43
1 New Improvements on Rate-Distortion Peformance of DPCM Using Multi-Rate Processing Anna N. Kim, Tor A. Ramstad
1
New Improvements on Rate-Distortion Peformance of DPCM Using Multi-Rate Processing
Anna N. Kim, Tor A. Ramstad
2
The Classic DPCM Structure
Uniform Quanitzer
Linear Predictor
+
+
)(nx
)(ˆ nx
)(ne )(neq
)(nz
-
3
Rate Distortion Performance of DPCM
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.5
1
1.5
2
Normalised Distortion
Rat
e in
Bits
/Sam
ple
DPCMR(D)
4
Previous Work
Farvardin & Modistino (1985) Øien & Ramstad (2001) Guleryuz & Orchard (2001)
5
Øien & Ramstad: DPCM and The Wiener Filter
UQ EC
P
P1
1WED+
+
)(nx )(ˆ nx
-
jeP )(
2)(
)()(
qxx
xx
S
SW
6
Øien & Ramstad: Simulation Results
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.5
1
1.5
2
Normalised Distortion
Rat
e in
Bits
/Sam
ple
DPCMR(D)Ø&R
7
Guleryuz & Orchard: System Structure
UQ EC
C
P1
1L2ED+
+
)(nx )(ˆ nx
1-P L1
jeP )(
8
Guleryuz & Orchard: Simulation Results
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.5
1
1.5
2
Normalised Distortion
Rat
e in
Bits
/Sam
ple
DPCMR(D)Ø&R O&M
9
Rate-Distortion Theory
Discrete Time Discrete Amplitude Source
Discrete Time Continuous Amplitude Source
Source With Memory
10
The Water-filling Principle
11
Rate-Distortion Bound of AR(1) Process
deS
Rj
xx
2log2
1,0max
2
1)(
deSD j
xx
,min2
1)(
12
Optimal Mapping
H +
)(nx )(ˆ nx
)(n
13
Approximations & Assumptions
No correlation between quantizer input and quantization noise
No correlation between samples of quantizer output
Quantization noise variance Entropy of quantizer output as bit rate
12
22 q
14
Proposed Codec (a)
UQ EC
P
P1
1WED+
+
)(nx )(ˆ nx
-
jeP )(
2)(
)()(
qxx
xx
S
SW
L L
15
Simulation Results: (a)
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.5
1
1.5
2
Normalised Distortion
Rat
e in
Bits
/Sam
ple
L+W R(D)DPCM
16
Proposed Codec (b)
UQ EC
P
P1
1WED+
+
)(nx )(ˆ nx
-
2)(
)()(
qxx
xx
S
SW
r rL L
jreP )(
17
Simulation Results: (b)
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.5
1
1.5
2
Normalised Distortion
Rat
e in
Bits
/Sam
ple
L+W R(D) DPCM Multi
18
Discussions
Source Spectral Shaping Bit Rate Reduction Prediction Coefficient Effects of Wiener Filter Multi-Rate Processing
19
Shaping of Source Signal Spectra
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
Normalised Frequency
Pow
er S
pect
ral D
ensi
ty in
[dB
]
Low-passed Source Quantization Noise Level
20
Bit Rate Reduction
Bit rate after down-sampling:
Additional bits for maintaining distortion level
Total bit rate reduction
RateSampingDownSampleCoded
Bits
SampleSource
Bits 1Ra :
Ra - Rb
Rb
21
The Linear Predictor
In classic DPCM
With down-sampling rate r
)()1()( nznxnx
)(~)()( nzrnxnx r
22
Effect of Prediction Coefficient
0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.260.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Normalised Frequency
Rat
e in
Bits
/Sam
ple
Down Sampling Rate: 4
Modified Prediction CoefficientOriginal Prediction Coefficient
23
Effects of Wiener Filter
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
0.5
1
1.5
2
2.5
3
Normalised Frequency
Rat
e in
Bits
/Sam
ple
Down Sampling Rate: 2
With Wiener Filter Without Wiener Filter
24
Multi-Rate Processing
Integer Sampling Rate Alteration
Fractional Sampling Rate Alteration
Hd(z) M Hu(z)L
L H(z) M
25
Conclusions
Rate Distortion Motivated Set-up Simple Configuration Superior Performance Robust System
26
Future Work
Non-monotonically decreasing spectrum Non-linear mapping Application in low bit rate image coding
27
28
29
30
PSD of The Source Signal (b)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-15
-10
-5
0
5
10
15
20
Normalised Frequency
Pow
er S
pect
ral D
ensi
ty in
[dB
]
31
PSD of the Low Passed Source (b)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
Normalised Frequency
Pow
er S
pect
ral D
ensi
ty in
[dB
]
32
PSD of Downsampled Signal
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-20
-15
-10
-5
0
5
10
15
Normalised Frequency
Pow
er S
pect
ral D
ensi
ty in
[dB
]
33
PSD of Quantizer Input (b)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
Normalised Frequency
Pow
er S
pect
ral D
ensi
ty in
[dB
]
34
PSD of Decoded Signal
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-15
-10
-5
0
5
10
15
Normalised Frequency
Pow
er S
pect
ral D
ensi
ty in
[dB
]
Decoded Signal Quantization Noise
35
PSD of Upsampled Signal
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-15
-10
-5
0
5
10
Normalised Frequency
Pow
er S
pect
ral D
ensi
ty in
[dB
]
Upsampled Signal Quantization Noise
36
PSD of Interpolated Signal
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-70
-60
-50
-40
-30
-20
-10
0
10
20
Normalised Frequency
Pow
er S
pect
ral D
ensi
ty in
[dB
]
Interpolated SignalQuantization Noise
37
PSD of Wiener Filtered Signal
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
Normalised Frequency
Pow
er S
pect
ral D
ensi
ty in
[dB
]
Wiener Filtered SignalWiener Filtered Noise Interpolated Signal Interpolated Noise
38
PSD of The Source Signal (a)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-15
-10
-5
0
5
10
15
20
Normalised Frequency
Pow
er S
pect
ral D
ensi
ty in
[dB
]
39
PSD of the Low Passed Source (a)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
Normalised Frequency
Pow
er S
pect
ral D
ensi
ty in
[dB
]
40
PSD of Quantizer Input (a)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-14
-12
-10
-8
-6
-4
-2
Normalised Frequency
Pow
er S
pect
ral D
ensi
ty in
[dB
]
41
PSD of Lowpassed Signal
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-60
-50
-40
-30
-20
-10
0
10
20
Normalised Frequency
Pow
er S
pect
ral D
ensi
ty in
[dB
]
Lowpassed SignalLowpassed Noise
42
PSD of Wiener Filtered Signal
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-70
-60
-50
-40
-30
-20
-10
0
10
20
Normalised Frequency
Pow
er S
pect
ral D
ensi
ty in
[dB
]
Wiener Filtered SignalLowpassed Signal Lowpassed Noise Wiener Filtered Noise
43
PSD of Decoded Signal
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-15
-10
-5
0
5
10
15
20
Normalised Frequency
Pow
er S
pect
ral D
ensi
ty in
[dB
]
Decoded Signal Quantization Noise
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