Post on 18-Dec-2021
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
2
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.
3
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.
4
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)
5
Delta Modulation (DM)
6
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)
8
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:
9
( ) ( ) ( )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
13
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)
17
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)
18
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)
19
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)
21
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)
24
Δ
Sol.
Example1
25
Δ
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Example 1
27
Example 2