delta modulation (dm)

Faculty of ENGINEERINGC H U L A L O N G K O R N U n i v e r s i t y

2102371 Principles of Communications

Delta Modulation (DM)(based on communication systems by Simon Haykin & Michael Moher)

Lecture 1:

October 27, 2020

Students understand about the DM signal.

Students get to know about the Time Division Multiplexing (TDM).

Objectives

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PCM

PCM quantizes and encode the “actual” amplitude of the

signal.

Voice or video signals are found to exhibit a high

correlation between adjacent samples. (i.e. the signal

does not change rapidly from one sample to the next.)

The encoding signal contains redundant information.

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DPCM

Differential PCM (DPCM) quantizes and encode the

“difference / change” of two adjacent samples.

The difference between adjacent samples has a smaller

variance than the variance of the signal.

Thus, the number of bits for encoding the signal can be

reduced.

Hence, lower bit rate is obtained.

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DM is the simplest form of Differential PCM (DPCM).(1-bit version of DPCM)

Advantages: lower complexity for circuits at both

transmitters and receivers.

Delta Modulation (DM, DM)

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Delta Modulation (DM)

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S Quantizer Encoder

S

Delay

( )m t

ˆ ( )m t

−

+( )e t ( )qe t

Output

Accumulator

( )qm t

++

Fig.1 (a) The transmitter of a DM system.

The delay time is

chosen to be Ts, the

sampling period.

Delta Modulation (DM)

The error signal, which is the difference between the

present sample value of the input signal and its

approximation is obtained as

The quantized error is given by

and is encoded as the DM signal / waveform.7

ˆ( ) ( ) ( )e t m t m t (1.1)

( ) ( )q s

m t m t T (1.2)

( ) sgn ( )qe t e t (2)

Ts : sampling period

step-size

Delta Modulation (DM)

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Input

Output

0( )e t

( )qe t

Fig.1 (b) The input/output characteristics of quantizer of the DM system.

Delta Modulation (DM)

The relationship of signals in the accumulator:

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( ) ( ) ( )q q s qm t m t T e t (3)

Input SDecoder

Delay

+

Output

Accumulator

+

Fig.1 (c) The receiver of a DM system.

10Principle of Communications

(a) Normal case (b) Slope overload case

Fig.2 A DM waveform.

D+

D−t

)(tm

)(ˆ tm

)(~ tm

)(tm )(tm

)(ˆ tm

D+

tD−

0 0

p0(t) p0(t)

Delta Modulation (DM)

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Fig.3 A DM waveform for the ‘Slope Overload’ case.

Quantizing (granular)

noise region

= step size

= pulse width

= sampling period

DM outputSlope overload

Input

Transmitted

sequence 1 1 1 1 1 0 1 0 1 0 1 0

Δ

TΔ

T

signal

region

sT

Delta Modulation (DM)

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Thus, it is required to have a large step-size to

accommodate a wide dynamic range, and a small step-

size for the low-level signals.

To overcome this slope-overload problem, an Adaptive

Delta Modulation (ADM) is necessary.

Delta Modulation (DM)

Differentialamplifier sampler

integrator Variable gainamplifier

Square-law device

R

C

Output signal

out

in

gain-control voltage

-

)(tm

)(ˆ tm

Fig. 4 Adaptive delta modulator (Analog type)

+ )(tD

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Adaptive Delta Modulation (ADM)

lowpass filter

If m(t) is nearly constant, the output signal will alternate in

sign. Thus the dc value is nearly zero. This small value

controls the gain of the variable gain amplifier such that it is

very small under this condition, thus a very small step size.

If m(t) is increased or decreased rapidly, the output signal

will have the same polarity over this period. Thus the

magnitude of the output of the RC lowpass filter will be

relatively large. The result is an increase in the gain of the

variable-gain amplifier and hence the increase of step size.

Adaptive DM

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Fig. 5 Adaptive delta modulator (Digital type)

∑

Integrator

Slope controllogic

N-bit storage

Transmitted bits

Input signal

Samplerand

1-bitquantizer

NB - 1NB NB B- 2 1

Adaptive DM

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Fig.6 Sample Output of Adaptive Delta Modulator

+6

+8

+16

-8

+6

-4

-4

+2 -1-1

-2

Step size = [1, 2, 4, 6, 8, 16]

Transmitted bit pattern111. . . 1 1 0 1 0 0 1 0 0 0 0

Input

signal

ADM output

data Step size

000 Increase

111 Increase

100 Hold

011 Hold

101 Decrease

010 Decrease

110 Decrease

001 Decrease

Adaptive DM

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Quantization Noise

ให้รปูคล่ืนของความผิดพลาด เป็น( )e t

ˆ( ) ( ) ( )e t m t m t (4)

โดยท่ี เป็นต้นก าเนิดของ quantization noise( )e t

Delta Modulation (DM)

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PDF ของ เป็นแบบ uniform distribution ดงันัน้( )e t

22 21{ ( )}

2 3E e t e de (5)

pdf( )e

eD+D−

D2

1

Delta Modulation (DM)

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มีการพิสูจน์มาแล้วว่า Normalized Power ของ มีการกระจายของความน่าจะเป็นแบบ Uniform Distribution ในช่วงความถ่ี โดยท่ี คืออตัราการสุ่มตัวอย่าง ดงันัน้ Spectral density ของจะเป็น

( )e t

],0[ sf sf

2 / 6( )

0s

e

fG f

sff ,

, ท่ีความถ่ีอ่ืน(6)

=

s

sT

f1

พืน้ท่ีใต้กราฟเท่ากบัก าลงัของ

f

sf +sf −

sf

D

6

2( )eG f

2

( )3

e t0

Delta Modulation (DM)

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20

Delta Modulation (DM)

เน่ืองจาก เป็นผลตอบสนองของ lowpass filter ต่อ ท่ีขาออกของเครื่องรบั Normalized Average Power ของ จะเป็น

)(tnq( )e t

)(tnq

2)}({ tnEN qq =

2

3x

qs

fN

f

( )x

x

f

efG f df

(7)

ดงันัน้ fsf +sf −

sf

D

6

2( )eG f

xf+xf− 0

)(tnqพืน้ท่ีใต้กราฟเท่ากบัก าลงัของ

Output Signal Power

ในการค านวณ Output Signal Power สมมติให้เป็นสญัญาณไซน์

tfAtm x2cos)( =

ดงันัน้ Output Signal Power เป็น

2)}({

22

0

AtmES == (8)

Delta Modulation (DM)

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เพ่ือไม่ให้เกิด Slope Overload เราต้องเลือก Maximum Slope ของสญัญาณซ่ึงเท่ากบั

xfAdt

tdm2

)(

max

=

เพื่อท าให้ Sample-to-Sample change ซ่ึงเท่ากบั

sxs TfATdt

tdm=

2

)(

มีค่าน้อยกว่าหรือเท่ากบั step size )(D

)(tm

sT

dt

dmSlope ณ ต าแหน่งน้ี = Sample-to-Sample change

)(D

Delta Modulation (DM)

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เลือกขนาดสงูสดุของสญัญาณก่อนเกิด Slope Overload

x

s

f

fA

D=

2

ดงันัน้

(9)

D=ATf sx2

Delta Modulation (DM)

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ดงันัน้2

2

022

1

2

D==

x

s

f

fAS

และ Signal-to-Quantization Noise ratio เท่ากบั3

02

3SQR

8s

q x

S f

N f(10)

Delta Modulation (DM)

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Δ

Sol.

Example1

25

Δ

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Example 1

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Example 2