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SKKU J.D. Cho1

Lecture on Communication Theory

Chapter 6. Pulse Modulation

6.1 Introduction

CW modulation - AM, DSB-SC,SSB,VSB,FM,PM

Pulse modulation

Analog: discrete in time & continuous in amplitude

Digital : discrete in time & amplitude

6.2 Sampling Process

Time domain Frequency domain

W T s

2

1<

W T s

2

1=

W T s 2

1

>

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1. Notation

- : original analog signal

- : sampling period

- : sampling rate

2. Sampling

Sampling with δ (t)

; ideal sampled signal

where

; discrete Fourier Transform

If

W T s

2

1=

( ) ∑∞

−∞=δ

π−

=

n W

nf j

W

n g f G

2exp

2

( ) ( ) ( )∑∞

≠−∞=

δ −+=

0mm

s s s mf f G f f G f f G

)(t g

sT

s

s

T f

1=

( ) ( ) ( )∑∞

−∞=

−=n

snfT jnT g f G s

π2expδ

)()( f Gt g

⇔

( ) ( ) )(δδ

f GnT f G f f g n

s s∑∞

−∞=

=−⇔ let

( ) ( ) ( )∑∞

−∞=

−=n

snT t nT g f g δδ

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Under following conditions

has all information contained in g(t)

3. Reconstruction

Reconstruction

=> Interpolation formula

where : Interpolation function

( )

=

≥=

W f

W f f G

s 2

0 for

( ) ( ) W f W f GW

f G <<−= δ 21

( ) D.F.T. , ;2

exp22

1W f W

W

nf j

W

n g

W f G

n

<<−

π−

=∴ ∑

∞

−∞=

( )W n g 2∴

W

n g t g

2)( from

( )Wt 2sinc

∑∞

−∞=

=n

dt ft j f Gt g )π2exp()()(

∑

∑ ∫ ∫ ∑

∞

−∞=

∞

−∞=−

−

∞

−∞=

−−

=

−

=

−

=

n

n

W

W

W

W n

nWt

nWt

W

n g

df W

nt f j

W W

n g

df ft jW

nf j

W

n g W

)ππ2(

)ππ2sin(

2

)]2

(π2exp[2

1

2

)π2exp(π

exp22

1

( )∑∞

−∞=

=

n

n-Wt W n g t g sinc So, 22

)(

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4.Sampling Theorem

1) BW=W 로 band-limited signal 은 1/2W seconds 로 샘플링

한

value 로 완전하게 describe 된다 .

2) BW=W 로 band-limited signal 은 1/2W 로 샘플링한 샘플값

으로

완전하게 복원 가능하다

5. Aliasing

Anti - aliasing filter

Aliasing 없애는 방법

Nyquist rate 보다 높게 sampling

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6. Reconstruction filter

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6.3 Pulse-Amplitude Modulation

1. PAM

Instantaneous sampling

1) Sample and hold :Lengthening(T)

- Impulse 대신 lengthening with T 를 사용한 이유

to avoid the use of an excessive channel BW

)( sT

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2. PAM signal

PAM signal

where

Instantaneous sampled version of

==

<<

=

otherwise ,0

,0 ,2

1

0 ,1

)( T t t

T t

t h

)(t m

∑∞

−∞=

δ −δ=n

s s nT t nT mt m )()()(

∫

∞

∞−δ ττ−τ=∗ δ d t hmt ht m )()()()(

∫ ∑∞

∞−

∞

−∞=

ττ−−τδ=n

s s d t hnT nT m )()()(

∫ ∑∞

∞−

∞

∞−

ττ−−τδ= d t hnT nT m s s )()()(

∑∞

−∞=

−=n

s s nT t hnT m )()(

)()()( δ t ht mt s ∗= ,So

)()()( f H f M f S δ=

∑∞−∞=

−=n

s s nT t hnT mt s )()()(

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여기서

3. 로 부터 를 recover 하는 방법

Sample and Hold 의 전 달 함 수

( ) f S

∑∞

−∞=

−=k

s s kf f M f f M )()(δ

)(t m

∑∞

−∞=

−=∴k

s s f H kf f M f f S )()()(

Reconstruction

filter outputEqualizer 특성

Amplitude distortion

)exp(sinc)( fT j(fT)T f H π−=

Delay of T/2

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6.5 Pulse-Position Modulation

1. PDM = Pulse-width mod. = Pulse-length mod.

: Pulse duration ∝2. PPM (Pulse Position Mod)

To make strictly nonoverlapping

)( snT m

∑∞

−∞=

−−=n

s p s nT mk nT t g t s ))(()(

Standard Pulse

2)(

max

s p

T t mk <

Sensitivity of PPM

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3. Generation of PPM waves

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4. Detection of PPM waves

)(t m PAM PDM PPM

sample&Integrator

→→→

↑

Integrator & Sample

elayed Version

sliver output

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5. Noise in PPM

1) If the received pulses were perfectly rectangular

→no effects on pulse position

but infinite channel BW

∴ Finite rise time →affected by noise

Ex1. SNR of a PPM system using sinusoidal mod.

Standard Pulse : raised cosine pulse

where

( )[ ] T t T t B A

t g T ≤≤−+= πcos12

)(

T BT

1=

nV

τ

)/41(π2

)/π2sin()(

22

T

T

B f f

B f A f G

−=

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For sinusoidal message

Peak-to-peak swing : by Ts

Rms value of receiver output

where K : constant

Avg. signal power (with 1Ω )

If noise amplitude = , τ : positive error, : noise

22

s KT =

822

222

s s T K KT =

nV nV

; g(t) 의 slope

; error

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Avg. noise power

Avg. transmitted power P

(Avg. noise power in message BW = W) = WNo

∫ − == 2

2

22

4

3)(

1 S

S

T

T

S T T B

Adt t g

T P

0

222

32

π(

N

AT B S T =∴ oSNR)

24

π

)(

)( 322 W T B

SNR

SNR FOM sT

C

O ==

[ ][ ]

22

0

2

222

2

n

2

222

2

n2

22

π

4

π

VE4

π

4V

τ

A B

N K

A B

K

A B

E K

E K

T T T

==

=

=

[ ] )2

2(V 00

2

n

N B N B E T T ×←=∴

oT S

C WN BT

ASNR)

4

3(

2

=∴

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Assume ; Nyquist sampling rate

FOM > 1 if

3) 결론

– PPM 에서는 FOM 은 크게 하려면 BT 가 커 야 된 다 .

– Noise 가 너 무 크 면 noise 를 message 로 오인

⇒Threshold Effect

6.6 Bandwidth - Noise trade off

1. PPM 과 FM 의 유 사 한 점

1) FOM 가 (BT/W)2 에 비례

2) Threshold Effect

2. FOM 와 BT 간의 trade - off

22

199

π

=W

B FOM T

W T s

2

1=

W BT 41.4>

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6.7 Quantization Process

1. Memoryless Quantizer

1) Quantizer

Signal amplitude

k = 1,2,…..,L

mk : decision levels or decision thresholds

vk : representation levels or reconstruction levels

∆ = mk - m

k-1: quantum or step - size

Figure 6.16 Description of a memoryless quantizer

1: +≤<→ k k mmmk m if index

ϕ k

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2)

3) Qunatization 하는 이유 : human sense (ear or eye )

can detect only finite intensity differences

Uniform Quantizer

Non-uniform Quantizer

Midthread

Midrise

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2. Quantization Noise

1) Q = M - V ; quantization Error ≈ thermal noise

uniformly distributed r.v.

( )⋅ g r.v. M

(continuous)

r.v. V

(discrete)

vm

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2) p.d.f of Q

where

∴ Variance

3. Bit 할당 및 SNR

R: bits per sample

=)( g f Q∆

1

22

∆≤<

∆− q

0 otherwise

L

mmax2Δ = , signal = [-m

max , m

max]

L = total # of levels

R

R

R

m

m

L R L

222

max

2

23

1Δ

12

1σ

2

2Δ

log,2

max

−==∴

=∴

==

2

Q

( ) [ ]

12

Δ

Δ

1

σ

2

2

Δ

2

Δ

2

2

Δ

2

Δ

222

==

==

∫

∫

−

−

dqq

Q E dqq f q QQ

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Output SNR (neglect receiver noise)

ex) Sinusoidal Modulating Signal

][68.1)(log10 10 dB RSNR o +=

R

Q

om P P 22max

2 23σ

( ==SNR)

)2(2

32

23

2

2

2

22

max

2

R

m

Rm

mm

A

A

Am A

P

==∴

==

O(SNR)

Level Bits SNR

32 5 31.8

64 6 37.8128 7 43.8

256 8 49.8

12 73.8

16 97.8

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6.8 PCM

1. PCM 구조

LPF : for anti-aliasing

2. Sampling : a train of narrow rectangular pulses,

Nyquist sampling theorem 에 의한 sampling

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3. Quantization : Discrete in both time & amplitude

1) Linear Quantizer

2) Nonlinear Quantizer

Voice signal 에서 사용하는 이유 :

loud talk : weak talk = 100 : 1

3) Nonlinear Q = Compressor + Uniform Quantizer

x µ - law

where µ > 0

- µ = 0 →Uniform Quantizer

weak talk →small step-size

loud talk → large step-size

⇒

)μ1log(

)μ1log(

++

=m

v

1 3 7 15 31

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- Reciprocal slope : quantum steps

≈→>>≈→<<

⇒

+×

+=

++

=

m

m

mmd

vd

m

vd

md

1

1μ

μ1

μ

)μ1log(

1

)μ1(

μ

)μ1log(

linear

logarithmic

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y A - law

- A = 1 →Uniform Quantizer

- Practical value of A ⇒A ≅ 100

- Reciprocal slope

z Compander = Compressor + Expander

=v

Am

A

m A 10,

log1≤≤

+

11

,log1

)log(1≤≤

++

m A

A

m A

=vd

md Am

A

A 10,

log1≤≤+

11

,)log1( ≤≤+ m A

m A

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4. Encoding

1) Encoding process : Translate the discrete set of

sample values to a more appropriate form of signal

to make the transmitted signal more robust to noise,

interference and other channel degradations

- Code : particular arrangement of discrete events

- Code element or symbol- Code word or character or alphabet

ex) 8 bit code word 256 levels.

R = 4 bits →16 level

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2) Binary symbols 을 나타내는 line codes.

x On - off signaling 1 →high, 0 →zero.y NRZ signaling 1 →+A

0 →-A

z RZ signaling 1 →half width

0 →no pulse

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Binary RZ 1 →+A,-A alternately

0 →no pulse

- no dc

| Split-phase (Manchester Code)

1 : +A → -A with half width

0 : -A →+A with half width

- no dc

Differential Encoding

1 : no transition

0 : transition

5. Regeneration ( Regenerative Repeater)

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Accumulation of distortion and noise Completely removed

if no error in decision making process.

Regenerated signal 에 error 가 생 길 경 우

Channel noise and interference →bit errors

Timing jitter →distortion →bit errors

6. Decoding →Quantized PAM signal

7. Filtering →LPF with BW = W (message BW)

8. Multiplexing ; TDM

9. Synchronization

- TDM 의 경우 수신기가 송신기에 synchronization- Frame Sync.

Example 3. T1 System

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x bit 193 : framing bit, for sync

y Voice channels

- 8 bit PCM used on five of six frames- 7 bit PCM used on every of six frames.

Bit 8 of each channel is a signaling bit

Voice signal : 300 ~ 3100 Hz

LPF for anti-aliasing = cut off = 3100 Hz

Standard Sampling = 8KHz

Compression law = approximate µ -law with µ =255

Mbps544.1125

1193 =

µ×

64

128

5121024

2048

4096

256

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MSB = 1 if +8bit word MSB = 0 if -

Next 3 bits : segments

LSB 4 bits : amplitude within a segment

6.9 Noise Considerations in PCM Systems

1. Channel noise

- Additive, white, Gaussian

- Bit error

- Error Rate = Average probability of symbol error

2. Quantization Noise : signal dependent

3. Error threshold

PCM using NRZ (Binary PCM)

Assume 105 bps = 100 Kbps

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Note: Thresho ld of visibility

(TOV) has been me asured to

occur at a S/N of 14.9dB w hen

there are 2.5 segme nt errors per

second which is a segment

error rate (SER) of 1.93x10-4.

SER at TOV

S/N db (RM S)

1E+01

1E+00

1E-01

1E-02

1E-03

1E-04

1E-05

1E-06

1E-07

1E-08

8 9 10 11 12 13 14 15

P r o b a b i l i t y

o f E r r o r

16 17 18

Segment error probability, 8 VSB with 4 state trellis, RS

(207,187)

Error threshold (at about 11dB)

≈ 60 - 70 dB required using AM

Regenerative Repeater

: Effects of amplitude, phase, and nonlinear

distortions in one link have practically no effect

on the regenerated input signal to the next link

Cliff Effects

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16 VSB error probability

S/N db (RM S)

1E+01

1E+00

1E-01

1E-02

1E-03

1E-04

1E-05

1E-06

1E-07

1E-08

8 12 16 20 24 28 32 36

16-VSB

Sym bol Error Rate

Segm ent Error

Rate After Reed-

Solom on FEC

P r o b a b i l i t y

o f E r r o r

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6.10 Virtues, Limitations, and Modifications of PCM1. PCM 의 장점

1) Ruggedness to channel noise & interference

2) Efficient regeneration of a coded signal

3) Efficient exchange of increase channel BW for improved SNR

4) Uniform format → integration with other forms of digital datain a common network

5) Comparative ease in TDM system

6) Secure communication ex ) encryption

2. 단점

1) Increased channel BW ←

2) Increased system complexity ←VLSI or Delta modulationwide band : satellite,optic fiber

data compression

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6.11 Delta Modulation

1. DM System

Oversampling → to increase the correlation between adjacent

samples.

Staircase approximation → ±∆

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))1()()](sgn[)(

)()()(

sq sq sq

s sq

s sq s s

nT eT nmnT mnT enT e

T nT mnT mnT e

+−=∆=

−−= ; error signal

; quantized error signal

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2. DM Quantization error

1) Slope overload distortion

2) Granular Noise3) Slope overload distortion 을 없애는 조건

4) Slope overload Granular Noise

5) Adaptive DM: 입력 신호의 크기에 따라 adaptive

하게 결정

dt

t dm

T s

)(max≥

∆

→← Trade off

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3. Delta-Signal Modulation (D -Σ M)

; 수신기의 integrator 을 송신기로 이동

1) DM 의 단점 : 수신기에 error 의 accumulation

2) Delta-Sigma Mod 의 영향

x Low frequency content of input signal is pre-emphasized

y Adjacent sample 간의 correlation 이 증가하여 overall

system performance 증가

z Receiver 가 simple

3) Smoothed version of 1 bit PCM

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6.12 Differential PCM

1. High correlation between adjacent samples

→ Redundant information

2. DPCM System

redundancy in spatial

redundancy in entropy

redundancy in temperal

ex) Video

pitch frequency

temporal redundancy Voice

sq T ne )1(( −

sq T nm )1( −

)( sq nT m

Ts

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Prediction error

Quantizer output

Prediction filter input

)(ˆ)()( s s s nT mnT mnT e −=

)()()( s s s nT qnT enT eq −=

Quantization

error

)()(ˆ)( s s sq nT enT mnT m q+=)()()(ˆ s s s nT qnT enT m

++=)()( s s nT qnT m +=

)( snT mq is independent of )(ˆ snT m

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3. Processing Gain

- Processing Gain ; Predictor 를 사용해서 생기는 Gain

- Optimum adv. of DPCM over PCM = ≈ 4 ~ 11dB

- For constant signal-to quantization noise ratio

DPCM → saving of about 8 - 16kbps over PCM (T s= 8KHz )

where : Variance of prediction error

E

2σ

: signal to quantization noise ratioQSNR )(

Q

M

OSNR 2

2

σ

σ

)( =

QO SNRGSNR p

Q

E

E

M )()(

2

2

2

2

=

σ

σ

σ

σ=

where : Variance of with zero mean)( snT m

: Variance of with zero mean)( snT q

E

M pG

2

2

σ

σ=

QSNR)(

QSNR)(

2σ M

2σ Q

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6.13 Coding Speech at low bit rates. (32, 16, 8, 4 kbps)

1. Design Philosophy

1) Remove redundancies by statistical characteristics

of speech waveforms

2) Properties of hearing →Phychoaccoustic Modeling

2. 오디오 압축의 원리

Psychoacoustic model

- Masking curve 이하는 감지 불능 .

- 큰 소리는 주변의 작은 소리를 Mask 함 .

1KHz sinewave masker

Masking threshold

Threshold in quiet

0.02 0.05 0.1 0.2 0.5 1 2 5 10 20KHz

0

10

20

30

40

50

60

70

frequency

Sound

pressure

lev

el(dB)

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2. Adaptive Differential PCM ; Time-domain codes

1) Adaptive quantization : time-varying step size

where φ : constant

: estimate of

x Adaptive quantization with forward estimation (AQF)

unquantized samples of the input signal

→ forward estimates of

Buffer 필요 : ∆ (nT s) 를 정할 동안 unquantized

sample 저장

TX of level information

Delay in encoding(≈ 16msec) ⇒AQB 를 더 많 이 사 용

y Adaptive quantization with backward estimation (AQB)samples of quantized output

→backward estimates of

Buffer 없음 , delay 없음

)(ˆ)( s s nT nT M σφ=∆

)(ˆ snT M σ )( snT M σ

)( snT M σ

)( snT M σ

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2) Adaptive Prediction

speech signals are inherently nonstationary

→ Autocorrelation function & p.s.d of speech signals are

time-varying

x Adaptive prediction with forward estimation (APF)

y Adaptive prediction with backward estimation (APB)

- Logic for adaptive prediction : LMS algorithm

- 32 kbps 에서 impressive

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3. Adaptive Subband Coding ; Freq-domain code

- 16kbps ≈ 64 kbps PCM

- 16kbps, 8kbps, 4kbps

Quasi - periodic nature of voice signal

pitch frequency →pitch prediction

Noise Masking Phenomenon

동일 band 의 signal 보다 15dB 낮은 noise 는 human ear 가

감지하지 못한다 .

즉format frequency (resonance freq of the vocal track tube)

근처에서는 large coding error 를 허용한다 .

Psychoaccoustic modeling

- Adaptive subband coding

- Complexity of 16kbps ≅ 100 × (that of 64k PCM)

- Processing delay ≈ 25 msec

¯ Voice mail 에는

문제없음

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