Optimization of Unequal Error Protection Rateless Codes for Multimedia Multicasting
Abstract
Rateless codes have been shown to be able to provide greater flexibility and efficiency than fixedrate codes for multicast applications. In the following, we optimize rateless codes for unequal error protection (UEP) for multimedia multicasting to a set of heterogeneous users. The proposed designs have the objectives of providing either guaranteed or besteffort quality of service (QoS). A randomly interleaved rateless encoder is proposed whereby users only need to decode symbols up to their own QoS level. The proposed coder is optimized based on measured transmission properties of standardized raptor codes over wireless channels. It is shown that a guaranteed QoS problem formulation can be transformed into a convex optimization problem, yielding a globally optimal solution. Numerical results demonstrate that the proposed optimized random interleaved UEP rateless coder’s performance compares favorably with that of other recently proposed UEP rateless codes.
ultimedia, error control coding, unequal error protection, raptor codes, video transmission
1 Introduction
Multimedia transmission is a main driver for explosive data traffic growth in wired and wireless networks. While decades of research have been conducted in designing reliable multimedia transmission over errorprone channels, multimedia multicast over lossy packet networks is still challenging due highly variable channel conditions among different users, QoS constraints and multimedia devices, e.g., smart phones, tablets, laptops. Scalable video coding (SVC) [3] is useful as it layers the source to enable efficient progressive reconstruction at the receiver.
Protection against channel impairments can be achieved by using codes that provide forward error correction (FEC): ReedSolomon [4], low density parity check (LDPC) [5], Turbo [6] and fountain [7] [8], as well as joint sourceandchannel coding (JSCC) [5] [6] [7]. Other approaches that exploit source scalability to provide UEP use hybrid automatic repeat request or crosslayer optimization [9] [10] [11]. The above approaches, however, were mainly envisioned for pointtopoint links and do not consider heterogeneous users’ QoS. As a result, adaptation of JSCC to multimedia multicast is often inefficient in that they cater to the lowest QoS user.
A critical aspect of robust multimedia multicast is channelcoding performance. Traditional fixedrate FEC encounters the problem of channel heterogeneity as in the case of ReedSolomon (RS) codes that are targeted for one specific loss rate [12]. Rateless fountain codes [13] are efficient and flexible for broadcasting or multicasting over erasure channels. The rateless property enables (1) a transmitter to generate, as needed, an unlimited number of encoded symbols, and (2) a receiver to successfully recover any subset of the encoded symbols of size slightly greater than the number of information symbols. Raptor codes [14] due to their high performance and low complexity are fountain codes that have been incorporated into the third generation partnership program (3GPP) Multimedia Broadcast/Multicast Services (MBMS) standard [15]. In [16], raptor codes have been extensively evaluated for MBMS download delivery. A more recent version appears in [17], and background can be found in [18]. Rateless codes have been applied to SVCbased multisource streaming [19], adaptive unicast streaming [8], and SVC streaming from multiple servers [20]. A JSCC rateless coding framework for scalable video broadcast appears in [21]. Applications to distributed video streaming for relay/cooperation based receiverdriven layered multicasting is found in [22], while [23] and [24] use fountain codes for distributed video caching via user cooperation.
While the raptor code itself is not suited for progressive decoding, multimedia has a hierarchical source symbol priority structure necessitating unequal error protection (UEP), sometimes referred to as priority encoding transmission (PET) [25]. Numerous UEP approaches to multimedia transmission have been proposed [4] [26] [27] [28]. In [4], Mohr proposes a PETbased packetization scheme for transmitting compressed images over noisy channels. In [26], the Mohr scheme is optimized to minimize endtoend distortion. Optimization of receiverdriven networks has also been investigated [29]. Ratedistortionbased optimization can be found in [30]. Rather than incorporate code performance into the optimization, these existing optimization approaches generally employ maximum distance separable (MDS) codes. In this paper, code performance is taken into account in the UEP rateless code optimization.
Not surprisingly, UEP rateless code design methods have recently appeared. In [31], message symbols are encoded by nonuniform selection of source symbols and applied to MPEGII video transmission in [32]. In [33], expanding window fountain (EWF) codes organize source symbols into a sequence of nested windows. In [34], EWF codes are applied to scalable video multicasting. Windowing approaches for rateless codes that achieve equal error protection (EEP) have been proposed in [35] and [36]. The slidingwindow (SW) rateless code design proposed in [36] is applied to wireless video broadcasting in [37]. Ahmad et. al [38] achieve UEP in video multicast using the Luby Transform (LT) [39] via block duplication. In [40], a UEP rateless code based on hierarchical graph coding is proposed for media streaming. However, these previous UEP rateless code design approaches [31] [33] may compromise performance as they alter the LT code [39] degree distribution unless the degree distribution is jointlyoptimized with UEP parameters.
Finally, previous approaches to UEP optimization for multimedia have focused almost exclusively on providing besteffort QoS, i.e., maximization of an average fidelity measure of video/image quality of end users for a given transmission rate [4] [26] [30] or with rateless codes [21] [32] [34] [37]. As rateless codes have no predetermined transmission rate, QoS may be achieved by transmitting enough coded symbols to meet users’ QoS demands. In contrast, our focus is on guaranteed QoS optimization, i.e., minimizing resource usage under the constraints of heterogeneous QoS guarantees. While [1] and [2] presented early versions of this approach, this paper provides more complete background, technical detail, a method to simplify constraints, as well as an example video multicasting application.
Our main contributions are summarized as follows:

a UEP scheme is proposed that uses random interleaving of raptor coders that enables direct application of already optimized standardized raptor codes used for 3GPP MBMS [15]. When applied to multicasting to heterogenous users, low bandwidth clients need not receive encoded symbols targeted to high bandwidth clients, which can significantly reduce receiver complexity and time to decode.

the proposed design, optimized for multimedia multicast to heterogeneous users, contains QoS guarantees and factors in rateless code performance. With standardized raptor codes, this guaranteed QoS optimization problem is shown to be convex with a simplified solution using KarushKuhnTucker (KKT) optimality conditions.

through a combination of simulation and analysis, performance of the proposed random interleaved UEP rateless design is compared to other EEP and UEP rateless coders.
The paper is organized as follows: Section 2 describes the system setup and proposed UEP rateless code design; Section 3 presents the problem formulations for guaranteed and besteffort QoS; Section 4 provides the solution for guaranteed QoS. Section 4 transforms the original problem formulation for guaranteed QoS into a convex optimization problem where optimal selection probabilities for interleaving are obtained in closed form for certain cases or else numerically. Comparisons with recent UEP rateless coding schemes are provided in Section 5.
2 System setup and proposed design
2.1 System setup
A multimedia server that transmits multimedia content simultaneously to multiple users is considered, which may include streaming with strict delay requirements. Multimedia content is divided into multiple coded blocks. The server first compresses each source block using a predefined source coder and then adds error protection to the source information using a rateless, e.g., raptor or LT code. Encoded symbols are then multicast over a wireless lossy packet network.
User subscribers are classified into classes according to reception capability. For Class users, reception capability is defined as the proportion of symbols that the receiver can successfully receive compared to the number of transmitted symbols, . Therefore, in each transmission session, the number of successfully received encoded symbols for each user in Class is , where is the number of symbols transmitted^{1}^{1}1For analytical simplicity, the number of received symbols for each user class is modeled as multiplied by the total transmitted as in [34]. . Without loss of generality, we order the classes according to reception quality, i.e., . For example, a Class 1 user may represent a mobile cell phone with limited reception quality due to size and power restrictions, while a Class 2 user may represent an automobile equipped with larger antenna and higher bandwidth service. Reception capabilities are determined by channel quality and bandwidth between server and receiver, and no distinction is made between overhead incurred by symbol erasures and lost symbol transmission opportunities due to client bandwidth restrictions. Users in different classes may also have different QoS requirements, and outage QoS guarantees are used to enable users to recover a given portion of source data with an achieved target probability. Without loss of generality, the term peak signaltonoise ratio (PSNR), a common measure used for visual media quality, is used to denote QoS.
Let represent the number of information symbols in a raptorcoded source block. Assume the server transmits encoded symbols in order to meet all users’ QoS demands, where is the total transmission overhead for all layers needed to combat losses of the heterogenous users in the multicast system. For scalability, the coded source block is partitioned into layers in decreasing order of importance: Layer contains the most important symbols while Layer contains the least important symbols. For example, in video or image compression terminology, Layer might represent the base layer (BL), and Layer the first enhancement layer (EL). The number of source symbols in Layer is denoted by , and .
Successful decoding of layer requires layers to be decodable. Rather than jointly optimizing the source and channel coders, we focus on optimizing channel coding parameters for a given source coder. Therefore, we assume that the values of are provided by a predetermined scalable source coder.
2.2 Proposed UEP rateless code
We propose a randomly interleaved UEP rateless encoder structure to provide FEC for multimedia multicast as shown in Fig. 1. The encoder assumes that source symbols have been allocated to the layers prior to encoding. Encoding is performed by randomly selecting layer with probability for where .
Encoded output symbols are generated by the raptor encoder for Layer with code dimension , degree distribution and precode . The overall encoded data stream consists of interleaved raptorencoded symbols from the encoders. From the above definitions, can be lower bounded by
(1) 
where denotes the code dimension of the selected layer for user Class .
The proposed rateless coded scheme uses the random interleaving to achieve UEP. While probabilistic encoding has been used in EWF rateless codes in [34], as well as in [41] and [42], an advantage of the proposed scheme in Fig. 1 is that the different layers can be encoded and decoded separately. In addition, in [31] and [34], degree distributions and selection probabilities need to be optimized jointly, which is a complicated task. A practical advantage of Fig. 1 when applied to a multicast system for users with different bandwidth constraints, low bandwidth clients need not receive symbols generated from source layers targeting high bandwidth clients, which reduces the complexity and timetodecode for low BW clients.
It is worth noting that one may alter the ordering of the output symbols from the random interleaved UEP raptor coder using scheduling algorithms while maintaining the priority of each layer. Investigations along these lines have been recently proposed in [43] and [44]. Unlike [45] [46], the proposed need not specify a packetization structure; the scheme may be applied to data packets rather than to symbols.
3 Problem formulations with QoS constraints
3.1 Guaranteed QoS formulation
We consider users that require playback media at a quality no lower than their own QoS requirement. Since the transmitter has to provide guaranteed QoS for all user classes before the start of transmission of the next source block, system delay and throughput for each source block is determined by the maximum number of transmitted symbols required to satisfy the QoS of each individual user class. As delay is a critical issue in multimedia multicast, the objective is to provide different levels of QoS guarantees according to users’ requirements while minimizing total transmission overhead :
Problem 1.0 (Guaranteed QoS):
(2) 
(3) 
where represents the PSNR of the successfully recovered source data of the Class user given transmitted symbols, and and denote target PSNR threshold and outage probability, respectively, for the Class user. The aim is to allocate coding rates across layers through optimization of the probabilities , .
The source (e.g., video, image) coder is assumed to be progressive, so that the reconstruction media quality is determined mainly by the symbol errors in the lowest layer encountered in the recovery process. Let represent the PSNR achieved when Layers to are successfully recovered by raptor decoding, where . For a given source coder, if the source PSNR is represented as nondecreasing function, , of the total number of source symbols decoded by the receiver, then . For each class , let be the minimum index that satisfies . In order to satisfy , users in Class require the raptor decoder to successfully decode, at minimum, Layers to . For a given UEP raptor code design, let represent the error probability that the Class decoder fails to decode layer given transmission overhead and reception quality . In the most stringent case when decoding errors across layers are independent, QoS requirements of end users can be simplified to
(4) 
3.2 Besteffort QoS formulations
While the above formulation focuses on minimizing transmission overhead subject to satisfying guaranteed user QoS, this subsection considers transmission overhead that is upper bounded due to delay constraints or cost. For this scenario, given a maximum transmission overhead, , the service provider attempts to provide users of different classes with the best possible QoS. The following besteffort QoS problem extends that in [34] by 1) considering both constrained and unconstrained cases, 2) allowing for allocating different weighting factors to different user classes as well as 3) possessing the previously mentioned advantages of the proposed random interleaved UEP raptor codes.
The expected PSNR of users in Class , which serves as a measure of the besteffort QoS, can be evaluated as , where is the PSNR achieved when Layers to are successfully recovered, where represents the probability that a Class user successfully recovers Layers to but fails to recover Layer . The optimization balances users of different classes with different channel qualities by assigning weighting coefficient for Class where , . The choice of depends on both the user class importance as well as the number of users in that class. The weighted average PSNR over all user classes then becomes the objective function:
Problem 2: (Besteffort QoS)
(5) 
(6) 
where
(7) 
In Problem , no guaranteed minimum QoS is provided. For a given maximum transmission overhead, the service provider may instead aim to provide besteffort QoS to multiple user classes, but under the additional constraint of a minimum QoS guarantee for each user class:
Problem 3: (Besteffort QoS with constraints on individual classes)
(8) 
(9) 
(10) 
where is given by (7) and is the minimum layer index that satisfies . Problem 2 is a special case of Problem 3 without user QoS constraints.
In the next section, we show that Problem 1.0 can be transformed to an equivalent convex optimization problem when standardized raptor codes are employed. Unfortunately, while Problems 2 and 3 cannot be similarly transformed due to the form of the expressions, they can still be solved numerically by searching the dimensional parameter space of , checking the constraints (10) and the resulting average PSNR (8). When , the numerical method is significantly simplified as only that gives the maximum average PSNR needs to be determined. Numerical results and comparisons for Problems 2 and 3 are provided later.
4 Solving the guaranteed QoS problem
4.1 Evaluation of decoding failure probability
In the proposed design, existing highperformance standardized raptor codes can be directly applied, which enable low encoding/decoding complexity and overhead. Details about the precode, degree distribution and code construction can be found in [15], (Annex B). When standardized raptor codes are employed with maximum likelihood (ML) decoding for code dimension greater than , the decoding failure probability, i.e., failure to decode source symbols after symbols are successfully received, have been shown, through extensive experimentation, to be accurately modeled by [16],
(11) 
where constants , . For , Eq. (11) underestimates the error probability due to short block length. One way to improve code performance for layers with fewer symbols is to merge source layers with similar optimized selection probabilities into larger layers. However, for video the condition is unlikely to occur.
We also remark that standardized raptor codes outperform the recently proposed SWraptor codes [37]. For example, according to Fig. 2(b) of [37], the SWraptor codes have a decoding failure probability of almost with code dimension and overhead while standardized raptor codes have negligible decoding failure probability at the same code dimension and lower overhead according to (11).
When more general LT or raptor codes using iterative decoding are employed, the decoding failure probability can be approximated by assuming that symbol errors in iterative decoding are mutually independent
(12) 
where is the symbol error probability of a Class user decoding Layer (also see (3) of [34]) which can be analytically determined by andor tree analysis [47]. Since each layer is encoded by a separate rateless code, evaluating the symbol error probability of each layer can be consider as a special case of (6) and (7) in [31] where uniform selection is used ( in [31]), and Eq. (12) can be approximated using
(13) 
where is the LT code degree distribution, denotes derivative with respect to , is the number of decoding iterations and is the total number of encoded symbols transmitted for Layer in each transmission block. The asymptotic symbol error probability of iterative decoding can be estimated by choosing a large value in Eq. (13) (see [47]).
4.2 Convexity analysis
For a given transmission overhead , , ^{2}^{2}2Strictly speaking, is a Binomialdistributed random variable with mean . However, the randomization of has little effect on the problem of interest when averaged over a large number of realizations. In addition, one can always schedule the selection of layers to make sure that is proportional to . and satisfies . When standardized raptor codes are used, substituting and into Eqs. (11) and (4), and taking the logarithm of the constraints described by (4), Problem is transformed to:
Problem 1.1:
(14) 
(15) 
where , and . The constraint that is nonnegative is implicitly guaranteed by the function. To ensure an integer solution, we compute as if realvalued, then round to the nearest larger integer. Although the above transformation uses the decoding failure probability evaluation of standardized raptor codes given by Eq. (11), a similar method can be applied to other decoding failure probability models that can be approximated by an exponential function.
To solve Problem , we first prove convexity. As the objective function is linear, we only need to prove that the constraint functions are convex with respect to . It can be shown that for the second derivatives of with respect to satisfy
(16) 
According to the second order condition of convex functions [48], is a convex function of . Since nonnegative weighted sums preserve convexity [48], the constraint functions (15) are convex functions of the vector . Problem 1.1 can therefore be solved numerically by available convex optimization algorithms [48]. We remark that the above convexity holds not just for values of and in the exponential model of Eq. (11) from [16] but also more generally over the range and which represent a wide family of exponential fountain code failure probability models.
Let and be the variable vectors of the primal and dual problems of Problem , respectively. If and represent sets of primal and dual optimal points, they must satisfy the KarushKuhnTucker (KKT) optimality conditions for the objective function and constraint functions :
(17) 
(18) 
(19) 
(20) 
where here and . Since the original Problem is convex and satisfies Slater’s condition, the above KKT optimality conditions provide the necessary and sufficient conditions for optimality [48]. In general, solving the KKT condition is not straightforward. However, if we can identify a set of inequality constraints that are most likely to be active, i.e., achieve equality at the optimal solution, then we can obtain a corresponding set of primal and dual solution points and verify the optimality with KKT condition.
A simplification to Problem 1.1 arises if we have a onetoone mapping between user classes and channel coding layers, i.e., for and , which is the assumption used in the formulation of [34], and if all the inequality constraints are active. Using the above assumption, the solution to Problem can be obtained by finding using the constraint for Class in Eq. (15) and substituting the solution of into the next constraint, solving for with the constraint for Class in Eq. (15) etc. until all of the variables are determined. However, since this simplification has not been proven to be equivalent to Problem 1.1 in general, the solution obtained in this manner has to be verified using the KKT optimality conditions. If all the inequality constraints are active, Eqs. (17) and (19) are automatically satisfied. Therefore, if we obtain a solution of Problem by solving , we can substitute the value of into Eq. (20) and obtain . If satisfies Eq. (18), i.e., , then we have proven that the value of we obtained is indeed an optimal solution of Problem . If the KKT optimality condition is not satisfied, then numerical methods can still be used to solve this convex optimization problem.
4.3 Classtolayer mapping algorithm
In the following, we propose an algorithm to transform a general guaranteed QoS problem into a problem with onetoone mapping between user classes and channel coding layers. The idea is to reduce the dimensionality of the problem by removing redundant user constraints and merging sourcecoding layers. The process is explained in the following algorithm:
Algorithm 1: (Classtolayer mapping algorithm)
Step 1 (User class amalgamation): Repeat the following class amalgamation operation until for every , where : for any pair of user class indices and where (hence ), if Class users have the same or higher target PSNR threshold than Class users (i.e., or ), we absorb Class into Class .
Step 2 (Source layer merging): Repeat until for every layer , there exists a class such that : if there exists a source layer where there is no corresponding user class (i.e., no exists such that ), Layers and are merged to form a new source layer with code dimension .
Step 1 finds a set of the most demanding user classes with respect to their channel conditions; Step 2 reduces the number of channel coding layers to the minimum without compromising the performance. After performing Algorithm 1, we can show the following fact:
Lemma 1: After performing Algorithm 1, and for . If for every Class that has been absorbed into Class in Step 1, is also satisfied, then the new optimization problem after performing Algorithm 1 is equivalent to Problem 1.1. In addition, any further partitioning of layers cannot reduce the minimum transmission overhead required to achieve the QoS requirements.
Proof: First we show that any QoS constraint dropped from Step 1 (user class amalgamation) is irrelevant. Suppose the QoS constraint of Class users is satisfied, i.e., . Since , we have . Hence, Class users receive more coded symbols than Class users. Therefore, the decoding failure probability for all . Then, because , from the assumption of Lemma 1, , and
(21)  
Hence, the QoS constraint for Class users is also satisfied.
Next we show that after performing Algorithm 1, the number of source layers and the number of user classes are equal. The class amalgamation procedure ensures that the set , is monotonically increasing with . This fact does not change after performing the source layer merging procedure. Since , we have . On the other hand, source layer merging ensures that for any , there exists an integer such that . Therefore, we also have . Thus, . Together with the fact that is monotonically increasing with , we can conclude that for .
Finally, to complete the proof, in the appendix we show that any further partitioning of layers cannot reduce the required minimum transmission overhead. QED.
Remark 1: The condition that for every Class that has been absorbed into Class , , is a sufficient condition for Lemma 1 but not a necessary condition. Even if this condition is not satisfied, it is possible that the transformed problem due to Algorithm 1 results in the optimal solution. In addition, if this condition is violated, to ensure that the optimal solution of the transformed problem is the optimal solution of the original problem, we can always verify if the obtained solution satisfies all the constraints of the user classes that have been amalgamated in Step 1. If not, the convex Problem 1.1 can be solved numerically. This is further illustrated in Section 4.4.
Remark 2: For besteffort QoS Problem 3, the transformation given in Algorithm 1 may not apply, as an optimal solution also depends on the fidelity measure of the multimedia source.
Remark 3: In the original general problem and are arbitrary, which means it is possible that a user with worse channel quality may have a higher QoS requirement. Lemma 1 and the mapping algorithm transform the original problem to a progressive transmission problem where there is a onetoone mapping between user classes and channel coding layers.
Remark 4: In the case of , the transmission overhead can be lower bounded by Eq. (1) which is independent of code optimization. Minimizing , as in Problems 1 and 1.1, maximizes code performance.
4.4 Video multicasting numerical example
We now illustrate the mapping process and solution to the guaranteed QoS problem for multicasting a H.264 SVC [3] videocoded stream which contains 15 layers: a base layer (BL) and 14 enhancement layers (ELs). Since our focus is on optimizing a channel coder for a given source coder, the number of information symbols and the corresponding PSNR values are taken from Table I of [34]. As in [34], each source symbol represents 400 source bits. The UEP rateless encoders and decoders operate at the symbol level. We assume there are four classes of users with reception capabilities and QoS requirements shown in Table 1.
User class index (j)  1  2  3  4 
User reception capability  0.4  0.5  0.6  1 
User QoS req. (PSNR thr. (dB))  25.79  29  27.25  40.28 
# Decoded symbols to achieve QoS  400  1155  700  3800 
# Decoded source layers required ()  1  4  2  15 
Probability threshold  0.8  0.9  0.85  0.95 
Using the previously described simplification strategy for class and layer mapping, we observe that while , which means that Class users have both better reception capabilities and lower PSNR requirements than Class users. Therefore, the QoS constraint from Class users can be dropped. Then, since the number of layers required by the three classes are , and , after the layermerging procedure of Algorithm 1, we obtain a new set of channel layers with Layer 1 comprising the BL, Layer 2 consisting of the first 3 ELs, and Layer 3 consisting of the fourth to fourteenth ELs. Since , from Lemma 1, the new problem after mapping is equivalent to the original problem. The parameters of the transformed problem after the mapping are shown in Table 2.
Combined classlayer index (j or l)  1  2  3 
Reception capability  0.4  0.5  1 
PSNR threshold (dB)  25.79  29  40.28 
Number of decoded symbols to achieve QoS  400  1155  3800 
Number of decoded layers required  1  2  3 
Probability threshold  0.8  0.9  0.95 
Number of symbols in each layer  400  755  2645 
To determine the interleaving probabilities for the standardized raptor codes for the three new layers, , and need to be determined to minimize such that
(22) 
Assuming all the inequality constraints are active, we obtain a minimum overhead , which is achieved when , and . The solution is then verified to be optimal using KKT conditions. In contrast, equal error protection (EEP) allocation requires a minimum overhead of , a factor of over four higher.
With the optimal selection parameters, we find that Class 3 users of the original problem (Table 1) can successfully decode the base layer and one enhancement layer with a probability higher than . This means that even if the target probability threshold in Table 1, which violates the assumption of Lemma 1, the problem transformed by Algorithm 1 still has the same optimal solution as the original problem. As a further remark, let us suppose that the conditions of Lemma 1 were violated, and we assume the extreme case of and vary the value of within the range . In that case, only when , our obtained solution does not satisfy the QoS constraint of Class users. In practice, however, distinct classes would have a greater reception capability difference than .
5 Numerical and simulation results
This section provides comparisons of the proposed random interleaved UEP rateless code design to EEP codes and to other recent UEP rateless codes. The parameters of the different scenarios are described in the corresponding figure captions. Performance of LT codes are evaluated using andor tree analysis while standardized raptor codes are evaluated using Eq. (12). Simulations are also used to confirm the andor tree analysis.
Figs. 2 to 4 compare the proposed UEP design to EEP design for the guaranteed QoS problem when standardized raptor
codes are employed. The minimum transmission overhead is evaluated using the method described in Section 4.4. For simplicity, only two layers are considered. The dimension of the standardized raptor code used in layer is . The inefficiencies incurred by the standardized raptor codes are characterized by the decoding failure probability in Eq. (11) and are small as expected. The optimal selection probability and minimum overhead for the UEP scheme are obtained by the simplified method described in Section 4 for solving Problem , i.e., by assuming that all inequality constraints are active. All results shown in Figs. 2 to 4 were verified to satisfy the KKT optimality conditions. To achieve EEP, the ratio is fixed to . Fig. 2 shows minimum transmission overhead, , required for optimized UEP and EEP raptor codes as the ratio between the numbers of bits in the two layers is varied. Fig. 3 compares UEP and EEP as a function of channel reception quality of the first user class, . It can be seen that UEP has a significant advantage over EEP whenever the channel reception qualities of the two classes differ appreciably. Fig. 4 plots minimum transmission overhead, , as a function of selection probability where it is observed that is very sensitive to the choice of . In particular, a nonoptimized allocation scheme may be significantly outperformed by EEP.
To enable comparison with the UEP raptor code from [31] as well as with EWF codes from [34], rather than use raptor codes, we employ iteratively decoded LT codes that have degree distribution [14]
(23) 
for all layers as used for UEP codes in [31] and for the EWF code [34]. That is, for analytical simplicity, no precode is used in any of the schemes. The decoding failure probability on the left side of the constraint functions in Eq. (4) is evaluated as follows: the symbol error probability of Layer for the UEP rateless codes in [31], the EWF code, and the proposed random interleaved scheme are estimated by andor tree analysis and obtained using Eqs. (6) and (7) in [31], Eq. (7) in [33], and Eq. (13) in this paper, respectively. The failure probability of decoding each layer is estimated as .
Parameter optimization of the other schemes can be found in [2] and are not reproduced here. Fig. 5 in [2] provides the minimum transmission overhead required to satisfy all the user constraints of the proposed random interleaved scheme as well as that in [31] using different values of , a parameter that governs the degree of nonuniformity of input symbol selection. Fig. 6 in [2] shows a similar comparison between the proposed scheme and the EWF code. The size of the first window in the EWF code is fixed to the number of symbols in Layer (). Parameter is the probability of choosing the more important first layer during encoding (see [34]). It can be observed that when all schemes are optimized, the proposed random interleaved rateless code performance matches that of [31] as well as [33]. The existence of two local minima in Fig. 6 in [2] is due to the symbol error rates of the more important bits not decreasing monotonically as increases (see Fig. in [34]).
An advantage of EWF codes [33] over those in [31] is flexibility in deploying different degree distributions for different windows. Fig. 5 plots transmission overhead as a function of numbers of symbols in the first layer or window for the three UEP schemes each using LT codes after optimization over their respective parameters where different degree distributions are applied to different EWF code windows as well as to different layers of the proposed UEP scheme.
Degree distributions, chosen for the more important bits (MIB) and less important bits (LIB), are denoted as and , respectively. Degree distribution described by Eq. (5) is used as well as a truncated robust soliton distribution (RSD) , where is the maximum degree, is applied to the MIB for the EWF and proposed random interleaved schemes. The truncated RSD has better error performance compared to at the cost of higher decoding complexity. It can be seen from Fig. 5 that the truncated RSD for the MIB provides a significant performance boost for both schemes. When the same degree distributions are used, the proposed random interleaved scheme matches the performance of existing schemes.
We note that in Fig. 4, = 0.432, indicating that the code performance nearly achieves minimum overhead while the code performance shown in Fig. 5 does not come close to the minimum overhead. This difference is mainly attributable to the use of standardized raptor codes, which includes a high performance precode as well as efficient maximum likelihood (ML) decoding in contrast to the iterative decoding used for Fig. 5. It can be argued that the performance of existing UEP designs in [31] and [33] can similarly benefit from a precode and ML decoding. However, ML decoding complexity of the proposed randominterleaved design would likely be lower due to a lower dimension decoding matrix obtained from separatelayer decoding. In addition, the code structure and generating matrix of systematic standardized raptor code implementation has been highly optimized, including the decoding schedule in the code constraint processor [49]. To the authors’ best knowledge, such techniques have not been applied to EWF codes, which may also be complicated by their overlapping structure.
Fig. 6 shows source reconstruction quality, in terms of PSNR, of the proposed random interleaved and EWF schemes for the besteffort QoS formulations of Problems and . Transmission of H.264 SVC coded CIF Stefan video sequence [34] is performed in two layers, with the first (base) layer containing symbols and all enhancement streams comprising the second layer with symbols.
Successfully decoding the first layer provides a PSNR of 25.79 dB while decoding both layers provides a PSNR of 40.28 dB. Performance is plotted as average PSNR versus selection probability for the proposed random interleaved scheme and the first window selection probability of the EWF code. Given or , average PSNR is obtained numerically by setting and substituting the corresponding decoding failure probabilities into (7) and (8). It should be noted that selection probabilities and for the two different schemes are not directly comparable. For the crossmarked and starmarked curves, we have used the LT code with an iterative decoder and degree distribution applied to all windows and layers. For these parameters, both the proposed random interleaved and EWF schemes provide a maximum average PSNR of around 32.4 dB. For Problem 3, the feasible regions of selection probabilities and are obtained by checking constraints (10). We note that for Problem 3, the maximum achievable average PSNRs remain the same since both optimal operating points of the proposed UEP scheme and the EWF code lie inside the feasible regions. The diamondmarked curve shows the results when standardized raptor codes are employed for the proposed random interleaved UEP scheme. A maximum average PSNR of 40.28 dB can be achieved for , which, as expected, is significantly higher than the other two LT coded curves. We can also observe from Fig. 6 that different choices of result in significant differences in average PSNR, showing the need for optimization. Finally, we observe that the steep performance curve of standardized raptor codes results in only two obtained PSNR values.
The above LT coding / iterative decoding results are obtained using andor tree analysis which assumes infinite block length. As a check, simulation of LT codes with degree distribution and iterative belief propagation (BP) decoding [39] [14] are provided in Fig 7. Layer selection parameter obtained from Eqs. (12) and (13) determine the constraints in Problem 1.0. The horizontal axis depicting transmission overhead includes the minimum overhead achieved by andor tree analysis (), as well as (1.525) and (1.575) greater than the minimum. The resulting PSNR for each user class is computed for each realization. The vertical axis shows the relative frequency that the PSNR is larger than the desired threshold () for each of the two user classes. It can be seen from the left side of Fig. 7 that the simulation results closely match the andor tree analysis. The probability of reaching target PSNR in simulation is very close to the desired probability threshold . Also, by increasing the overhead to , a higher probability in reaching target QoS can be obtained.
6 Conclusions
A randomly interleaved rateless coder for scalable multimedia multicasting systems with heterogeneous users is optimized for guaranteed and besteffort QoS. The resulting design achieves unequal error protection. Further, guaranteeing QoS is shown to be a convex optimization problem, which can be solved analytically in practical scenarios. Numerical results show the transmission overhead required for the optimized proposed UEP rateless codes to be significantly less than that for EEP design and at least as low as recent optimized EWF and nonuniformselection UEP rateless code designs. Significant gains for the proposed UEP scheme can be obtained by employing standardized raptor codes. For example, in the besteffort QoS example in Fig. 6, the maximum achievable average PSNR using the proposed design employing standardized raptor codes is about dB higher than that of either the proposed or EWF designs based on LT codes with iterative decoding.
We prove the last part of Lemma 1, i.e., after performing Algorithm 1, transmission overhead cannot be further reduced with additional layer partitioning and selection probability reassignment. Let Scheme A denote the sourcetochannel layer mapping produced by Algorithm 1 and denote Scheme B as one which further partitions Layer into Layers and with dimensions and , respectively. Denote the resulting optimal selection probabilities for Scheme B which minimize the transmission overhead as and for Layers and , respectively. We now show that the minimum required transmission overhead is no larger by using Scheme A with selection probability assigned to Layer . For the same number of total transmitted symbols , the effective average raptor code rates for Layer in Scheme , Layer in Scheme and Layer in Scheme are , and , respectively. Without loss of generality, we assume . Then it can be shown that . As the decoding failure probability of the raptor codes is monotonically increasing with code rate for the same user class, we have for any class index , where is the same decoding failure probability function as defined in (4). This means that for the same number of transmitted symbols, the original mapping scheme (Scheme A) has higher probability of successfully decoding all the symbols in Layer than Scheme B for all user classes. Therefore, for the same QoS constraints described by (4), Scheme A requires less minimum transmission overhead compared to Scheme B. Finally, raptor codes with larger dimension have better performance for the same code rate, which also implies no further layer partitioning.
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Photo_Cao.epsYu Cao received his B.S. degree from Department of Electronic Engineering, Tsinghua University, Beijing, China and his M.S. and Ph.D. degrees from Department of Electrical and Computer Engineering, Queen’s University, Kingston, Canada. From 2011 to 2012, he was a postdoctoral fellow at Queen’s University, Canada. Since 2012, he has been working as a research engineer at Huawei Canada Research Center, Ottawa, Ontario, Canada. His research interests are forward error correction codes and optimization for wireless communications. His current research focus is air interface design for next generation radio access networks.
Photo_Blostein.epsSteven D. Blostein received his B.S. degree in Electrical Engineering from Cornell University, Ithaca, NY and the M.S. and Ph.D. degrees in Electrical and Computer Engineering from the University of Illinois, UrbanaChampaign. He has been with the Department of Electrical and Computer Engineering Queen’s University since 1988. He was MultiRate Wireless Data Access Major Project leader in the Canadian Institute for Telecommunications Research, a consultant to industry and government in image compression, target tracking, radar imaging and wireless communications, and spent sabbatical leaves at Lockheed Martin Electronic Systems and at Communications Research Centre of Industry Canada. His interests lie in the application of signal processing to wireless communications systems, including synchronization, network MIMO and physical layer optimization for multimedia transmission. He has been a member of the Samsung 4G Wireless Forum and an invited distinguished speaker. He served as Department Head, IEEE Kingston Section Chair, Biennial Symposium on Communications Chair, and Editor for IEEE Transactions on Image Processing and IEEE Transactions on Wireless Communications. He is a registered Professional Engineer in Ontario.
Photo_Chan.epsWaiYip Chan, also known as Geoffrey Chan, received his B.Eng. and M.Eng. degrees from Carleton University, Ottawa, and his Ph.D. degree from University of California, Santa Barbara, all in Electrical Engineering. He is currently with the Department of Electrical and Computer Engineering, Queen’s University, Canada. He has held positions with the Communications Research Centre, Bell Northern Research (Nortel), McGill University, and Illinois Institute of Technology. His research interests are in speech processing and multimedia coding and communications. He is an associate editor of IEEE/ACM Transactions on Audio, Speech, and Language Processing, and of EURASIP Journal on Audio, Speech, and Music Processing. He has helped organize IEEE sponsored conferences on speech coding, image processing, and communications. He held a CAREER Award from the U.S. National Science Foundation.