Optimal Multiplexed Hierarchical Modulation for Unequal Error Protection of Progressive Bit Streams

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Title Optimal Multiplexed Hierarchical Modulation for Unequal Error Protection of Progressive Bit Streams

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Authors Chang, S.-H. Rim, M Cosman, P C et al.

Publication Date 2009-11-01

Peer reviewed

eScholarship.org Powered by the California Digital Library University of California Optimal Multiplexed Hierarchical Modulation for Unequal Error Protection of Progressive Bit Streams

Seok-Ho Chang†, Minjoong Rim‡, Pamela C. Cosman† and Laurence B. Milstein† †ECE Dept., University of California at San Diego, La Jolla, CA 92093, USA ‡ICE Dept., Dongguk University, Seoul, Korea

Abstract—Progressive image and scalable video have gradual quality. Since these progressive transmissions have gradual differences of importance in their bitstreams, which can benefit differences of importance in their bitstreams, a large number from multiple levels of unequal error protection (UEP). Though of error protection levels are required. However, hierarchical hierarchical modulation has been intensively studied as an UEP approach for digital broadcasting and multimedia transmission, modulation can achieve only a limited number of UEP levels methods of achieving a large number of UEP levels have rarely for a given constellation size. For example, hierarchical 16 been studied. In this paper, we propose a multilevel UEP system QAM provides two levels of UEP, and hierarchical 64 QAM using multiplexed hierarchical quadrature amplitude modulation yields at most three levels [11]. In the DVB-T standard, video (QAM) for progressive transmission over mobile radio channels. data encoded by MPEG-2 consists of two different layers, We suggest a specific way of multiplexing, and prove that multiple levels of UEP are achieved by the suggested method. When and thus the use of hierarchical 16 or 64 QAM meets the the BER is dominated by the minimum Euclidian distance, we required number of UEP levels. However, if scalable video derive an optimal multiplexing approach which minimizes both is to be incorporated in a digital video broadcasting system, the average and peak powers. An asymmetric hierarchical QAM hierarchical 16 or 64 QAM may not meet the system needs. which reduces the peak-to-average power ratio (PAPR) without Most of the work about hierarchical modulation up to now performance loss is also proposed. Numerical results show that the performance of progressive transmission over Rayleigh fading has considered only two layered source coding, and to the channels is significantly enhanced by the proposed UEP systems. best of our knowledge, methods of achieving an arbitrarily large number of UEP levels have not been studied. I. INTRODUCTION In this paper, we propose a multilevel UEP system using When a communication system transmits messages over multiplexed hierarchical modulation for progressive transmis- mobile radio channels, they are subject to errors, in part be- sion over mobile radio channels. We propose a specific way cause mobile channels typically exhibit time-variant channel- of multiplexing, and prove that multiple levels of UEP are quality fluctuations. For two-way communication links, these achieved by the proposed method. These results are presented effects can be mitigated using adaptive methods. However, in Section II. When the BER is dominated by the minimum the adaptive schemes require a reliable feedback link from the Euclidian distance, we derive an optimal multiplexing ap- receiver to the transmitter. Moreover, for a one-way broadcast proach which minimizes both the average and peak powers, as system, those schemes are not appropriate because of the presented in Section III. While the suggested methods achieve nature of broadcasting. When adaptive schemes cannot be multilevel UEP, the PAPR typically will be increased when used, the way to ensure communications is to classify the data constellations having distinct minimum distances are time- into multiple classes with unequal error protection (UEP). multiplexed. To mitigate this effect, an asymmetric hierarchical Since the theoretical and conceptual basis for UEP was QAM constellation, which reduces the PAPR without perfor- initiated by Cover [1], much of the work has shown that mance loss, is proposed in Section IV. It is also shown that one method of achieving UEP is based on a constellation of asymmetric hierarchical QAM can provide multilevel UEP nonuniformly spaced signal points [2]–[5], which is called a even when multiplexed constellations need to have constant hierarchical constellation. In this constellation, more important power. In Section V, the performance of the suggested UEP bits in a symbol have larger minimum Euclidian distance than system for the transmission of progressive images is analyzed less important bits. Hierarchical constellations were intensively in terms of the expected distortion, and Section VI presents studied for digital broadcasting systems and multimedia trans- numerical results of performance analysis. mission [2][4]–[7]. Moreover, the Digital Video Broadcasting (DVB-T) standard [8], which is now commercially available, II. MULTILEVEL UEP BASED ON MULTIPLEXING incorporated hierarchical QAM for layered video data trans- HIERARCHICAL QAM CONSTELLATIONS mission. Progressive image and scalable video encoders [9][10], A. Hierarchical 16 QAM Constellation which are expected to have more prominence in the future, Fig. 1 shows a hierarchical 16 QAM constellation with employ a progressive mode of transmission such that as more Gray coded bit mapping [8]. The two most significant bits bits are transmitted, the source can be reconstructed with better (MSBs), i1 and q1, determine one of the four clusters, and

978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings. Data classes i1 00 11 Hierarchical i 2 01 10Cluster Channel 16 QAM Class 1 Encoder 1 (1) Hierarchical symbols q1 q2 (1) 16 QAM Constellation Average power 1 Mapper 1 Channel 0 0 Class 2 (5) Distance factor 1 Encoder 2 (2) d L 0 1 Channel Class 3 Encoder 3 (3) Hierarchical (2) 16 QAM d Constellation M Average power 2 Mapper 2 Channel Class 4 (6) Distance factor 2 Encoder 4 Less (4) Multiplexer 1 1 important Channel Class 5 Encoder 5 (5) Hierarchical (3) 16 QAM 1 0 Constellation Average power 3 Mapper 3 Channel Class 6 (7) Distance factor 3 Encoder 6 (6) d M d L Channel Class 7 Encoder 7 (7) Hierarchical (4) 16 QAM Constellation Average power 4 Mapper 4 Fig. 1. Hierarchical 16 QAM constellation. Channel Class 8 (8) Distance factor 4 Encoder 8 (8)

their minimum Euclidian distance is dM . The two least sig- Fig. 2. The multilevel UEP system using multiplexed hierarchical 16 QAM niﬁcant bits (LSBs), i2 and q2, determine which of the four constellations based on Theorem 1. signal points within the cluster is chosen, and their minimum Euclidian distance is d . The distance ratio α = d /d (> 1) L M L that 2N levels of UEP can be achieved by multiplexing N determines how much more the MSBs are protected against hierarchical 16 QAM constellations. errors than are the LSBs. Since hierarchical 16 QAM has one Theorem 1: For N hierarchical 16 QAM constellations, embedded QPSK subconstellation consisting of four clusters P and P , given by (3), satisfy and provides two levels of UEP, it is denoted by 4/16 QAM. M,i L,i We consider multiplexing N hierarchical 16 QAM constel- PM,1

N dM,1 >dM,2 > ···>dM,N >dL,1 >dL,2 > ···>dL,N . S 1 S avg = N avg,i (1) (5) i=1 Proof: The proof of this theorem as well as the proofs of all other results are not included here due to space limitations, where Savg,i is the average power per symbol of constellation but they can be found in [12]. i. Savg,i is given by Fig. 2 depicts the multilevel UEP system using multiplexed d 2 d 2 d2 hierarchical 16 QAM constellations based on Theorem 1 for S M,i M,i d M,i d d d2 avg,i = + + L,i = + M,i L,i+ L,i eight data classes (N =4). 2 2 2 2 (2) B.Hierarchical 2 K (K ≥ 3) QAM Constellation N 2K K ≥ where dM,i and dL,i are minimum distances for the MSBs and Next, we consider multiplexing hierarchical 2 ( i d ≤ n ≤ K LSBs of constellation , respectively. The BERs of the MSBs 3) QAM constellations. Let Mn,i (1 ) denote the and LSBs of hierarchical 16 QAM constellation i, denoted by minimum distance for the nth MSBs of constellation i (1 ≤ i ≤ N PM,i and PL,i, respectively, are given by [11] ). Theorem 2: If the SNR of interest for the nth MSBs d γ ≤ n ≤ K P 1Q M,i 2 s (2 ) is sufﬁciently large so that the probability of the M,i = d 1 d 2 2 Savg noise exceeding the Euclidian distance of Mn−1,i + 2 Mn,i is 1 insigniﬁcant compared to that of the noise exceeding dM ,i d γ 2 n 1 M,i 2 s (note that the distance ratio of the hierarchical constellation, + Q + dL,i 2 2 Savg d /d is greater than unity), the BER of the nth Mn−1,i Mn,i MSBs (2 ≤ n ≤ K), P , becomes d γ d γ Mn,i P Q L,i 2 s 1Q d L,i 2 s L,i = S + M,i + S P app = 2 avg 2 2 avg ⎧Mn,i − − 2K n−1 1 d K +2K q 2 ⎪ Q Mn,i p d γs d γ ⎪ =0 2K−n 2 + = +1 2K−q+1 M ,i −1Q d 3 L,i 2 s ⎪ p q n q Savg M,i + S (3) ⎨ ≤ n ≤ K − 2 2 avg for 2 1 d 2 1 d 2 (6) ⎪Q MK ,i γs Q d MK ,i γs ⎪ 2 + 2 MK−1,i + 2 where γs is the signal-to-noise ratio (SNR) per symbol, and ⎪ Savg Savg √ ∞ 2 ⎩⎪ Q x / π e−y /2dy n K ( )=1 2 x . The following theorem states for =

978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings. Data classes x x Hierarchical where denotes the largest integer less than or equal to . Channel 16 QAM Class 1 Encoder 1 symbols n (1) Hierarchical Note that for the MSBs (i.e., =1), the top line of (6) is the (1) 16 QAM app Constellation n P Average power 1 exact BER expression when is set to unity (i.e., = Mapper 1 M1,i Channel Class 2 (8) Distance factor 1 P Encoder 2 M1,i). (2) Channel 2K Class 3 N Encoder 3 Theorem 3: For hierarchical 2 QAM constellations, (3) Hierarchical app (2) 16 QAM P , given by (6), satisfy Constellation M ,i Average power 2 n Mapper 2 Channel Class 4 (7) Distance factor 2 app app app app Encoder 4 P < ···

···>d >d > ···>d > ··· Encoder 6 M1,1 M1,N M2,1 M2,N (6) >d > ···>d MK ,1 MK ,N (7) Channel Class 7 Encoder 7 (7) Hierarchical (4) 16 QAM Constellation Average power 4 Mapper 4 N Channel Theorem 3 tells us that, by multiplexing hierarchical Class 8 (5) Distance factor 4 Encoder 8 (8) 22K (K ≥ 3) QAM constellations having minimum distances satisfying (7), KN levels of UEP are achieved under the assumption that the SNR of interest for the nth MSBs Fig. 3. The multilevel UEP system using multiplexed hierarchical 16 QAM (2 ≤ n ≤ K) is reasonably large so that the condition of constellations based on Theorems 6 and 7 . Theorem 2 is satisfied. III. OPTIMAL MULTIPLEXING OF HIERARCHICAL QAM Corollary 5: From Theorem 4, regardless of how the mini- CONSTELLATIONS FOR HIGH SNR mum distances are permuted, the BERs given by (10) stay the In this section, we define high SNR as an SNR which is same. sufficiently large so that the BER is dominated by the error Theorem 6: After the distances are permuted as described function term having the minimum Euclidian distance. in Theorem 4, the average power, Savg, given by A. Hierarchical 22J /22K (K>J≥ 1) QAM Constellation N d˜2 We first analyze a hierarchical 4/16 QAM (or 16 QAM) 1 M,i 2 2 2 S d˜ d˜ d˜ J / K avg = + M,i L,i + L,i (11) constellation, as a simple example of the hierarchical 2 2 N 2 QAM constellations [11]. For high SNR, from (3), the BERs i=1 i ≤ i ≤ N of a hierarchical 16 QAM constellation (1 ) are is minimized if and only if distances are permuted such that given by d d M,i is combined with L,N+1−i in the same constellation. That is, 1 dM,i 2γs dL,i 2γs PM,i ≈ Q and PL,i ≈ Q . 2 2 Savg 2 Savg d˜ = d and d˜ = d +1− (1 ≤ i ≤ N). (12) (8) M,i M,i L,i L,N i Theorem 4: Suppose that there are N multiplexed hierarchical 16 QAM constellations, and the minimum distances Corollary 5 and Theorem 6 indicate that the average power satisfying (5) are given. Also, suppose the given minimum is minimized by permuting distances according to (12), while distances can be permuted such that d 1, ··· ,d for the M, M,N the BERs are unchanged for high SNR. MSBs can be arbitrarily combined with d 1, ··· ,d for the L, L,N Next, we consider the peak signal power of the multi- LSBs. After the distances are permuted, the resultant minimum plexed hierarchical constellations. If we assume that all the distances for the MSBs and LSBs of constellation i, denoted constellations are time-multiplexed, the peak power of all the by d˜M,i and d˜L,i, respectively, can be expressed as multiplexed hierarchical constellations, Speak, is given by d˜ = d and d˜ ( ) = d (9) M,i M,i L,π i L,i Speak = max Speak,i1 ≤ i ≤ N (13) where π(i) is the index of the constellation to which dL,i is permuted. Then, with the permuted distances given by (9), we X X have where max[ ] denotes the maximum element of the set , and S is the peak power of a hierarchical constellation i P < ···

978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings. i1 00 1 1 1st cluster Theorem 7: After the distances are permuted as described i2 01 1 0 in Theorem 4, the peak power, Speak, given by 2nd cluster q1 q2 d˜2 3rd cluster M,i 2 S d˜ d˜ d˜ ≤ i ≤ N 00 peak = max +2 M,i L,i +2 L,i 1 A, Q 2 dL 01 (15) A, Q dM is minimized if the distances are permuted according to (12) 11 of Theorem 6. Theorems 6 and 7 tell us that the permutation of the dis- 10 tances that minimizes the average power also, coincidentally, minimizes the peak power. Fig. 3 depicts the multilevel UEP

system using multiplexed hierarchical 16 QAM constellations A, I A, I dM d based on Theorems 6 and 7 for eight data classes (N =4). L It can be shown that the results for hierarchical 4/16 QAM (or 16 QAM) can be generalized to hierarchical Fig. 4. Asymmetric hierarchical 16 QAM constellation . 22J /22K (K>J≥ 1) QAM. dA,I ,dA,I ,dA,Q, dA,Q IV. ASYMMETRIC HIERARCHICAL QAM CONSTELLATION where M,i L,i M,i and L,i are the minimum distances for the inphase MSB and LSB, and quadrature MSB We propose asymmetric hierarchical QAM which reduces and LSB, respectively, and x(i) and y(i) satisfy the PAPR of time-multiplexed hierarchical constellations without performance loss. From here onwards, we refer to con- x(i),y(i) ∈{1, ··· ,N}, ventional hierarchical QAM, which has been presented in {x(i),y(i)|1 ≤ i ≤ N/2} = {1, ··· ,N}. (17) Sections II and III, as symmetric hierarchical QAM, in order to distinguish it from asymmetric hierarchical QAM. With the minimum distances given by (16), the average power A.Hierarchical 22K (K ≥ 2) QAM Constellation and BERs of N/2 multiplexed asymmetric hierarchical 16 For an asymmetric hierarchical 22K QAM, the minimum QAM constellations are the same as those of N multiplexed Euclidian distances for the inphase and quadrature compo- symmetric hierarchical 16 QAM constellations, regardless of nents are different from each other. We present asymmetric the choice of x(i) and y(i) satisfying (17). hierarchical 16 QAM, depicted in Fig. 4, as a simple example. Theorem 9: Suppose that the minimum distances of the N The MSB i1 for the inphase component determines the first multiplexed symmetric hierarchical 16 QAM constellatoins dA,I q cluster, and its minimum distance is M .TheMSB 1 for the satisfy (5) of Theorem 1. Then, with the minimum distances quadrature component determines the second cluster within given by (16), the peak power of all N/2 multiplexed asym- the first cluster that i1 determined, and its minimum distance metric hierarchical 16 QAM constellations is less than that dA,Q i N S is M .TheLSB 2 for the inphase component determines of all multiplexed symmetric hierarchical 16 QAM, peak, dA,I x i the third cluster, and its minimum distance is L , and the given by (13) and (14), regardless of the choice of ( ) and LSB q2 for the quadrature component determines the specific y(i) satisfying (17). signal point within the third cluster, and has minimum distance dA,Q L . Asymmetric hierarchical 16 QAM has three embedded Theorems 8 and 9 tell us that when asymmetric hierarchical subconstellations, and it provides four levels of UEP when 16 QAM is used instead of symmetric hierarchical 16 QAM, dA,I >dA,Q >dA,I >dA,Q M M L L . the PAPR is reduced without performance loss. Note that this In order to provide 2N levels of UEP, we consider multi- result holds for all SNR. plexing N/2 (N is assumed to be even) asymmetric hierar- We now consider the case where it is desirable for the chical 16 QAM constellations. multiplexed hierarchical QAM constellations to have the same Theorem 8: Suppose that there are N multiplexed sym- average power (i.e., constant power), either due to the limited metric hierarchical 16 QAM constellations whose minimum capability of a power amplifier, or for cochannel interference distances are given by dM,1, ··· ,dM,N and dL,1, ··· ,dL,N . control. Also, suppose that there are N/2 asymmetric hierarchical 16 Theorem 10: Suppose that N/2 multiplexed asymmetric hi- QAM constellations, and the minimum distances for the in- erarchical 16 QAM constellations are required to have constant phase and quadrature components of asymmetric hierarchical power, and their minimum distances are given by (16). If x(i) constellation i are the same as those of two distinct sym- and y(i) are chosen as x(i)=i and y(i)=N +1− i metric hierarchical constellations x(i) and y(i), respectively (1 ≤ i ≤ N/2), 2N levels of UEP still can be achieved. (1 ≤ i ≤ N/2). In other words, for 1 ≤ i ≤ N/2, A,I A,I It can be shown that the results for asymmetric hierarchi- d = d ( ) and d = d ( ) M,i M,x i L,i L,x i cal 16 QAM can be generalized to asymmetric hierarchical dA,Q d dA,Q d 2K K ≥ M,i = M,y(i) and L,i = L,y(i) (16) 2 ( 2) QAM.

978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings. V. T HE PERFORMANCE OF THE PROPOSED UEP SYSTEM 35 FOR PROGRESSIVE BITSTREAM TRANSMISSION We analyze the performance of the proposed UEP system 34 for progressive image source transmission over Rayleigh fad- 33 ing channels. We ﬁrst consider the UEP system depicted in 32

Fig. 2. The system takes successive blocks (data classes) of 31 the compressed progressive bitstream, and transforms them 30 into a sequence of channel codewords of ﬁxed length lc with error detection and correction capability. Then, the coded 29 PSNR, dB classes are mapped to the multiplexed symmetric hierarchical 28

16 QAM constellations, whose minimum distances satisfy (5) 27 of Theorem 1 to achieve 2N levels of UEP. At the receiver, Uniformly Spaced QPSK 26 Uniformly Spaced 16QAM if a received class is correctly decoded, then the next class is Single Symm. H−16QAM considered by the decoder. Otherwise, the decoding is stopped. 25 Multiplexed Symm. H−16QAM (16 UEP Levels) r i ≤ Multiplexed Symm. H−16QAM (32 UEP Levels) Let i be an error correction code rate for class (1 24 Multiplexed Symm. H−16QAM (64 UEP Levels) i ≤ N d d ,d 2 ), and i =( M,c(i) L,c(i)) be a pair of minimum 20 22 24 26 28 30 32 34 36 38 40 42 distances of some specific constellation c(i) (1 ≤ c(i) ≤ N) SNR per symbol, dB i p r ,d ,γ to which class is mapped. Let ( i i s) denote the probability of a decoding error of class i. Then, the probability Fig. 5. PSNR performance of UEP system using multiplexed symmetric that no decoding errors occur in the first i classes with an error hierarchical 16 QAM (H-16QAM denotes hierarchical 16 QAM). in the next one, Pc,i (1 ≤ i ≤ 2N − 1), is given by i P p r ,d ,γ − p r ,d ,γ c,i = ( i+1 i+1 s) 1 ( j j s) (18) VI. NUMERICAL RESULTS j=1 We evaluate the performance of the suggested UEP system P p γ ,d ,γ Note that c,0 = ( 1 1 s) is the probability of an error for the progressive source coder SPIHT [9] as an example. P 2N − p r ,d ,γ in the first class, and c,2N = j=1 1 ( j j s) is We provide the results for the standard 8 bits per pixel (bpp) the probability that all 2N classes are correctly decoded. The 512×512 Lena image with a transmission rate of 0.375 bpp. end-to-end performance can be measured by the expected We define a frame as a group of constellation symbols to distortion, E[D], given by which one image bitstream is mapped. It is assumed that the 2N transmitted signal experiences block Rayleigh fading in which E[D]= Pc,iDi (19) channel coefficients are nearly constant over a frame. Lastly, to i=0 compare the image quality, we use peak-signal-to-noise ratio 2/E D E D where Di is the reconstruction error using the first i classes (PSNR) defined as 255 [ ], where [ ] is given by (19). (1 ≤ i ≤ 2N), and D0 is a constant. For the case of We present the PSNR performance for the uncoded case by an uncoded system, Di is given by Di = V (ilc), where numerically evaluating (18) – (22) as follows: We first compute V (x) denotes the operational rate-distortion function of the (21) using the expected distortion, E[D], derived from (18) – p r ,d ,γ source coder. Also, for the uncoded system, ( i i s) can (20) for the block Rayleigh fading channel, and the weight ω γ d , ··· ,d be obtained analytically: function, ( s) , given by (22). Next, with 1 2N (or l d 1, ··· ,d and d 1, ··· ,d ) obtained from (21), we p(r ,d ,γ )=p(d ,γ )=1−{1 − P (d ,γ )} c (20) M, M,N L, L,N i i s i s i i s evaluate PSNR over a range of expected SNRs given by (22). P d γ i where i, a function of i and s, is the BER of data class . Fig. 5 shows the PSNR performance of the multiplexed γ E D Note that for a given SNR of s, [ ] is the conditional symmetric hierarchical 16 QAM as well as that for single sym- expected distortion. In situations when exact SNR informa- metric hierarchical 16 QAM. The PSNR of single symmetric tion is not available at the transmitter, one can find the hierarchical 16 QAM is evaluated in the same way as that d , ··· ,d d , ··· ,d minimum distances, 1 2N (or M,1 M,N and for multiplexed symmetric hierarchical 16 QAM. From Fig. d , ··· ,d L,1 L,N ), which minimize the expected distortion over 5, it is seen that multiplexed symmetric hierarchical 16 QAM a range of expected SNRs using the weighted cost function improves the performance more than does single symmetric ∞ ω γ E D dγ hierarchical 16 QAM. It is also seen that 32 multiplexed 0 ( s) [ ] s arg max ∞ (21) symmetric hierarchical 16 QAM constellations, which provide d ,··· ,d ω(γ )dγ 1 2N 0 s s 64 levels of UEP, have almost saturated performance in this where ω(γs) in [0, 1] is the weight function. For example, evaluation. Note that the performance of N/2 multiplexed ω(γ ) can be given by N s asymmetric hierarchical 16 QAM is the same as that of N , , , γa ≤ γ ≤ γb multiplexed symmetric hierarchical 16 QAM ( =8 16 32), 1 for s s s ω(γs)= (22) as stated by Theorem 8, though the former is not depicted 0, otherwise. here. Table I shows the PAPRs of the multiplexed symmetric or

978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings. studied for digital broadcasting and multimedia transmission, 35 methods of achieving an arbitrarily large number of UEP 34 levels, to the best of our knowledge, have not been studied. 33 In this paper, we proposed a multilevel UEP system using

32 multiplexed hierarchical modulation for progressive transmission over mobile radio channels. We suggested a speciﬁc 31 way of multiplexing N hierarchical 22K QAM constellations 30 (K ≥ 2) and proved that the suggested method achieves KN 29 levels of UEP. When the BER is dominated by the error PSNR, dB 28 function term having the minimum Euclidian distance, we derived an optimal multiplexing approach which minimizes 27 Single Symm. H−16QAM 2J / 2K Multiplexed Asym. H−16QAM (16 UEP Levels) both the average and peak powers for hierarchical 2 2 26 Having Constant Power QAM (K>J≥ 1) constellations (typical examples are Multiplexed Asym. H−16QAM (64 UEP levles) 25 Having Constant Power 4/16 QAM and 4/64 QAM, which are employed in the DVB- Multiplexed Symm. H−16QAM (16 UEP Levels) T standard). We designed an asymmetric hierarchical QAM 24 Multiplexed Symm. H−16QAM (64 UEP Levels) constellation, which reduces the PAPR without performance 20 22 24 26 28 30 32 34 36 38 40 42 SNR per symbol, dB loss. The asymmetric hierarchical QAM also can be used to provide multilevel UEP even when multiplexed constellations are required to have constant power. Numerical results showed Fig. 6. PSNR performance of UEP system using multiplexed asymmetric that the proposed multilevel UEP system based on multiplexed hierarchical 16 QAM having constant power (H-16QAM denotes hierarchical 16 QAM). modulation signiﬁcantly enhances the performance for progressive transmission over Rayleigh fading channels.

ACKNOWLEDGMENT asymmetric hierarchical 16 QAM. For reference, the PAPRs of single symmetric hierarchical 16 QAM and uniformly spaced This work was partially supported by the SETsquared 16 QAM are listed in Table II. From Tables I and II, it is seen UK/US Programme for Applied Collaborative Research and that when symmetric hierarchical 16 QAM constellations are by the US Army Research Office under MURI, grant number time-multiplexed, they have larger PAPR than does uniformly W911NF-04-1-0224. spaced 16 QAM. Table I also shows that the PAPR is reduced REFERENCES when an asymmetric hierarchical constellation is used, as [1] T. Cover, “Broadcast channels,” IEEE Trans. Inform. Theory, vol. IT-18, stated in Theorem 9. pp. 2–14, Jan. 1972. Fig. 6 shows the PSNR performance of the multiplexed [2] K. Ramchandran, A. Ortega, K.M. Uz, and M.Vetterli, “Multiresolution broadcast for digital HDTV using joint source/channel coding,” IEEE J. asymmetric hierarchical 16 QAM constellations having con- Select. Areas Commun., vol. 11, pp. 6–23, Jan. 1993. stant power. It is seen that the performance is degraded when [3] L.-F. Wei, “Coded modulation with unequal error protection,” IEEE constellations are required to have constant power. However, Trans. Commun., vol. 41, pp. 1439–1449, Oct. 1993. [4] A. R. Calderbank and N. Seshadri, “Multilevel codes for unequal error the PAPR problem is completely solved. protection,” IEEE Trans. Inform. Theory, vol. 39, pp. 1234–1248, July 1993. VII. CONCLUSIONS [5] M. Morimoto, H. Harada, M. Okada, and S. Komaki, “A study on power assignment of hierarchical modulation schemes for digital broadcasting,” Progressive image and scalable video encoders employ IEICE Trans. Commun., vol. E77-B, pp. 1495–1500, Dec. 1994. progressive transmission, which benefits from multiple levels [6] M. B. Pursley and J. M. Shea, “Adaptive nonuniform phase-shift-key of UEP. Though hierarchical modulation has been intensively modulation for multimedia traffic in wireless networks,” IEEE J. Select. Areas Commun., vol. 18, pp. 1394–1407, Aug. 2000. [7] Y. Pei and J. W. Modestino, “Cross-Layer Design for Video Transmis- sion over Wireless Rician Slow-Fading Channels Using an Adaptive TABLE I Multiresolution Modulation and Coding Scheme,” EURASIP J. Advances PAPR OF MULTIPLEXED HIERARCHICAL 16 QAM Signal Processing, vol. 2007, article ID 86915. [8] “Digital Video Broadcasting (DVB); Framing Structure, Channel Coding Number of UEP Levels 4 16 64 and Modulation for Digital Terrestrial Television,” ETSI EN 300 744 Multiplexed Symm.H-16QAM 3.31 6.87 9.43 V1.5.1, Nov. 2004. Multiplexed Asym. H-16QAM 1.11 4.18 6.60 [9] A. Said and W. A. Pearlman, “A new, fast, and efficient image codec Multiplexed Asym. H-16QAM 1.11 1.43 1.46 based on set partitioning in hierarchical trees,” IEEE Trans. Circuits Having Constant power Syst. Video Technol., vol. 6, pp. 243–249, Jun. 1996. [10] J. Reichel, H. Schwarz, and M. Wien (eds.), “Scalable Video Coding TABLE II -Working Draft 1,” Joint Video Team of ITU-T VCEG and ISO/IEC MPEG, Doc. JVT-N020, Hong Kong, CN, Jan. 2005. PAPR OF UNIFORMLY SPACED 16 QAM AND SINGLE SYMMETRIC [11] P. K. Vitthaladevuni and M.-S. Alouini, “A recursive algorithm for the HIERARCHICAL 16 QAM exact BER computation of generalized hierarchical QAM constellations,” IEEE Trans. Inform. Theory, vol. 49, pp. 297–307, Jan. 2003. PAPR (dB) [12] S.-H. Chang, M. Rim, P. C. Cosman and L. B. Milstein, “Optimized Uniformly Spaced 16 QAM 2.55 Unequal Error Protection using Multiplexed Hierarchical Modulation,” Single Symmetric Hierarchical 16 QAM 0.90 submitted to IEEE Trans. Inform. Theory.