Optimal Layered Transmission Over Quasi-Static Fading Channels Farzad Etemadi

Hamid Jafarkhani

Department of EECS University of California Irvine, CA 92697, USA Email: [email protected]

Department of EECS University of California Irvine, CA 92697, USA Email: [email protected]

Abstract— We consider layered transmission of a successively refinable complex Gaussian source over a quasi-static fading channel. For a given number of source coding layers, we propose an efficient algorithm to calculate the optimal rate assignment for each layer, as well as the optimal size of each layer. The optimality of the algorithm is proved and numerical results for a multiple antenna Rayleigh fading channel are presented. It is numerically shown that a small number of layers is usually sufficient to achieve most of the layering gain.

I. I NTRODUCTION A successively refinable source provides a rate-distortion trade-off mechanism for lossy transmission of the source over a noisy channel. In situations where the Shannon’s sourcechannel separation theorem does not hold, successive refinement allows one to jointly optimize the rate allocation between the source and channel coding bits in order to minimize the expected distortion of the received signal. A quasi-static block fading channel is an example of such a scenario. If delay constraints limit the coding interval to a single block and no channel state information is available to the transmitter, there is always a non-zero probability of outage and the Shannon capacity of such a channel is zero. The expected received distortion thus have to be minimized by joint source-channel coding. From a practical point of view, joint source-channel coding of successively refinable sources has been widely studied in the literature [1], [2], [3], [4] (and references therein). Various optimal and near-optimal rate allocation strategies have been proposed with an emphasis on the transmission of progressively encoded multi-media bitstreams. As such, this body of work is primarily focused on the operational aspects of the problem and not on the information-theoretical ones, such as the theoretically achievable minimum expected distortion. Moreover, the scope of the latter work is limited to simple channels such as the binary symmetric channel (BSC), erasure channel, or finite-state channels for which well-known coding strategies exist. Recent advances in coding for multiple-input, multiple output (MIMO) fading channels has motivated the researchers to study the joint source-channel coding problem for such channels [5], [6], [8]. Distortion exponent was defined by Laneman et. al [5] as a means to describe the high signal to

noise ratio (SNR) behavior of the optimal expected distortion. Various transmission strategies have been evaluated based on their distortion exponents [5], [8]. The distortion exponent provides a useful tool for comparing various transmission strategies. It does not, however, completely characterize a given system since the scope of its definition is limited to the high-SNR regime only. Moreover, for a given system and an arbitrary SNR, one still has to solve the associated distortion minimization problem to fully characterize the optimal solution. In this paper, we propose an efficient algorithm to find the optimal rate assignment as well as the optimal size of layers in a layered transmission system. We prove the optimality of the proposed solution and present numerical results for the case of a MIMO Rayleigh fading channel. The structure of this paper is as follows. In Section II, we formulate the optimization problem. The proposed solution and the proof of its optimality are presented in Section III. Numerical results are presented in Section IV. Finally, Section V concludes this paper. II. P ROBLEM F ORMULATION The goal is to transmit a successively refinable complex Gaussian source over a quasi-static fading channel. The distortion-rate (D-R) function of the source is given by D(R) = 2−R , where R is the source coding rate in bitsper-symbol, and mean square error (MSE) distortion measure is assumed. The channel fading coefficients are known to the receiver but not to the transmitter, and are assumed to be constant over a block of M channel symbols. To satisfy the delay constraints, coding is assumed to be limited to a single block. A sequence of K source symbols are mapped into a single block, resulting in a bandwidth expansion factor of b = M K [8]. The transmission block is divided into N partitions,P as shown in Fig. 1. Partition i contains αi M channel N symbols ( i=1 αi = 1, 0 ≤ αi ≤ 1), and is transmitted at rate Ri bits-per-symbol. Since the source is successively refinable, layer n has to be decoded before layer n+1 can be decoded. To ensure this, we assume that Rn ≤ 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 defined as Pout (R, SN R) =

This algorithm can be formalized as the following (superscripts represent the iteration number). Algorithm Dmin Fig. 1.

•

Layered Transmission

•

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 SN R. The outage probability of a layer with rate r is denoted by Pr . The probability of successful decoding of the first n layers is prob{Rn ≤ C < RP n+1 } = PRn+1 − PRn , n and the associated distortion is 2−b( k=1 αk Rk ) . The expected distortion is given by ED (R, α) =

N X

(PRn+1 − PRn ) 2−b(

Pn k=1

αk Rk )

(1)

•

Initialize: Set k = 0, choose an arbitrary partition α0 Optimize: Rk+1

= arg min ED (R, α = αk )

(3)

αk+1

= arg min ED (R = Rk+1 , α)

(4)

R α

Iterate: Set k = k + 1, go to (3)

In each iteration, the optimization sub-problems (3) and (4) reduce the expected distortion. As a result, the sequence of non-negative values of ED eventually converges to a local minima. The Dmin algorithm is thus guaranteed to converge to an optimal solution. In what follows, we consider each subproblem separately.

n=0

where PR0 = 0 and PRN +1 = 1. In the above equation, rate and partition vectors are defined 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. We can now define the distortion minimization problem as min ED (R, α) R, α

(2)

0 ≤ Ri ≤ Ri+1 , 0 ≤ αi ≤ 1, i ∈ {1, · · · , N − 1} For very small values of N , this optimization problem can be solved through an exhaustive search [8]. If we limit the search for rates to a discrete set of rates, R, of size |R| and the optimal partition α is given, then the search for optimal R has a worst case complexity of O(|R|N ). Joint search for the optimal α and R is thus infeasible even for moderate values of N . On the other hand, solving this optimization problem of 2N − 1 variables through the well-known nonlinear programming techniques is numerically difficult because of the sensitivity to the choice of initial conditions, high time-complexity, and slow convergence. In what follows, we introduce an efficient minimization algorithm and prove its optimality. We emphasize that the layered transmission scheme discussed in this paper is not the only transmission strategy for a fading channel. Consequently, the solution to the optimization problem (2) does not necessarily represent the best achievable performance among all the transmission alternatives, such as the broadcast strategy [7], [8]. Our goal is to find the optimal solution within the particular framework of Fig. 1. Moreover, any practical coding scheme can only approach the transmission rates defined by the outage capacity. As a result, the solution to (2) is in fact an upper bound on the performance of any practical implementation. III. D ISTORTION M INIMIZATION A LGORITHM We propose an iterative algorithm to solve the optimization problem (2). The basic idea is to alternately optimize the rate and partition vectors and repeat this process until convergence.

A. Optimal Rate Assignment N Denote an N -layer expected distortion by ED (R, α) and let

D1 (β, r) , Pr + (1 − Pr )2−bβr D2 (β, r, t) , (1 − 2−bβr )Pr + 2−bβr t

(5) (6)

The following two lemmas form the basis of our rate assignment algorithm. Lemma 1: The expected distortion is given by 1 ED (R, α) 2 ED (R, α)

= D1 (1, R1 ) = D2 (α1 , R1 , D1 (α2 , R2 )) ···

N ED (R, α)

=

(7) (8)

D2 (α1 , R1 , D2 (α2 , R2 , · · · , D2 (αN −1 , RN −1 , D1 (αN , RN ))

(9)

Proof. We prove the lemma using mathematical induction on the number of layers. For N = 1, α1 = 1 and (7) can be easily seen to be equivalent to (1). Now we assume that (9) holds for N − 1 layers and prove the lemma for N layers. By expanding the first two terms of (1), the expected distortion of an N -layered system can be written as: N ED (R, α)

=

PR1 + (PR2 − PR1 )2−bα1 R1 + N Pn X (PRn+1 − PRn ) 2−b( k=1 αk Rk ) n=2

= =

(1 − 2−bα1 R1 )PR1 + 2−bα1 R1 C D2 (α1 , R1 , C)

(10)

where C = PR2 +

N X

(PRn+1 − PRn ) 2−b(

Pn

k=2

αk Rk )

(11)

n=2

From (9) and (10), we see that the proof is complete if we show that C

= D2 (α2 , R2 , D2 (α3 , R3 , · · · , D2 (αN −1 , RN −1 , D1 (αN , RN ))

(12)

We note that C is independent of R1 and α1 . For k ≥ 1, define Rk0 αk0

, ,

Rk+1 αk+1

(13) (14)

1

and PR00 , 0. Using the latter definitions and re-indexing the sum in (11), C can be written as: C=

N −1 X

0 (PRn+1 − PRn0 ) 2−b(

Pn

k=1

0 α0k Rk )

(15)

n=0

By comparing (1) and (15), we see that C is an (N − 1)layer cost function for layers 2 to N . The inductive assumption on N − 1 layers then implies that C can be expressed recursively as in (12), and this completes the proof. ¤ Lemma 2: For N ≥ 2 and a given α, the optimal value ∗ (R2∗ , · · · , RN ) is given by ∗ ) = arg (R2∗ , · · · , RN

min

R2 , ··· , RN

C

(16)

where C is given by (12). Proof. From (10), we have N ED (R, α) = (1 − 2−bα1 R1 )PR1 + 2−bα1 R1 C

(17)

The first term in (17) is independent of {R2 , · · · , RN }, and from (11) we see that C is independent of R1 and only ∂E N (R,α) depends on {R2 , · · · , RN }. Since D∂C = 2−bα1 R1 > 0, N ED (R, α) is an increasing function of C. To minimize the expected distortion, therefore, {R2 , · · · , RN } should be chosen such that C is minimized. ¤ Lemma 2 implies that layers 2 through N can be optimized independently of layer 1 by minimizing the quantity C. The recursion (12) given in Lemma 1 shows that C, in turn, is a cost function defined over layers 2 through N and as a result, layers 3 through N can be optimized independently of layer 2. Applying this optimization break-down N − 1 times, we conclude that layer N can be optimized independently of ∗ all layers. The optimal RN so found can be held constant, and the two-layer cost function over {RN −1 , RN } can be optimized over RN −1 . Layers N −2 through 1 can be similarly optimized using this backward procedure. We have thus proved the optimality of the following N -step optimization algorithm. Algorithm Ropt ∗ For a given α, the optimal rate assignment (R1∗ , · · · , RN ) is given by: ∗ RN

Rn∗

The Ropt algorithm replaces an N -dimensional search with N single-dimensional searches, and consequently, reduces the optimization complexity from O(|R|N ) to O(N |R|). B. Optimal Partitioning We now consider the problem of finding an optimal partition α, when the rate vector R is given. From (1), we see that each term of the expected distortion is an exponential function of the partition variables αk . As a result, the expected distortion is a convex function of α and sub-problem (4) can be efficiently solved using constrained convex optimization techniques [9]. To simplify this constrained optimization, we prove the following intuitive result. ∗N Lemma 3: The optimal expected distortion ED is a nonincreasing function of N . Proof. Merge layers 1 and 2 and define R10 , R1 = R2 , α10 , α1 + α2 , and Ri0 , Ri+1 , αi0 , αi+1 for 3 ≤ i ≤ N . By imposing the additional constraint R1 = R2 on (2), this optimization problem becomes an N − 1layer problem defined by the R0 and α0 vectors, that can ∗N −1 achieve ED . Removing the latter constraint could only ∗N ≤ improve the expected distortion, and as a result, ED ∗N −1 ED . ¤ We will now show that the constraints 0 ≤ αk ≤ 1 can be safely removed from (4). Let us assume that N > 1 and we solve (4) using an unconstrained minimization technique. Also assume that the initial partition satisfies 0 < αk < 1 for all k. If some αk approaches 1 at any iteration, then the N -layer problem becomes a single-layered one. On the other hand, if αk approaches 0 at any iteration, then the N -layer problem becomes an N − 1-layered one. In either case, to ∗N −1 ∗N ∗1 achieve ED ≤ min (ED , ED ) (required by Lemma 3), each αk has to stay within the [0, 1] interval. As a result, an unconstrained optimization technique suffices to solve (4) as long as the initial partition satisfies 0 < αk < 1 for all k. In what follows, we give an explicit solution to the latter unconstrained optimization problem. Algorithm αopt For a given R, the optimal partition is given by: α∗k

=

1 S1 − S2 log2 , bRk (PRk − PRk−1 )Rk−1

α1∗

=

1−

N X

αk∗

RN

∗ = arg min D2 (αn , Rn , Cn+1 ), 1 ≤ n ≤ N − 1 Rn

where =

(19)

k=2

where

= arg min D1 (αN , RN )

Cn∗

2 ≤ k ≤ N (18)

∗ D2 (αn , Rn∗ , D2 (αn+1 , Rn+1 ,··· , ∗ ∗ D2 (αN −1 , RN −1 , D1 (αN , RN ))

1 Note that (13) for k = 0 would result in P 0 = P R1 . However, PR1 R0 does not appear in C and as a result, we arbitrarily define PR0 to be zero. 0

S1

=

S2

=

PN

∗

(RN − Rk−1 )(1 − PRN )2−b j=k+1 αj Rj N −1 Pi X ∗ Rk−1 (PRi+1 − PRi )2−b p=k+1 αp Rp

(20) (21)

i=k

Proof. For a given R, the global minima of the unconstrained N convex cost function ED can be found by setting the derivaN ∂E N ∂ED tives ∂αk to zero. We start with ∂αND−1 = 0 and taking into PN −1 account the relationship αN = 1 − k=1 αk . The resulting

equation can be solved for αN , giving us ∗ αN =

(1 − PRN )(RN − RN −1 ) 1 log2 bRN (PRN − PRN −1 )RN −1

Rayleigh Fading Channel, 1x1, b=1 0

(22)

∂E N

IV. N UMERICAL R ESULTS In this section, we present the numerical results of our proposed optimization algorithm. We consider a Rayleigh fading channel. The outage probability for the single-input, single output (SISO) case is given by [8]: PR = 1 − e

R − 2SN−1 R

(23)

and for the MIMO case, it has been obtained through simulations. Throughout this section we assume that b = 1. For N = 1, 2, 5, 500 layers, Figs. 2 and 3 show the expected distortion of a one-transmit, one-receive (1x1) antenna system, and a two-transmit, two receive (2x2) antenna system, respectively. As we can see from the figures, five layers achieve almost all of the layering gain, and most of this gain is realized by using only two layers. For the SNR range shown in the figures, the maximum layering gain is about 5 dB. Figure 4 compares the expected distortions of 1x1, 2x2, and 4x4 systems for five layers. As expected, increasing the number of antennas increases the slope of the expected distortion curve, and consequently, its distortion exponent. Figure 5 shows the optimal rate assignment and the optimal partitioning of the 2x2 system for N = 5 layers and different

Expected Distortion (dB)

−4

−6

−8

−10

−12

−14

−16

−18

−20 0

5

10

15

20 SNR (dB)

25

30

35

40

Fig. 2. Expected distortion of a 1x1 system for b = 1 and different number of layers Rayleigh Fading Channel, 2x2, b=1 0 N=1 N=2 N=5 N=500

−5

−10 Expected Distortion (dB)

Similarly, the equation ∂αND−2 = 0 can be solved for αN −1 , and the result contains αN for which we use the optimal value given by (22). This backward procedure can be continued for ∂E N k = {N − 3, · · · , 1}, and each equation ∂αDk = 0 yields a ∗ solution for αk+1 in terms of the already calculated optimal ∗ ∗ values αk+2 , · · · , αN . For 2 ≤ k ≤ N , each αk∗ is obtained according to (18) by solving the corresponding algebraic equation. The α1∗ solution given by (19) is a direct consequence of the definition of α. ¤ In deriving (18), we have implicitly assumed that Rk−1 6= Rk for all k. If two consecutive layers k and k − 1 happen to have the same rates Rk−1 = Rk = R0 , they should be considered as a single layer with rate R0 , and the two partition variables αk−1 and αk must be replaced with a single variable α0 = αk−1 +αk . Once the optimal value α0∗ has been found, it ∗ should be divided into two partitions αk−1 and αk∗ of arbitrary non-zero sizes. This is to ensure that the number of layers has not been reduced and consequently, the Ropt algorithm in the next iteration could still exploit the additional degree of freedom offered by having two layers instead of one. We conclude this section by summarizing our results. For a fixed partition, the Ropt algorithm offers a globally optimal rate assignment. For a given rate assignment, the αopt algorithm finds a globally optimal partition. The Ropt and αopt algorithms have complexities of O(N |R|) and O(N ), respectively. The Dmin algorithm iterates between the rate and partition optimization steps, and is guaranteed to converge to a local minima of the expected distortion with a complexity that is linear in the number of layers.

N=1 N=2 N=5 N=500

−2

−15

−20

−25

−30

−35

−40

−45 0

5

10

15

20 SNR (dB)

25

30

35

40

Fig. 3. Expected distortion of a 2x2 system for b = 1 and different number of layers

SNRs. Fig. 5(a) indicates that unequal rate assignment gains more significance as the SNR increases. V. C ONCLUSION In this paper, we considered layered transmission of a successively refinable complex Gaussian source over a quasistatic fading channel. We proposed an efficient algorithm that iteratively optimizes the rate assignment and the partitioning of the layers. It was analytically shown that the rate assignment and partitioning steps are each globally optimal and have a complexity that is linear in the number of layers. Numerical results for a multiple antenna Rayleigh fading channel were

Rayleigh Fading Channel, N=5, b=1 0 1x1 2x2 4x4

Expected Distortion (dB)

−10

−20

−30

−40

−50

−60

5

10

15

20 SNR (dB)

25

30

35

20

40 R (bits/symbol)

−70 0

Fig. 4. Expected distortion of 1x1, 2x2, and 4x4 systems for b = 1 and N = 5 layers

presented and it was numerically shown that a small number of layers is usually sufficient to achieve most of the layering gain.

15

0 dB 20 dB 40 dB

10

5

0 1

1.5

2

2.5

R EFERENCES

3.5

4

4.5

5

0.35 0 dB 20 dB 40 dB

0.3 0.25 α

[1] P.G. Sherwood, K. Zeger, “Progressive Image Coding for Noisy Channels,” IEEE Signal Processing Letters, July 1997. [2] V. Chande, H. Jafarkhani, and N. Farvardin, “Joint Source-Channel Coding of Images for Channels with Feedback, ” Information Theory Workshop, 1998. [3] V. Stankovic, R. Hamzaoui, Y. Charfi, Z. Xiong, “Real-Time Unequal Error Protection Algorithms for Progressive Image Transmission,” IEEE Journal on Selected Areas in Communications, December 2003. [4] F. Etemadi, H. Yousefi’zadeh, H. Jafarkhani, “A Linear-Complexity Distortion Optimal Scheme for Transmission of Progressive Packetized Bitstreams,” IEEE Signal Processing Letters, May 2005. [5] J. N. Laneman, E. Martinian, G. W. Wornell, J. G. Apostolopoulos, “Source-Channel Diversity for Parallel Channels, ” IEEE Transactions on Information Theory, October 2005. [6] M. Effros, R. Koetter, A. J. Goldsmith, M. Medard, “On Source and Channel Codes for Multiple Inputs and Outputs: Does Multiple Description Beat Space Time?, ” Information Theory Workshop, 2004. [7] S. Shamai, A. Steiner, “ A Broadcast Approach for a Single-User Slowly Fading MIMO Channel, ” IEEE Transactions on Information Theory, October 2003. [8] D. Gunduz, E. Erkip, “Source and Channel Coding for Quasi-Static Fading Channels, ” Thirty Ninth Annual Asilomar Conference on Signals, Systems and Computers, November 2005. [9] S. Boyd, L. Vandenberghe. Convex Optimization. Cambridge University Press, 2004.

3 Layer index (a)

0.2 0.15 0.1 1

1.5

2

2.5

3 Layer index (b)

3.5

4

4.5

5

Fig. 5. Optimal solution for a 2x2 system with b = 1 and N = 5 layers (a) optimal rate assignment (b) optimal partition

Hamid Jafarkhani

Department of EECS University of California Irvine, CA 92697, USA Email: [email protected]

Department of EECS University of California Irvine, CA 92697, USA Email: [email protected]

Abstract— We consider layered transmission of a successively refinable complex Gaussian source over a quasi-static fading channel. For a given number of source coding layers, we propose an efficient algorithm to calculate the optimal rate assignment for each layer, as well as the optimal size of each layer. The optimality of the algorithm is proved and numerical results for a multiple antenna Rayleigh fading channel are presented. It is numerically shown that a small number of layers is usually sufficient to achieve most of the layering gain.

I. I NTRODUCTION A successively refinable source provides a rate-distortion trade-off mechanism for lossy transmission of the source over a noisy channel. In situations where the Shannon’s sourcechannel separation theorem does not hold, successive refinement allows one to jointly optimize the rate allocation between the source and channel coding bits in order to minimize the expected distortion of the received signal. A quasi-static block fading channel is an example of such a scenario. If delay constraints limit the coding interval to a single block and no channel state information is available to the transmitter, there is always a non-zero probability of outage and the Shannon capacity of such a channel is zero. The expected received distortion thus have to be minimized by joint source-channel coding. From a practical point of view, joint source-channel coding of successively refinable sources has been widely studied in the literature [1], [2], [3], [4] (and references therein). Various optimal and near-optimal rate allocation strategies have been proposed with an emphasis on the transmission of progressively encoded multi-media bitstreams. As such, this body of work is primarily focused on the operational aspects of the problem and not on the information-theoretical ones, such as the theoretically achievable minimum expected distortion. Moreover, the scope of the latter work is limited to simple channels such as the binary symmetric channel (BSC), erasure channel, or finite-state channels for which well-known coding strategies exist. Recent advances in coding for multiple-input, multiple output (MIMO) fading channels has motivated the researchers to study the joint source-channel coding problem for such channels [5], [6], [8]. Distortion exponent was defined by Laneman et. al [5] as a means to describe the high signal to

noise ratio (SNR) behavior of the optimal expected distortion. Various transmission strategies have been evaluated based on their distortion exponents [5], [8]. The distortion exponent provides a useful tool for comparing various transmission strategies. It does not, however, completely characterize a given system since the scope of its definition is limited to the high-SNR regime only. Moreover, for a given system and an arbitrary SNR, one still has to solve the associated distortion minimization problem to fully characterize the optimal solution. In this paper, we propose an efficient algorithm to find the optimal rate assignment as well as the optimal size of layers in a layered transmission system. We prove the optimality of the proposed solution and present numerical results for the case of a MIMO Rayleigh fading channel. The structure of this paper is as follows. In Section II, we formulate the optimization problem. The proposed solution and the proof of its optimality are presented in Section III. Numerical results are presented in Section IV. Finally, Section V concludes this paper. II. P ROBLEM F ORMULATION The goal is to transmit a successively refinable complex Gaussian source over a quasi-static fading channel. The distortion-rate (D-R) function of the source is given by D(R) = 2−R , where R is the source coding rate in bitsper-symbol, and mean square error (MSE) distortion measure is assumed. The channel fading coefficients are known to the receiver but not to the transmitter, and are assumed to be constant over a block of M channel symbols. To satisfy the delay constraints, coding is assumed to be limited to a single block. A sequence of K source symbols are mapped into a single block, resulting in a bandwidth expansion factor of b = M K [8]. The transmission block is divided into N partitions,P as shown in Fig. 1. Partition i contains αi M channel N symbols ( i=1 αi = 1, 0 ≤ αi ≤ 1), and is transmitted at rate Ri bits-per-symbol. Since the source is successively refinable, layer n has to be decoded before layer n+1 can be decoded. To ensure this, we assume that Rn ≤ 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 defined as Pout (R, SN R) =

This algorithm can be formalized as the following (superscripts represent the iteration number). Algorithm Dmin Fig. 1.

•

Layered Transmission

•

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 SN R. The outage probability of a layer with rate r is denoted by Pr . The probability of successful decoding of the first n layers is prob{Rn ≤ C < RP n+1 } = PRn+1 − PRn , n and the associated distortion is 2−b( k=1 αk Rk ) . The expected distortion is given by ED (R, α) =

N X

(PRn+1 − PRn ) 2−b(

Pn k=1

αk Rk )

(1)

•

Initialize: Set k = 0, choose an arbitrary partition α0 Optimize: Rk+1

= arg min ED (R, α = αk )

(3)

αk+1

= arg min ED (R = Rk+1 , α)

(4)

R α

Iterate: Set k = k + 1, go to (3)

In each iteration, the optimization sub-problems (3) and (4) reduce the expected distortion. As a result, the sequence of non-negative values of ED eventually converges to a local minima. The Dmin algorithm is thus guaranteed to converge to an optimal solution. In what follows, we consider each subproblem separately.

n=0

where PR0 = 0 and PRN +1 = 1. In the above equation, rate and partition vectors are defined 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. We can now define the distortion minimization problem as min ED (R, α) R, α

(2)

0 ≤ Ri ≤ Ri+1 , 0 ≤ αi ≤ 1, i ∈ {1, · · · , N − 1} For very small values of N , this optimization problem can be solved through an exhaustive search [8]. If we limit the search for rates to a discrete set of rates, R, of size |R| and the optimal partition α is given, then the search for optimal R has a worst case complexity of O(|R|N ). Joint search for the optimal α and R is thus infeasible even for moderate values of N . On the other hand, solving this optimization problem of 2N − 1 variables through the well-known nonlinear programming techniques is numerically difficult because of the sensitivity to the choice of initial conditions, high time-complexity, and slow convergence. In what follows, we introduce an efficient minimization algorithm and prove its optimality. We emphasize that the layered transmission scheme discussed in this paper is not the only transmission strategy for a fading channel. Consequently, the solution to the optimization problem (2) does not necessarily represent the best achievable performance among all the transmission alternatives, such as the broadcast strategy [7], [8]. Our goal is to find the optimal solution within the particular framework of Fig. 1. Moreover, any practical coding scheme can only approach the transmission rates defined by the outage capacity. As a result, the solution to (2) is in fact an upper bound on the performance of any practical implementation. III. D ISTORTION M INIMIZATION A LGORITHM We propose an iterative algorithm to solve the optimization problem (2). The basic idea is to alternately optimize the rate and partition vectors and repeat this process until convergence.

A. Optimal Rate Assignment N Denote an N -layer expected distortion by ED (R, α) and let

D1 (β, r) , Pr + (1 − Pr )2−bβr D2 (β, r, t) , (1 − 2−bβr )Pr + 2−bβr t

(5) (6)

The following two lemmas form the basis of our rate assignment algorithm. Lemma 1: The expected distortion is given by 1 ED (R, α) 2 ED (R, α)

= D1 (1, R1 ) = D2 (α1 , R1 , D1 (α2 , R2 )) ···

N ED (R, α)

=

(7) (8)

D2 (α1 , R1 , D2 (α2 , R2 , · · · , D2 (αN −1 , RN −1 , D1 (αN , RN ))

(9)

Proof. We prove the lemma using mathematical induction on the number of layers. For N = 1, α1 = 1 and (7) can be easily seen to be equivalent to (1). Now we assume that (9) holds for N − 1 layers and prove the lemma for N layers. By expanding the first two terms of (1), the expected distortion of an N -layered system can be written as: N ED (R, α)

=

PR1 + (PR2 − PR1 )2−bα1 R1 + N Pn X (PRn+1 − PRn ) 2−b( k=1 αk Rk ) n=2

= =

(1 − 2−bα1 R1 )PR1 + 2−bα1 R1 C D2 (α1 , R1 , C)

(10)

where C = PR2 +

N X

(PRn+1 − PRn ) 2−b(

Pn

k=2

αk Rk )

(11)

n=2

From (9) and (10), we see that the proof is complete if we show that C

= D2 (α2 , R2 , D2 (α3 , R3 , · · · , D2 (αN −1 , RN −1 , D1 (αN , RN ))

(12)

We note that C is independent of R1 and α1 . For k ≥ 1, define Rk0 αk0

, ,

Rk+1 αk+1

(13) (14)

1

and PR00 , 0. Using the latter definitions and re-indexing the sum in (11), C can be written as: C=

N −1 X

0 (PRn+1 − PRn0 ) 2−b(

Pn

k=1

0 α0k Rk )

(15)

n=0

By comparing (1) and (15), we see that C is an (N − 1)layer cost function for layers 2 to N . The inductive assumption on N − 1 layers then implies that C can be expressed recursively as in (12), and this completes the proof. ¤ Lemma 2: For N ≥ 2 and a given α, the optimal value ∗ (R2∗ , · · · , RN ) is given by ∗ ) = arg (R2∗ , · · · , RN

min

R2 , ··· , RN

C

(16)

where C is given by (12). Proof. From (10), we have N ED (R, α) = (1 − 2−bα1 R1 )PR1 + 2−bα1 R1 C

(17)

The first term in (17) is independent of {R2 , · · · , RN }, and from (11) we see that C is independent of R1 and only ∂E N (R,α) depends on {R2 , · · · , RN }. Since D∂C = 2−bα1 R1 > 0, N ED (R, α) is an increasing function of C. To minimize the expected distortion, therefore, {R2 , · · · , RN } should be chosen such that C is minimized. ¤ Lemma 2 implies that layers 2 through N can be optimized independently of layer 1 by minimizing the quantity C. The recursion (12) given in Lemma 1 shows that C, in turn, is a cost function defined over layers 2 through N and as a result, layers 3 through N can be optimized independently of layer 2. Applying this optimization break-down N − 1 times, we conclude that layer N can be optimized independently of ∗ all layers. The optimal RN so found can be held constant, and the two-layer cost function over {RN −1 , RN } can be optimized over RN −1 . Layers N −2 through 1 can be similarly optimized using this backward procedure. We have thus proved the optimality of the following N -step optimization algorithm. Algorithm Ropt ∗ For a given α, the optimal rate assignment (R1∗ , · · · , RN ) is given by: ∗ RN

Rn∗

The Ropt algorithm replaces an N -dimensional search with N single-dimensional searches, and consequently, reduces the optimization complexity from O(|R|N ) to O(N |R|). B. Optimal Partitioning We now consider the problem of finding an optimal partition α, when the rate vector R is given. From (1), we see that each term of the expected distortion is an exponential function of the partition variables αk . As a result, the expected distortion is a convex function of α and sub-problem (4) can be efficiently solved using constrained convex optimization techniques [9]. To simplify this constrained optimization, we prove the following intuitive result. ∗N Lemma 3: The optimal expected distortion ED is a nonincreasing function of N . Proof. Merge layers 1 and 2 and define R10 , R1 = R2 , α10 , α1 + α2 , and Ri0 , Ri+1 , αi0 , αi+1 for 3 ≤ i ≤ N . By imposing the additional constraint R1 = R2 on (2), this optimization problem becomes an N − 1layer problem defined by the R0 and α0 vectors, that can ∗N −1 achieve ED . Removing the latter constraint could only ∗N ≤ improve the expected distortion, and as a result, ED ∗N −1 ED . ¤ We will now show that the constraints 0 ≤ αk ≤ 1 can be safely removed from (4). Let us assume that N > 1 and we solve (4) using an unconstrained minimization technique. Also assume that the initial partition satisfies 0 < αk < 1 for all k. If some αk approaches 1 at any iteration, then the N -layer problem becomes a single-layered one. On the other hand, if αk approaches 0 at any iteration, then the N -layer problem becomes an N − 1-layered one. In either case, to ∗N −1 ∗N ∗1 achieve ED ≤ min (ED , ED ) (required by Lemma 3), each αk has to stay within the [0, 1] interval. As a result, an unconstrained optimization technique suffices to solve (4) as long as the initial partition satisfies 0 < αk < 1 for all k. In what follows, we give an explicit solution to the latter unconstrained optimization problem. Algorithm αopt For a given R, the optimal partition is given by: α∗k

=

1 S1 − S2 log2 , bRk (PRk − PRk−1 )Rk−1

α1∗

=

1−

N X

αk∗

RN

∗ = arg min D2 (αn , Rn , Cn+1 ), 1 ≤ n ≤ N − 1 Rn

where =

(19)

k=2

where

= arg min D1 (αN , RN )

Cn∗

2 ≤ k ≤ N (18)

∗ D2 (αn , Rn∗ , D2 (αn+1 , Rn+1 ,··· , ∗ ∗ D2 (αN −1 , RN −1 , D1 (αN , RN ))

1 Note that (13) for k = 0 would result in P 0 = P R1 . However, PR1 R0 does not appear in C and as a result, we arbitrarily define PR0 to be zero. 0

S1

=

S2

=

PN

∗

(RN − Rk−1 )(1 − PRN )2−b j=k+1 αj Rj N −1 Pi X ∗ Rk−1 (PRi+1 − PRi )2−b p=k+1 αp Rp

(20) (21)

i=k

Proof. For a given R, the global minima of the unconstrained N convex cost function ED can be found by setting the derivaN ∂E N ∂ED tives ∂αk to zero. We start with ∂αND−1 = 0 and taking into PN −1 account the relationship αN = 1 − k=1 αk . The resulting

equation can be solved for αN , giving us ∗ αN =

(1 − PRN )(RN − RN −1 ) 1 log2 bRN (PRN − PRN −1 )RN −1

Rayleigh Fading Channel, 1x1, b=1 0

(22)

∂E N

IV. N UMERICAL R ESULTS In this section, we present the numerical results of our proposed optimization algorithm. We consider a Rayleigh fading channel. The outage probability for the single-input, single output (SISO) case is given by [8]: PR = 1 − e

R − 2SN−1 R

(23)

and for the MIMO case, it has been obtained through simulations. Throughout this section we assume that b = 1. For N = 1, 2, 5, 500 layers, Figs. 2 and 3 show the expected distortion of a one-transmit, one-receive (1x1) antenna system, and a two-transmit, two receive (2x2) antenna system, respectively. As we can see from the figures, five layers achieve almost all of the layering gain, and most of this gain is realized by using only two layers. For the SNR range shown in the figures, the maximum layering gain is about 5 dB. Figure 4 compares the expected distortions of 1x1, 2x2, and 4x4 systems for five layers. As expected, increasing the number of antennas increases the slope of the expected distortion curve, and consequently, its distortion exponent. Figure 5 shows the optimal rate assignment and the optimal partitioning of the 2x2 system for N = 5 layers and different

Expected Distortion (dB)

−4

−6

−8

−10

−12

−14

−16

−18

−20 0

5

10

15

20 SNR (dB)

25

30

35

40

Fig. 2. Expected distortion of a 1x1 system for b = 1 and different number of layers Rayleigh Fading Channel, 2x2, b=1 0 N=1 N=2 N=5 N=500

−5

−10 Expected Distortion (dB)

Similarly, the equation ∂αND−2 = 0 can be solved for αN −1 , and the result contains αN for which we use the optimal value given by (22). This backward procedure can be continued for ∂E N k = {N − 3, · · · , 1}, and each equation ∂αDk = 0 yields a ∗ solution for αk+1 in terms of the already calculated optimal ∗ ∗ values αk+2 , · · · , αN . For 2 ≤ k ≤ N , each αk∗ is obtained according to (18) by solving the corresponding algebraic equation. The α1∗ solution given by (19) is a direct consequence of the definition of α. ¤ In deriving (18), we have implicitly assumed that Rk−1 6= Rk for all k. If two consecutive layers k and k − 1 happen to have the same rates Rk−1 = Rk = R0 , they should be considered as a single layer with rate R0 , and the two partition variables αk−1 and αk must be replaced with a single variable α0 = αk−1 +αk . Once the optimal value α0∗ has been found, it ∗ should be divided into two partitions αk−1 and αk∗ of arbitrary non-zero sizes. This is to ensure that the number of layers has not been reduced and consequently, the Ropt algorithm in the next iteration could still exploit the additional degree of freedom offered by having two layers instead of one. We conclude this section by summarizing our results. For a fixed partition, the Ropt algorithm offers a globally optimal rate assignment. For a given rate assignment, the αopt algorithm finds a globally optimal partition. The Ropt and αopt algorithms have complexities of O(N |R|) and O(N ), respectively. The Dmin algorithm iterates between the rate and partition optimization steps, and is guaranteed to converge to a local minima of the expected distortion with a complexity that is linear in the number of layers.

N=1 N=2 N=5 N=500

−2

−15

−20

−25

−30

−35

−40

−45 0

5

10

15

20 SNR (dB)

25

30

35

40

Fig. 3. Expected distortion of a 2x2 system for b = 1 and different number of layers

SNRs. Fig. 5(a) indicates that unequal rate assignment gains more significance as the SNR increases. V. C ONCLUSION In this paper, we considered layered transmission of a successively refinable complex Gaussian source over a quasistatic fading channel. We proposed an efficient algorithm that iteratively optimizes the rate assignment and the partitioning of the layers. It was analytically shown that the rate assignment and partitioning steps are each globally optimal and have a complexity that is linear in the number of layers. Numerical results for a multiple antenna Rayleigh fading channel were

Rayleigh Fading Channel, N=5, b=1 0 1x1 2x2 4x4

Expected Distortion (dB)

−10

−20

−30

−40

−50

−60

5

10

15

20 SNR (dB)

25

30

35

20

40 R (bits/symbol)

−70 0

Fig. 4. Expected distortion of 1x1, 2x2, and 4x4 systems for b = 1 and N = 5 layers

presented and it was numerically shown that a small number of layers is usually sufficient to achieve most of the layering gain.

15

0 dB 20 dB 40 dB

10

5

0 1

1.5

2

2.5

R EFERENCES

3.5

4

4.5

5

0.35 0 dB 20 dB 40 dB

0.3 0.25 α

[1] P.G. Sherwood, K. Zeger, “Progressive Image Coding for Noisy Channels,” IEEE Signal Processing Letters, July 1997. [2] V. Chande, H. Jafarkhani, and N. Farvardin, “Joint Source-Channel Coding of Images for Channels with Feedback, ” Information Theory Workshop, 1998. [3] V. Stankovic, R. Hamzaoui, Y. Charfi, Z. Xiong, “Real-Time Unequal Error Protection Algorithms for Progressive Image Transmission,” IEEE Journal on Selected Areas in Communications, December 2003. [4] F. Etemadi, H. Yousefi’zadeh, H. Jafarkhani, “A Linear-Complexity Distortion Optimal Scheme for Transmission of Progressive Packetized Bitstreams,” IEEE Signal Processing Letters, May 2005. [5] J. N. Laneman, E. Martinian, G. W. Wornell, J. G. Apostolopoulos, “Source-Channel Diversity for Parallel Channels, ” IEEE Transactions on Information Theory, October 2005. [6] M. Effros, R. Koetter, A. J. Goldsmith, M. Medard, “On Source and Channel Codes for Multiple Inputs and Outputs: Does Multiple Description Beat Space Time?, ” Information Theory Workshop, 2004. [7] S. Shamai, A. Steiner, “ A Broadcast Approach for a Single-User Slowly Fading MIMO Channel, ” IEEE Transactions on Information Theory, October 2003. [8] D. Gunduz, E. Erkip, “Source and Channel Coding for Quasi-Static Fading Channels, ” Thirty Ninth Annual Asilomar Conference on Signals, Systems and Computers, November 2005. [9] S. Boyd, L. Vandenberghe. Convex Optimization. Cambridge University Press, 2004.

3 Layer index (a)

0.2 0.15 0.1 1

1.5

2

2.5

3 Layer index (b)

3.5

4

4.5

5

Fig. 5. Optimal solution for a 2x2 system with b = 1 and N = 5 layers (a) optimal rate assignment (b) optimal partition