

Source Coding

Modulation

Multiplexing

Audio

Video

Training material of NHK-CTI

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Training material 2007 NHK-CTI

Binary and Hexadecimal Notation

Training material of NHK-CTI

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Training material of NHK-CTI

Binary, Decimal, Hexadecimal Training material of NHK-CTI

• Binary number – Computers can distinguish only two states; high and low voltage

• Hexadecimal number – Integrate every 4 digits of a binary number to shorten the length (easy to understand for us)

Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Binary 0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111 10000

Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F 10 12

Training material of NHK-CTI

Convert decimal into binary Training material of NHK-CTI

• What is (1,234)d in binary number? 2)1,234 2)617 …0 2)308 …1 2)154 …0 2)77 …0 2)38 …1 2)19 …0 2)9 …1 2)4 …1 2)2 …0 1 …0

(10011010010)b

• What is (567)d in binary number? 2)567 2)283 …1 2)141 …1 2)70 …1 2)35 …0 2)17 …1 2)8 …1 2)4 …0 2)2 …0 1

(100110111)b 13

Training material of NHK-CTI

Convert binary into decimal Training material of NHK-CTI

• What is (10011010010)b in decimal number? 1 0 0 1 1 0 1 0 0 1 0 210+ 29+ 28+ 27+ 26+ 25+ 24+ 23+ 22+ 21+ 20 =1,024+128+64+16+2 = (1,234)d

• What is (1000101011)b in decimal number? 1 0 0 0 1 0 1 29 + 25 + 23 =512+32+8+2+1 = (555)d

0

1 1 + 21 + 20

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Training material of NHK-CTI

Convert decimal into hexadecimal Training material of NHK-CTI

• What is (1,234)d in hexadecimal number? – (1,234)d = (10011010010)b – 100 1101 0010 4 D 2

(4D2)h • Another solution

• What is (573)d in hexadecimal number? 16)573 16)35 …13=D 2 …3

(23D)h

16)1,234 16)77 …2 4 …13 (4D2)h (Note that (13)d=(D)h) 15

Training material of NHK-CTI

Convert hexadecimal into decimal Training material of NHK-CTI

• What is (4D2)h in decimal number? 4 x 162 +D x 161+2 x 160 = 4x256+13x16+2x1 = (1,234)d

• What is (1ABC)h in decimal number? 1x163+Ax162+B x161+Cx160 = 1x4,096 +10x256 +11x16+12x1= (6,844)d

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Training material 2007 NHK-CTI

Analog to Digital Conversion

Training material of NHK-CTI

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Training material of NHK-CTI

Analog to Digital Conversion (ADC) Training material of NHK-CTI

• Video and audio are ANALOG signal Convert them to DIGITAL signal A/D Conversion

Source Coding

Modulation

Sampling Quantization

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Training material 2007 NHK-CTI

Sampling Training material of NHK-CTI

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Training material of NHK-CTI

What is sampling? Training material of NHK-CTI

• Measure the level of input analog signal at every certain temporal (or spatial) intervals Switch on/off periodically series of samples

input signal level

level time (or position)

time (or position) 20

Training material of NHK-CTI

Sampling interval and sampling frequency

 fs=1/Ts • Amplitude of samples is analog value  PAM (Pulse Amplitude Modulation)

input signal

time level

– Sampling interval Ts – Sampling frequency fs

Ts

sampling pulse

time level

• Pick out samples from a continuous signal at every temporal interval Ts

level

Training material of NHK-CTI

series of samples

time 21

Training material of NHK-CTI

How much is the better sampling frequency? Training material of NHK-CTI

• High sampling freq.

• Low sampling freq. level

level time

level

level

D/A conversion

time

– Able to reproduce original waveform – Increase the number of samples

time D/A conversion

time

– Impossible to restore original input signal 22

Training material of NHK-CTI

Effect of sampling frequency for images Training material of NHK-CTI

• Resolution is equivalent to sampling freq. – Sampling freq.: No. of samples per unit time – Resolution: No. of samples per unit length

x4 resolution 23

Training material of NHK-CTI

Sampling theorem (Nyquist theorem) Training material of NHK-CTI

• To reproduce original waveform from a series of samples, sampling frequency fs must be at least TWICE as much as the highest frequency fh of input analog signal – Sampling theorem: fs ≥ 2fh

(2fh: Nyquist freq.) input signal: 2kHz sine wave

Sampling point

Sampling frequency: 4kHz 24

Training material of NHK-CTI

Aliasing Training material of NHK-CTI

• If the sampling frequency is less than 2fh, distortion will appear when these samples are D/A converted – Reproduced waveform is lower frequency than original signal level

:6kHz time

:2kHz

Sampling freq.: 4kHz 25

Training material of NHK-CTI

Why aliasing occurs? Mathematical analysis

• Temporal waveform: ∞

x (t ) = ∫ X ( f ) exp( j 2πft ) df −∞

Training material of NHK-CTI

• Frequency waveform: ∞

X ( f ) = ∫ x( t ) exp( − j 2πft )dt −∞

level

level time

fh

frequency


x(t) and X(f) are convertible each other  x(t)  X(f): Fourier Transformation  x(t)  X(f): Inverted Fourier Transformation 26

Training material of NHK-CTI

Frequency waveform of samples Training material of NHK-CTI

• Temporal Waveform

level

level

x(t): input signal

• Frequency Waveform X(f)

fh : highest frequency of input signal

fh frequency

time level

level

y(t): sampling pulse

Y(f)

Ts time

time

level

level

z(t): series of samples

fs : sampling frequency (fs=1/Ts)

fs

2fs frequency

Z(f)

frequency

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Training material of NHK-CTI

Nyquist frequency and aliasing Training material of NHK-CTI

• fs > 2fh

• fs < 2fh

fh

2fh

frequency

fh

level

level

Original waveform is reproducible with LPF

fh

fs

level

level

fs

frequency

Original waveform is NOT reproducible because of aliasing

frequency

fs/2

2fh

fs/2

fh

frequency

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Training material of NHK-CTI

Example of aliasing distortion Training material of NHK-CTI

Original image

Image with aliasing distortion

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Training material of NHK-CTI

Avoid aliasing Training material of NHK-CTI

• Low Pass Filter (LPF) can limit the highest frequency of input analog signal A/D Conversion

LPF

Source Coding

level

level frequency

frequency

– Sampling frequency is usually higher than Nyquist frequency
Compact Disk (fh=20kHz)  fs=44.1kHz
Telephone (fh=3.6kHz)  fs=8kHz

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Training material of NHK-CTI

Summary: Sampling Training material of NHK-CTI

• Measure the level of input analog signal at every certain temporal (or spatial) intervals • If sampling frequency is not high enough, the original signal is not reproducible because of aliasing • Sampling theorem: To avoid aliasing, sampling frequency must be at least twice as much as the higher frequency of input signal 31

Training material 2007 NHK-CTI

Quantization Training material of NHK-CTI

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Training material of NHK-CTI

What is quantization? Training material of NHK-CTI

• PAM signal is not digital – Temporally (or spatially) quantized – Level of the signal is still analog quantity

• Approximate the signal level to a quantized value  PCM (Pulse Coded Modulation) level

PAM

time (or position)

level 7 6 5 4 3 2 1 0

PCM

time (or position)

7 7 6 3 2 2 4 5 4 ... 111 111 110 011 010 010 100 101 100 … 33

Training material of NHK-CTI

Quantization error Training material of NHK-CTI

• Difference between the analog signal and the digital representation • A quantization error causes quantization noise when the signal is D/A converted

PAM

PCM

D/A converted level

level

level time

time

time

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Training material of NHK-CTI

How much is the better quantization level? Training material of NHK-CTI

• 4-bit quantization (16 levels: 0~15)

• 2-bit quantization (4 levels: 0~3) level

level time

time

14 15 12 5 3 4 8 9 7 ...

3 3 3 1 1 1 2 2 2...

1110 1111 1100 0101 0011 0100 1000 1001 0111...

11 11 11 01 01 01 10 10 10...

– Less quantization error – More amount of data

– More quantization error – Less amount of data 35

Training material of NHK-CTI

Effect of quantization level Training material of NHK-CTI

• 4-bit (16 levels) • Contouring appears in roughly quantized pictures • Size of picture data is proportional to the quantization level (only if no compression applied)

• 8-bit (256 levels)

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Training material of NHK-CTI

Effect of quantization level Training material of NHK-CTI

• 6-bit quantization – R/G/B 2-bit each – 64 levels

• 24-bit quantization – R/G/B 8-bit each – 16,777,216 levels

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Training material of NHK-CTI

How to relieve the effect of quantization error Training material of NHK-CTI

• Non-linear quantization • Scale factor • Dithering

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Training material of NHK-CTI

Non--linear quantization Non Training material of NHK-CTI

• Non-linear quantization level

level

• Linear quantization

– Quantization error is larger in low level signals
6.5”7” … 8%
1.5”2” … 33% (Quantity of error is same, but ratio or error is different)

– Solution of an affection of quantization error
High level signal  Coarse quantization
Low level signal  Fine quantization

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Training material of NHK-CTI

Quantization with scale factor Training material of NHK-CTI

• Quantization of consecutive low level samples level

time

– High quantization error – Waste of higher bits

Scale factor

time

– Quantize signals based on maximum level in certain interval – Need to transmit scale factor value

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Training material of NHK-CTI

Dithering Training material of NHK-CTI

• Add a random (or certain patterned) noise to input signal before quantization – Without dithering level

level

level time

time

– With dithering

Sampling

Quantize level

level

level time

time

time

time 41

Training material of NHK-CTI

Example of dithering Training material of NHK-CTI

• Without dithering

– Contouring appears at quantizing thresholds

• With dithering

– Represent pseudo gradations

42

Training material of NHK-CTI

Example of dithering Training material of NHK-CTI

• Convert to 64 colors without dithering

without dithering

• Convert to 64 colors with dithering

original

with dithering 43

Training material of NHK-CTI

Summary: Quantization Training material of NHK-CTI

• Convert analog signal quantity into quantized value (convert PAM to PCM) • Increasing the quantization level will reduce quantization error, but increase the amount of data • Non-linear quantization and scale factor may improve low level signals suffering quantization error 44

Training material 2007 NHK-CTI

Source Coding (or Encoding)

Training material of NHK-CTI

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Training material of NHK-CTI

What is Coding? Training material of NHK-CTI

• Process the sequence of binary data (PCM data) A/D Conversion

Source Coding

Modulation

Data compression

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Training material 2007 NHK-CTI

Data compression Training material of NHK-CTI

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Training material of NHK-CTI

What is data compression? Training material of NHK-CTI

• Eliminate redundancy of the data • What is redundancy? (example of a video) – Spatial redundancy
Neighbor pixels are similar

– Temporal redundancy
Neighbor frames are similar

– Psychological redundancy
Details are not significant

– Statistical redundancy
Data have certain trends 48

Training material of NHK-CTI

How huge is a data without compression? Training material of NHK-CTI

• Example of 60 minutes HDTV-quality video – A picture with 1,920x1,080 pixels, 8 bit quantized for each color (RGB)
1,920 x 1,080 x (8+8+8) = 49,766,400 bits (50Mbits)

– Video with 30 frames per second
49,766,400 x 30 = 1,492,992,000 bits/sec. (1.5Gbps)

– If this video continues for 60 minutes…
1,492,992,000 x 3,600 = 5,374,771,200,000 bits (5.3Tb ≈ 670GByte)

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Training material of NHK-CTI

Categories of compression techniques Training material of NHK-CTI

Entropy Encoding

Source Coding

Hybrid Coding

Run-length Coding Huffman Coding Arithmetic Coding DPCM Prediction DM FFT Transformation DCT Vector Quantization JPEG MPEG H.261 DVI RTV, DVI PLV 50

Training material of NHK-CTI

Reversible compression and Non--reversible compression Non

Training material of NHK-CTI

• Reversible compression (loss-less compression) – Decoded data is the same as original – Applied for computer, medical data

• Non-reversible compression (lossy compression) – Decoded data is not the same as original – Compression ratio is better than reversible compression – Applied for audio and video 51

Training material of NHK-CTI

Example of reversible compression Training material of NHK-CTI

• Run-length Encoding (RLE) – 000ppppppXXXXaaa 3(0), 6(p), 4(X), 3(a) Ratio:16/8 – Easy to implement – Do not fit for data alternating 0 and 1 frequently

• Huffman coding – Assign shorter words for frequently appearing patterns – Efficient if the trend of appearing pattern is well-known

• Arithmetic coding – Adaptively assign words according to the statistics of object data

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Training material of NHK-CTI

Example of Huffman coding Training material of NHK-CTI

Word

Symbol Probability A

0.50

0.50

0.50

0.50

B

0.15

0.15

0.15

0.15 1

C

0.15

D E F G

0.50 0.30

0.50

1 0.15 0 0.50 0.09 0.09 0.09 1 0 0.20 0.20 0.08 0.08 1 0.11 0 1 0.02 0.03 0 0.01 0 0.15

0.15

1 1 1.00 011 010 0 001 0001 00001 00000

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Training material of NHK-CTI

Example of Huffman coding Training material of NHK-CTI

• Original data AABACEAAACABGAAADBCACAF DEDABAA

• Fixed length code

A

B

C

D

E

F

G

001

010

011

100

101

110

111

001001010001011 011101 101001001001011001010 111 001001001100010011001011001110100101100001010001001

110111010000111101010110000011100101101010101000010010001001101111

• Huffman code

A

B

C

D

E

1

011

010

001

0001

90bits

66bits F

G

00001 00000

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Training material of NHK-CTI

Coding in Facsimiles Training material of NHK-CTI

• Scan pixels horizontally and compress data with run length coding and modified Huffman coding 11 1000 11 1100 – Terminating

1 2 3 4 5 6

white 000111 0111 1000 1011 1100 1110

code

–0101111011 Make up code 192196 010111 whites white + pixels 1011 4 whites (For more than 64 pixels in a row)

black 010 11 10 011 0011 0010

62 00110011 000001100110 63 10100 000001100111 0 00110101 0000110111

64 128 192 256

white 11011 10010 010111 0110111

black 0000001111 000011001000 000011001001 000001011011

1728 010011011 0000001100101
Example Continuous 196 pixels of white 192 whites + 4 whites word is “0101111011” 55

Training material of NHK-CTI

Audio compression considering audibility Training material of NHK-CTI


Perceptual coding • Aural masking effect

81

– the presence of a strong audio signal makes a temporal or spectral neighborhood of weaker audio signals imperceptible


Threshold of audibility

 Discard inaudible data to compress

level (dB)

 Signals are inaudible if its level is lower than certain threshold Aural masking Threshold of audibility

1k

10k 20k frequency (Hz) 56

Training material of NHK-CTI

Summary: Data compression Training material of NHK-CTI

• Eliminate redundancy of data to reduce the size of a data • Two category of compression: – Reversible compression
Restorable

– Non-reversible compression
Higher compression ratio
Non-restorable, but difference is insignificant because of human audio/visual characteristics

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Training material 2007 NHK-CTI

Digital to Analog Conversion

Training material of NHK-CTI

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Training material of NHK-CTI

Digital to Analog Conversion (DAC) Training material of NHK-CTI

• Output signal of audio/video must be analog  Convert digital to analog

D/A Conversion

Decoding

Demodulation

59

Training material of NHK-CTI

Comparison of DAC and ADC Training material of NHK-CTI

level

PAM

level

Quantization error 7 6 5 4 3 2 1 0

time

A/D conversion D/A conversion

PCM

time

7 7 6 3 2 2 4 5 4 ... not same!

111 111 110 011 010 010 100 101 100 …

7 7 6 3 2 2 4 5 4 ... level

level

= time

7 6 5 4 3 2 1 0

PCM

time

60

Training material of NHK-CTI

Convert PCM to analog signal Training material of NHK-CTI

• Analog signal will be obtained filtering a PCM signal with a low pass filter – Cutoff frequency of the LPF should be as same as LPF for A/D converter level

level Ts 2Ts

Ts 2Ts

time

time

level

LPF

freq.

61

Training material of NHK-CTI

LPF (interpolation) Training material of NHK-CTI

•

Principal of interpolation – When the pulse signal passes through LPF, it becomes a –

sampling function (sinc function) Assumed that cutoff frequency of LPF is same as the sampling frequency, it is theoretically completely restorable （However, ideal LPF cannot be produced) 1

1

0

‐T ‐2T

time 1 0

LPF

T=1/2f 0 T

2T time

sinc = sin(x)/x

frequency 62

Training material of NHK-CTI

LPF (interpolation) Training material of NHK-CTI

• Interpolation – Interpolate by passing the PAM signal through LPF and reproduce the original shape of waves – Ideal LPF cannot be produced, therefore the interpolation error is inevitable PAM signal

0

T

2T 3T 4T

Analog signal

0

time

1

LPF

0

frequency

T

2T 3T 4T

time

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Training material of NHK-CTI

Summary: D/A Conversion Training material of NHK-CTI

• Convert PCM to analog signal using LPF • Reproduced waveform is not same as the original signal – Quantization error in A/D conversion – Distortion of filtered signal in D/A conversion

freq.

level

level

ideal filter

actual filter freq. 64

Training material 2007 NHK-CTI

Informative reference

Training material of NHK-CTI

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Training material of NHK-CTI

Type of pulse code conversion Training material of NHK-CTI

Example of linear quantization Output

Output 3 2 -3 -2

-1

1

1 0

2

3

Input

Input

-1 -2 -3

Mid tread type

Mid riser type

No output in a small level

A small signal is not suppressed

66

Training material of NHK-CTI

Example of quantization Training material of NHK-CTI

2 1.5 1

V

0.5 0

-0.5 -1 -1.5 -2

0

0.01

0.02

0.03

0.04

Time [sec]

0.05

0.06

2-bit midriser 67

Training material of NHK-CTI

Quantization noise Training material of NHK-CTI

2 1.5

Quantization error

1

V

0.5 0

-0.5 -1 -1.5 -2

0

0.01

0.02

0.03

0.04

0.05

0.06

Time [sec]

2-bit midriser 68

Training material of NHK-CTI

Linear quantization of sine wave Training material of NHK-CTI

Level

V t

approximated by stairs

∆V

∆V The difference between true value and quantized value becomes like a saw-toothed wave 69

Training material of NHK-CTI

Quantization noise Training material of NHK-CTI

• Quantization error becomes quantization noise • It is thought that the sample value can take the given value within one quantization step (equal probability), therefore quantization noise is treated as variance, Nq = 1 ∆V

Signal power is

∆V / 2

2 ∆ V x dx = 2

– ∆V / 2

12

V 2 V2 S =( ) = 8 2 2

V 2 /8 3 V2 3 2b S / Nq = = = 2 2 2 ∆V / 12 2 ∆V 2

V2 Where number of quantization step: 2 = ∆V 2 2b

Therefore 6dB is improved by increasing one bit of b S / N (dB ) = 10 log(3 / 2 * 22b ) = 6b + 1.8 70

Training material of NHK-CTI

Non--linear quantization (µNon (µ-low) Training material of NHK-CTI

QUANTIZER CHARACTERISTIC 1

y = sign( x ) 0.8

log ( µ x + 1 ) e

log ( µ + 1 ) e

Where、x is normalized input

0.6

0≤ x ≤1

OUTPUT

0.4

: output y is quantized

0.2 0

− 1( x < 0)  sign( x) =  0 ( x = 0)  1( x > 0) 

255 128 32

-0.2

ITU standard :µ=255 (Japan, USA)

-0.4 -0.6 -0.8

In Europe: A-Low -1 -1

-0.5

0

INPUT

0.5

1

G.711 A-LAW/µ-LAW(8 bits/sample) 71

Training material of NHK-CTI

Compression and extension of signal Training material of NHK-CTI

• Even if a small signal level, compression and extension is performed to maintain S/N ratio. Output

Output

0

0

Input

Input

Compression

Extension

Transmission side

Reception side 72

Training material of NHK-CTI

Secret to reduce bit error Training material of NHK-CTI

Error!

Voltage 1.75 1.25 0.75 0.25 -0.25 -0.75 -1.25 -1.75

Transmitting signal

Received signal（Wave is distorted）

T 1.75V 1.25V 0.75V 0.25V -0.25V -0.75V -1.25V -1.75V

= 000 = 001 = 010 = 011 = 100 = 101 = 110 = 111

Two bits or three bits error occurs by the noise

1.75V 1.25V 0.75V 0.25V -0.25V -0.75V -1.25V -1.75V

= 100 = 101 = 111 = 110 = 010 = 011 = 001 = 000

This allocation allows error stay only one bit

Changing the relation of data and the voltage makes bit error minimum

73

Training material of NHK-CTI

Type of code Training material of NHK-CTI

Example of binary code: 8 steps quantization Quantization level

Natural

Gray

Reflected

0

000

000

000

1

001

001

001

2

010

011

010

3

011

010

011

4

100

110

111

5

101

111

110

6

110

101

101

7

111

100

100 74

Training material of NHK-CTI

Symbol allocation Training material of NHK-CTI

10

Q

00 Q 0100

0101

10

0110

0111

0011

0010

1

3

1 -3

Q

QPSK

0000

3

I 11

0001

-1

I

011 001

010

000 I

100 101

110

-1 1110

1111

1011

1010

1100

1101

1001

1000

-3

16 QAM

111

8PSK

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Training material of NHK-CTI

Type of pulse code Training material of NHK-CTI

 Video/audio signal can be express by the change in voltage by a/d conversion NRZ （Uni-polar） ） NRZ （Bi-polar） ） NRZI RZ （Uni-polar） ） RZ （Bi-polar） ） Manchester

+1 0 0 +1 0 -1

1

0

0

1

1

1

0

1

0

+1 0 +1 0 +1 0 -1 +1 0 -1 76

Training material of NHK-CTI

Secret of digital transmission (limitation of frequency band)

Training material of NHK-CTI

• When one symbol is transmitted by a rectangular pulse, the frequency element is shown in figure below. Since the high frequency element infinitely remains, and it is not possible to use in a severe band limiting channel • If the pulse is rolled off, the high frequency element decreases, however, interfering between symbols occurs (the error occurs in decoding the symbol) • Required both frequency band-limitation and no interference between symbols Fourier transformation f(t)

Higher frequency element remains (It might influence other signals)

1.5

1.25

F(f)

Fourier transformation

1

1

0.75

0.75

0.5

0.5

0.25

0.25 -1

-0.5

0.5

One pulse

t

1 -10

-5

5

10

f After Fourier transformation of one pulse

77

Training material of NHK-CTI

Nyquist pulse Training material of NHK-CTI

1.5

f(t) Every T cycle amplitude becomes zero 1

F(f)

Inverse Fourier transformation

1.25 1

0.75

…..

0.5

0.75

0.25

0.5

t

0.25 -1

-0.5

0.5

1

f

W/2

-W/2

-10

T=1/W

Frequency characteristic of band-limited signal

-5

-5

10

Signal continues from the infinite past to the infinite future ("Sampling function" in the sampling theorem. )

When each pulse is overlapped, the Nyquist pulse is convenient because every T cycle amplitude becomes 0, although it has the disadvantage that the vibration of around the main pulse lasts long. -10

5

1 0.75 0.5 0.25 5

10 78

Training material of NHK-CTI

Nyquist pulse Training material of NHK-CTI

(1 + a)  > 0 , f W  2  1 1 W (1 + a ) (1 − a) F ( f ) =  (1 − sin( π ( f − ))), W> f > W 2 2 aW 2 2  (1 − a ) 1, f > W  2

F: Nyquist function a: Roll off rate

The Nyquist pulse becomes zero every T cycle, and has the feature with a small amplitude in the part of the tail of the vibration. The occupied frequency band becomes (1+a) W. When a roll off rate (a) is small, although the amplitude in the part of the tail becomes large, the occupied frequency band narrows. When a roll off rate (a) is large, although the amplitude in the part of the tail is small, the occupied frequency band extends. Usually it is used 0.1.. ..<.. a<0.8. Occupied bandwidth

F(f)

-1

-0.5

F(f)

F(f)

1

1

1

0.8

0.8

0.8

0.8

0.6

0.6

0.6

0.6

0.4

0.4

0.4

0.4

0.2 -1.5

F(f)

1

0.2

f 0.5

a = 0.1

1

1.5

-1.5

-1

-0.5

0.2

f 0.5

a = 0.3

1

1.5

-1.5

-1

-0.5

0.5

a = 0.5

1

f

1.5

0.2 -1.5

-1

-0.5

f 0.5

a = 0.8

1

1.5

79

Training material of NHK-CTI

IFT of Nyquist pulse Training material of NHK-CTI

1

1

0.75

-10

0.75

0.5

0.5

0.25

0.25

-5

5

10

-10

-5

5

10

a=0.1

a=0.0

Waveform of Nyquist pulse 1

1

1

0.75

0.8

0.8

0.6

0.6

0.5

0.4

0.4

0.25 -10

-5

5

a=0.3

0.2

0.2 -10 10

-5

5

a=0.5

10 -10

-5

5

10

a=0.8 80

Training material of NHK-CTI

Frequency masking SPL Sound Pressure Level (in dB)

Training material of NHK-CTI

60

A (threshold in quiet) B (masking threshold)

40

C (masker)

D (masked sound)

20

0 0.02

0.05

0.1

0.2

1 2 0.5 Frequency(kHz)

5

10

A

Normal threshold of hearing

B

Modified threshold due to tone C

D

Band of noise rendered inaudible by the presence of tone C

20

81

Training material of NHK-CTI

Temporal masking Training material of NHK-CTI

SPL (in db) Masker on Pre-masking Simultaneous Masking

Post-masking

Time ～20ms

～200ms

～150ms

82

Training material of NHK-CTI

Scrambler Training material of NHK-CTI

 Even if data 1 and data 0 is consecutive in NRZ code, it is possible to regenerate clock easily due to scrambler
Scrambler/Descrambler block diagram Exclusive Or (Modulo 2)

Scrambler Pulse sequence

Transmitter side

To Channel

Shift register

Descrambler From channel pulse sequence

Reception side Same as original pulse sequence 83

Training material of NHK-CTI

Maximal--length sequences code generator Maximal Training material of NHK-CTI

• Maximal-length sequences – Longest cyclic code generated by using shift register and feedback

The most random code can be generated (period:2n -1） n: number of shift register Scrambled output

Input data

D

D

D

Clock

84