problem and that of packet transmission over erasure channels. The optimal rate ..... if we apply an erasure code across multiple fading blocks, in which case the ...
On the Duality of Layered Transmission for Fading and Packet Erasure Channels Farzad Etemadi, Hamid Jafarkhani Center for Pervasive Communications and Computing Electrical Engineering and Computer Science Department University of California, Irvine
[fetemadi,hamidj]@uci.edu Abstract We consider layered transmission of a successively
fading channels is the pair-wise error probability [10]. One
renable source over a quasi-static fading channel, where the goal
can optimize other design criteria if additional degrees of
is to minimize the expected end-to-end distortion of the received signal. The existing literature on this problem has been limited to either Gaussian sources or design metrics that are valid only for
freedom are available. Maximizing the average throughput using transmit power and rate allocation is an example of
asymptotically high signal to noise ratios (SNR). In this paper, we
such an optimization [11], [12]. The relevant design metric for
explicitly solve the rate allocation problem for an arbitrary source
multimedia applications is the distortion of the received signal
and a nite SNR. We establish a duality relationship between this
and optimizing the latter cost function for fading channels
problem and that of packet transmission over erasure channels. The optimal rate allocation problem is then solved using the dual solution for the erasure channel. Numerical results for a
has been a recent research area [14][21]. Motivated by the concept of diversity and its trade-off with transmission rate in
MIMO Rayleigh fading channel and a practical image coder are
MIMO systems [13], Laneman et. al [14] introduced distortion
presented.
exponent as a means to describe the high signal to noise ratio I. I NTRODUCTION
Shannon's separation theorem states that one can trans-
(SNR) behavior of the optimal expected distortion. Various transmission strategies have been evaluated based on their distortion exponents [14][18].
mit an analog source over a noisy channel by performing
The distortion exponent is a useful tool for comparing
the source and channel coding tasks separately. The source-
various transmission strategies. It does not, however, com-
channel separation assumes large block lengths over which
pletely characterize a given system since the scope of its
the channel variations are averaged. The latter assumption is
denition is limited to the high-SNR regime only. For a given
not realistic in the case of multimedia transmission over a
system and a nite SNR, one still has to solve the associated
slowly fading channel. The delay requirement of the system
distortion minimization problem to fully characterize the op-
may limit the coding interval to a single block of fading. In
timal solution. Another limitation of the existing work is the
the absence of channel state information at the transmitter
Gaussian assumption on the source, which does not represent
(CSIT), there is always a non-zero probability of outage and
an accurate model in multimedia applications.
the Shannon capacity of such a channel is zero. The overall
In this paper, we consider a layered transmission strategy in
system performance then has to be optimized by the joint
which the transmission block is partitioned, and each partition
design of the source and channel coders.
is transmitted at a different rate. The distortion exponent of the
Successive renement is a rate-distortion trade-off mech-
latter scheme was derived in [15], and a rate allocation strategy
anism that allows one to jointly optimize the rate alloca-
was derived in [16] based on maximizing the distortion expo-
tion between the source and channel coders [1]. Practical
nent at an asymptotically high SNR. In the nite-SNR regime,
implementations of successively renable source coders in-
an efcient algorithm was proposed in [21] to nd the optimal
clude SPIHT [2] and EBCOT (JPEG2000) [3] for still image
rate allocation, as well as the optimal partitioning when the
coding, and MPEG-4 FGS [4] for video coding. The joint
source is Gaussian. In the current paper, however, we solve
source-channel coding problem for packet transmission of
the associated optimization problem for non-Gaussian sources
successively renable sources has been widely studied in the
that represent practical source coders. We show that this
literature [5][9] (and the references therein). This body of
information theoretical, joint source-channel coding problem is
work is primarily focused on the design of source-channel
equivalent to the practical problem of packetized transmission
coding schemes for packet transmission over binary symmetric
mentioned earlier. The duality relationship between the two
channel (BSC), erasure channel, or nite-state channels.
problems is then used to solve the non-Gaussian rate allocation
Recent demand for wireless communications has motivated
problem using the well-known optimization techniques of
the researchers to develop advanced coding and modulation
packet transmission over erasure channels. Numerical results
techniques for such a channel. The classical design metric for
for a MIMO Rayleigh fading channel and a practical image
data transmission over multiple-input multiple-output (MIMO)
coder are also presented.
II. P ROBLEM F ORMULATION We consider a successively renable source with a mean square error (MSE) distortion measure. The distortion-rate (DR) function of the source is denoted by
D(R),
where
R
is
the source coding rate in bits-per-symbol. For a non-ergodic source, such as an image, we assume that
D(R)
Fig. 1.
Layered source (LS) coding with progressive transmission over a
quasi-static fading channel
represents
the operational D-R function of the source. It is assumed that a source realization of
K
symbols is to be encoded and
transmitted over the channel. In what follows, we dene the expected distortion cost functions for the fading and packet erasure channels. These denitions will then be used in Sec. III
An alternative representation of (2) can be found using the complement of the outage probability,
1 − PR . P¯Rn
the
Substituting this latter quantity in (2) and factoring terms, it can be shown that
to establish a duality relationship between the two problems.
ED (R, α) = D0 +
A. Quasi-Static Fading Channel
transmit and
NR
receive antennas. The channel fading
coefcients are known to the receiver but not to the transmitter, and are assumed to be constant over a block of
M
N X
P¯Rn (Dn − Dn−1 )
(3)
n=1
We rst consider a quasi-static fading MIMO channel with
NT
P¯R , Prob{C ≥ R} =
In other words, recovering layer
n
reduces the distortion by
Dn−1 − Dn .
channel
symbols. To satisfy the delay constraints, coding is assumed
B. Packet Erasure Channel
to be performed over a single block of fading. The block We rst consider a packet transmission system with
length M is assumed to be large enough such that any rate below the instantaneous channel capacity can be transmitted reliably. The
K
source symbols are mapped into a single
fading block, resulting in a bandwidth expansion factor of
b= M K . The transmission block is divided into N partitions, as shown in Fig. 1. Partition i contains αi M channel symbols PN ( i=1 αi = 1, 0 ≤ αi ≤ 1), and is transmitted at rate Ri bits-per-symbol. This transmission scheme is referred to as layered source (LS) coding with progressive transmission [15]. Since the source is successively renable, layer decoded before layer we assume that
Rn
n
has to be
n + 1 can be decoded. To ensure this, ≤ Rn+1 . The outage capacity is used
as a measure of successful decoding when transmission over a single block is of concern. Outage probability at a given transmission rate R and SN R is dened as Pout (R, SN R) = Prob{C < R}, where C is the instantaneous channel capacity. In what follows, we drop the dependence on the SN R to simplify the notation, and formulate the equations at a given
system. The total transmission budget is xed and is equal
BT = P L
to
the packets, where
mi source symbols and fi = P − mi Ne denote the number of packet erasures. with f parity symbols can recover at most
An RS codeword
f
erasures [22]. Consequently, the probability of successful
t data symbols is given by Qt , Prob{P − t ≥ Ne }. Unequal protection is provided by assuming that fi ≥ fi+1 for 1 ≤ i ≤ L − 1. The probability of recovering the rst i codewords is then Prob{fi+1 < Ne ≤ fi } = Qmi − Qmi+1 , and the associated distortion Pi s 0 1 0 is Di , D(b j=1 mj ), where b = K . The expected decoding of an RS codeword with
distortion is given by
ED (m) =
L X
where
σ2
Qm0 , 1, QmL+1 , 0,
ED (R, α) = PR0 , 0
and
and
m = (m1 , · · · , mL ).
The
L X
Qmi (Di − Di−1 )
(5)
i=1 We will now extend the FLP system to a variable-length
(PRn+1 − PRn ) Dn
(2)
n=0 where
ED (m) = D0 +
is the variance of the source. The expected distortion
N X
(4)
alternative representation of (4) is given as
D0 , σ 2 ,
is given by
(Qmi − Qmi+1 ) Di
i=0
k=1 since the source is successively renable. Note that
across
is the symbol size in bits. Each RS
codeword consists of
where
(1)
s
GF (2s )
parity symbols. Let
associated distortion is
αk Rk )
symbols. Packet erasure protection is provided
by applying a Reed-Solomon (RS) code over
SN R. The outage probability of a layer with rate r is denoted Pr . The probability of successful decoding of the rst n layers is Prob{Rn ≤ C < Rn+1 } = PRn+1 − PRn , and the n X
P
symbols as shown in Fig. 2(a). This
scheme is referred to as the xed-length packetized (FLP)
by
Dn , D(b
L
packets of length
PRN +1 , 1.
codeword now has length satised through
In the above equation, rate
and partition vectors are dened as R = (R1 , · · · , RN ) and α = (α1 , · · · , αN −1 ), respectively. Note that αN = 1 − PN −1 i=1 αi is excluded from the list of independent variables.
ith RS P , and the same overall budget is i PL P = B . We note that some packets T i=1 i
packetized (VLP) scheme shown in Fig. 2(b). The
may not include symbols from all the codewords, and as 1
The distinction between
clear from the context.
Di
for the fading and erasure channels should be
TABLE I D UAL
PARAMETERS OF THE FADING AND ERASURE CHANNELS
Fading
Erasure
LS
LSE
VLP
FLP
C N Ri αi P¯Ri b
C N Ri
Ne L ri Pi Qri ,Pi b0
Ne L mi P Qmi b0
1 N
P¯Ri b N
a one-to-one correspondence between the parameters of the two systems, as shown in Table I. An LS system with equal partitions is denoted as LSE, and is easily seen to be the dual of the FLP system. We note that an outage can be logically seen as an erasure if we apply an erasure code across multiple fading blocks, in which case the erasure probability is the same as the outage probability [20]. What we have achieved here, however, Fig. 2.
Packet transmission with (a) xed-length packets (FLP), and (b)
variable-length packets (VLP). Shaded regions represent a single packet.
is to go beyond traditional erasure coding and to show the equivalence of an outage-based transmission strategy for fading channels (LS) to a packet-based transmission scheme for
a result, packets can have different lengths. The expected distortion for the VLP system is given by
ED (r, P ) = D0 +
L X
Qri ,Pi (Di − Di−1 )
Di , D(b0
(6)
i X
above
theoretical bound on the achievable performance. The VLP performance can be practically realized. A manifestation of
rj Pj )
(7)
P = (P1 , · · · , PL ), r = i (r1 , · · · , rL ), where ri = m Pi is the channel coding rate th of the i codeword. Qri ,Pi = Prob{(1 − ri )Pi ≥ Ne } is the probability of successful decoding of codeword i. It is assumed that fi = (1 − ri )Pi ≥ (1 − ri+1 )Pi+1 = fi+1 . the
in terms of the outage capacity, and as such, provides a problem, on the other hand, is a practical coding scheme whose
j=1 In
have fundamental differences. First, the underlying channels are clearly different. Second, the LS scheme is modeled
i=1 where
erasure channels (VLP). We note that the latter two problems
equation,
III. D UALITY
this latter distinction is the fact that
P¯Ri
is independent of
αi ,
since the capacity does not depend on the block length. The
Qri ,Pi
quantity, on the other hand, depends on
Pi ,
since the
performance of a practical code depends on its block length. Finally, the VLP system has integer variables, and as a result, the solution to its optimization problem is exact, and has a worst case complexity that is only a function of
L and P . The
LS problem has continuous variables, its numerical solution is approximate, and the desired numerical accuracy affects the optimization complexity.
A maximum distance separable (MDS) erasure decoder,
We will now show the application of duality in optimizing
such as an RS decoder, can fully recover the codeword as
the cost functions of Sec. II. An efcient algorithm for the
long as the number of erasures is not greater than the number
fading rate allocation problem was derived in [21] for the case
of parity symbols. As such, an MDS erasure decoder either
of a Gaussian source. In the general case when the source
decodes correctly, or fails to decode. Similarly, if the trans-
is non-Gaussian, we note that the equal partition problem
mission rate over a quasi-static fading channel is less than the
(LSE) is the dual of the xed-length packet (FLP) transmission
outage capacity, then reliable transmission (in an information
problem. In [9], an optimal algorithm, known as Algorithm A,
theoretical sense) is possible. Layered transmission over fading
for the FLP problem with an arbitrary source was proposed.
and packet erasure channels exhibit a similar threshold effect
Using Table I, the latter algorithm can be easily translated to
and as a result, it is reasonable to expect a similar formulation
the LSE domain. This allows us to nd the optimal solution
for the corresponding cost functions. An inspection of (3) and
to the LSE problem with a complexity of
(6), as well as (1) and (7), reveals that the two problems are
R
in fact, identical. In the LS scheme, the instantaneous channel
of non-equal partitions with non-Gaussian sources cannot be
capacity,
C,
is the search space for a rate variable
O(N 2 |R|2 ), where Ri . The general case
is a random index of the channel that determines
solved using algorithm A, since the latter algorithm is only
the number of decoded layers. In the VLP system, the number
applicable to the xed-length packet problem, and its equal-
of the erasures,
Ne ,
plays a similar role. We can establish
partition dual. The practical value of solving the unequally
Rayleigh Fading Channel, 1x1, b=1
partitioned scenario depends on the amount of gain provided
50 N=1 N=2 N=5 N=10
by using unequal partitions. For instance, it is known that for Gaussian sources and sufciently large
N,
using equal
45
partitions is nearly optimal [16], [21], suggesting that the 40 Expected PSNR (dB)
equally partitioned system may indeed be near-optimal for more general sources as well. In the context of packet transmission, duality can be exploited to reduce the optimization complexity when the source is Gaussian. Algorithm
Ropt
35
30
of [21] solves the rate allocation
problem of the fading channel with linear complexity, and 25
can be applied to the xed length packet (FLP) problem. The known solutions for the packet erasure channel are either
20 −5
optimal with quadratic complexity [9], or sub-optimal with
0
5
10
15
20
25
30
SNR (dB)
linear complexity [7]. Using duality, we have shown that optimality and linear complexity can be achieved at the same time if the source is Gaussian.
Fig. 3.
Expected PSNR of the Lena image for a 1x1 system with different
number of layers
Rayleigh Fading Channel, 2x1, b=1 50
IV. N UMERICAL R ESULTS
N=1 N=2 N=5 N=10
In this section, we present the numerical results for the case
45
of a Rayleigh fading channel. The outage probabilities for the single-input single-output (SISO), single-input multiple-output
Expected PSNR (dB)
40
(SIMO), and multiple-input single-output (MISO) cases were analytically calculated [10]. The outage probability for the MIMO case can be obtained through off-line simulations. A
K = 512 × 512
gray scale Lena image is encoded using the
35
30
SPIHT source coder [2], and Algorithm A is used to optimize the LSE system. The expected peak signal to noise ratio
25
2552 ED , is used as the performance measure. The results for 1x1 and 2x1 systems are shown in (PSNR), dened as
10 log10
20 −5
0
5
10
15
20
SNR (dB)
in Figs. 3 and 4, respectively. It is seen from the gures that layered transmission provides signicant gains, and that
Fig. 4.
most of the gain can be achieved using a small number of
number of layers
layers (N
= 5
Expected PSNR of the Lena image for a 2x1 system with different
in Figs. 3 and 4). We note that unlike the
Gaussian case [21], a signicant improvement (about 2dB) can be achieved by going from
N =2
layers to
N =5
layers.
This is because an image is a highly correlated source and
the joint source-channel coding problem for packet erasure channels can be signicantly reduced if the source is Gaussian.
can be compressed much more efciently than a source with
R EFERENCES
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