Source Coding and Parallel Routing
Azadeh Faridi, Anthony Ephremides
TECH2004
Introduction
DDC Results
End-to-End Distortion
strategy #1
SDC:
⎧2
DSDC = ⎨
⎩1
Path 2
Destination
DSDC = 2−2 R Pr [T ≤ ∆ ] + Pr [T > ∆ ]
Path k
{
Problem: Choose the routing parameters together
with the coding parameters to minimize average
end-to-end distortion
Objective
( R*, q*) = arg min DSDC
„
{
DDC:
DDDC
Parallel routing (layer 3): uses diversity but introduces extra delays
Source coding (layer 6): compression
Source: Gaussian, Delay-sensitive: packets that
arrive later than ∆ sec are useless
„
if T > ∆ & T > ∆
1
Π = (1 − d1 )(1 − d 2 )
λo
λ
Queue #2
1-q
Decoder
0
Expected packet length: R bits
„
greater R Æ smaller distortion Æ longer delay
{
DDC:
-40
Network Parameters
„
-5
-10
Source
λ1
-25
Queue #2
Destination
Decoder 2
λ2
100
200
300
400
λ : Arrival rate (bits/sec)
500
600
0.9
Joint
Decoder
q
Other Sources
Problem setup:
Two queues
{
Unlimited buffer size
{
Balanced capacities C1=C2
„
{
0.8
0.7
150
200
250
300
350
λ : Arrival rate (bits/sec)
400
450
500
12
0.4
0.3
0.2
optimal routing helps save one
of two queues instead loosing
both
{
0.1
0
100
200
300
400
λ : Arrival rate (bits/sec)
500
600
6.5
DDC: more flexibility without adding extra traffic load
6
Optimal rate R* (bits/symbol)
If R1 + R2 = R , both descriptions can jointly carry as much
info as an SDC packet
„
{
R = R1+ R2
{
λ1 = λ2= λ
„
{
8
Observations:
6
2
0
50
100
150
200
250
300
350
λ : Arrival rate (bits/sec)
400
450
500
{
δ1* = δ2*
{
q*= 0, q*= 0
{
R* decreases with arrival rate λ
{
2 strategies for R-optimal DDC
0.9
strategy #1: long packets Æ high
delay probability (both packets less
likely to arrive) Æ low encoder
redundancy
0.8
0.7
0.6
0.5
strategy #2: shorter packets Æ low
delay probability (both packets likely
to arrive) Æ high redundancy
0.4
0.3
0.2
R1*= R2*
0
50
100
150
200
250
300
350
λ : Arrival rate (bits/sec)
400
450
500
DDC outperforms SDC under similar
conditions
{
2.5
2
1.5
SDC with two queues and optimal R
SDC with single queue and optimal R
DDC with two queues and optimal R
DDC with two queues and R=6 from [ASEF]
SDC with two queues and R=6 from [ASEF]
1
0.5
0
0
50
100
150
200
250
300
350
λ : Arrival rate (bits/sec)
400
450
500
[ASEF] M. Alasti, K. Sayrafian-Pour, A. Ephremides, N. Farvardin, “Multiple
Description Coding in Networks with Congestion Problem,” IEEE Transaction on
Information Theory, Vol. 47, No.3, March 2001
Concluding Remarks
Fixed R case:
One queue carries as much
traffic it can handle. The rest of
traffic is dumped to the other
queue.
0.5
Each source symbol encoded into 2 packets
Each DDC description carries less info than SDC packet
Exponential packet length
10
Observations:
{
0.6
0
expected description lengths: R1 , R2 < R bits
Balanced capacities C1= C2 =1000 bits/s
{
-30
0
Queue #1
1-q
100
-20
1
Decoder 1
Other Sources
Encoder 2
q
Demux
Encoder 1
50
1
{
-15
C1
q , γ =
C1 + C 2
R1
, α =
R1 + R 2
{
-50
-35
Coding Parameters
δ1 , δ 2
„
Unlimited buffer size
4
SDC Results
Destination
„
„
−2( R1 + R2 )
q
Average end-to-end distortion (dB)
Encoder
(R)
λs
{
-30
x 10
3
Each source symbol encoded into one packet (description)
„
Λ = d1d 2 − 2
( R1 , R2 , q ,δ1 ,δ 2 )
„
„
&
2
Queue #1
Optimal queuing probability q*
Source
if T 1 > ∆ & T 2 ≤ ∆
( R1*, R2 *, q*, δ1*, δ 2 *) = arg min DDDC
SDC:
-20
14
2
δi : redundancy of encoder i
Encoder:
Other Sources
Two queues
16
if T ≤ ∆ & T > ∆
1
small δ : good individual descriptions that jointly contribute little
large δ : not individually good descriptions but jointly can achieve
same distortion as SDC
{ Objective
Model
{
if T 1 ≤ ∆ & T 2 ≤ ∆
Problem setup:
{
0
1
Π − Λ ]+ ) 2
DDDC = d 0 P ⎣⎡T 1 ≤ ∆, T 2 ≤ ∆ ⎦⎤ + d1 P ⎣⎡T 1 ≤ ∆, T 2 > ∆ ⎦⎤ + d 2 P ⎣⎡T 1 > ∆, T 2 ≤ ∆ ⎦⎤ + P ⎣⎡T 1 > ∆, T 2 > ∆ ⎦⎤
want less distortion Î need longer packets Î more delay
{
„
−2( R1 + R2 )
⎧
1− ([
⎪d0 = 2
⎪⎪
−2 R1 (1−δ1 )
= ⎨ d1 = 2
⎪ d = 2−2 R2 (1−δ 2 )
⎪ 2
⎩⎪1
0 ≤ δ1 , δ 2 ≤ 1 ,
Trade-off: Distortion vs. Delay
{
„
-10
-60
„
Motivation: Coupling of different network layers
{
„
( R ,q )
Optimal Pr [ T>∆ ]
„
if T ≤ ∆
if T > ∆
Optimal total rate R* (bits)
Decoder
Optimal redundancy δ∗
Encoder
strategy #2
0
Average end-to-end distortion (dB)
„
Path 1
Source
−2 R
{
Optimal R case:
5.5
5
{
q* = 0.5
{
R* decreases with λ
4.5
4
3.5
3
SDC with 2 queues, R=6, q* [ASEF]
SDC with 2 queues, R* , q*
2.5
2
0
100
200
300
400
λ : Arrival rate (bits/sec)
500
600
„ Interesting Observation:
Known queuing theory result: splitting the capacity between 2 links
can only increase the delay and in our case increase the distortion.
{
Known information theoretic result: MDC does not reduce the
distortion compared to SDC.
{
Our result: For a delay-sensitive source combining parallel routing
and MDC in an optimal way can result in less distortion than an
optimal SDC system with a single queue.
{
„ For a general memory-less source, explicit inner and outer bounds for
the multiple description rate-distortion region have been found by Zamir.
These bounds maintain the form of the distortion functions we discussed
and so we expect our analysis to be applicable for any memory-less
source.