Sparse code multiple access: An energy efficient uplink ... - IEEE Xplore

Globecom 2014 - Wireless Networking Symposium

Sparse Code Multiple Access: An Energy Efficient Uplink Approach for 5G Wireless Systems Shunqing Zhang, Xiuqiang Xu, Lei Lu, Yiqun Wu, Gaoning He and Yan Chen Huawei Technologies, Co. Ltd., Shanghai, China Email: {zhangshunqing, xuxiuqiang, lulei, wuyiqun, hegaoning, bigbird.chenyan}@huawei.com

Abstract—The rapid traffic growth and ubiquitous access requirements make it essential to explore the next generation (5G) wireless communication networks. In the current 5G research area, non-orthogonal multiple access has been proposed as a paradigm shift of physical layer technologies. Among all the existing non-orthogonal technologies, the recently proposed sparse code multiple access (SCMA) scheme is shown to achieve a better link level performance. In this paper, we extend the study by proposing an unified framework to analyze the energy efficiency of SCMA scheme and a low complexity decoding algorithm which is critical for prototyping. We show through simulation and prototype measurement results that SCMA scheme provides extra multiple access capability with reasonable complexity and energy consumption, and hence, can be regarded as an energy efficient approach for 5G wireless communication systems. Index Terms—5G, sparse code multiple access (SCMA), energy efficiency, non-orthogonal multiple access, low complexity implementation, prototype

I. I NTRODUCTION Wireless communications have been developed for more than three decades, e.g., from the traditional Global System for Mobile communications (GSM) to the newly launched Long Term Evolution Advanced (LTE-A) networks. With huge amount of traffic growth and ubiquitous access requirements in the recent years, people have reached to a common view on exploring the next generation wireless communication networks, which is also known as “5G”. For example, the European Union has initiated an integrated project “Mobile and wireless communications Enablers for the Twenty-twenty Information Society (METIS) [1]” in the seventh framework program (FP7) to lay the foundation of 5G and the U.S. National Science Foundation (NSF) has granted two projects to “gain a deep understanding” of 5G wireless communication, including NYU-Wireless Lab and Auburn University [2]. Multiple access technology, as one of the most important categories in the 5G research, is undergoing a paradigm shift from orthogonal to non-orthogonal based approaches. Low density signature OFDM (LDS-OFDM) [3] can be one example to realize the non-orthogonality. Due to the low density spreading property, we are able to squeeze more users to the limited number of time-frequency resources and maintain reasonable performance in the uplink scenarios. As a generalization of LDS-OFDM, [4] is recently proposed to achieve a better block error rate (BLER) performance in the overloaded OFDM systems. However, in order to make it as a candidate waveform for 5G wireless systems, we

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may need to further address the following two questions: 1) Whether SCMA scheme is an energy efficient approach since green is a key aspect of 5G [5]? 2) Is it possible to implement SCMA scheme in the real hardware environment with the high complexity message-passing decoding algorithm (MPA) [6]? In this paper, we shall provide a comprehensive study on the above questions and the major contributions of this paper are in two-fold as summarized below: • Energy Efficiency Analysis In order to perform the green analysis of SCMA scheme, we extend the traditional point-to-point outage definition to the multiple access scenarios, define an aggregate energy efficiency metric to cope with non-orthogonal utilization of wireless resources and propose a unified framework for energy efficiency analysis in the fading environment. The analytical framework is then applied to uplink SCMA scheme with 6 users under the symmetric conditions. Moreover, we build up a demo system to show the effectiveness of proposed energy efficiency analysis. • Low Complexity Implementation A low complexity logarithm-domain MPA decoding algorithm is also proposed to satisfy the run-time limitation in the prototyping systems. The proposed algorithm is able to turn most of the multiplication and exponent operations into the simple addition and maximization, which saves significant computational time and facilitate the real hardware implementation in the prototyping system. The rest of the paper is organized as follows. Section II describes the transceiver structure of SCMA scheme and the corresponding system models. In Section III, we propose a generalized energy efficiency definition for the non-orthogonal multiple access schemes and analyze the energy efficiency performance of SCMA in the symmetric channel environment. In Section IV, we propose an implementation-friendly decoding algorithm to achieve more than 50% decoding complexity reduction. Simulation as well as prototyping measurement results are shown in Section V to co-verify the above analysis, followed by some concluding remarks in Section VI. II. SCMA S YSTEM M ODEL Consider an uplink multiple access system as shown in Fig. 1 with K users spreading over N resource elements (REs). In the orthogonal scenario, K is less than or equal to N to make sure each user can enjoy an orthogonal resource element, while in the non-orthogonal case, K is greater than

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Codebook 1 00 01 10 0 a 0

detection of X = [x1 x2 . . . xK ] can be written by [4], 11

ˆ = arg X

0 0 0 -b* b* a* 0 0 0 a -a* -b*

b

1

(b )

Channel Coding

User 1

(b2)

Channel Coding

User 2 Encoded bits

SCMA Encoder

Variable Node

X1 X2 X3 X4

SCMA Encoder

Freq. h2

...

User 5

Channel Coding

User 6

Channel Coding Codebook 6 00 01 10 0 0 0 c d* f e c* d 0 0 0

... (b5)

(b6)

SCMA Encoder

SCMA Encoder

When the receiver can perfectly obtain the channel knowledge H, the non-orthogonal multiple access capacity region C(U0 ) can be expressed as [7], . C(U0 ) = C 1 , . . . , C K : C(S) ≤ log 1 +

SCMA Decoder

f* 0

N XX

Fig. 1. Illustration example of a non-orthogonal uplink multiple access system with K users spreading over N resource elements. In this example, we have 6 users spreading over 4 resource elements, and the overloading factor is 150%.

N and the ratio K/N is defined to be the overloading factor [3]. The following illustration and analysis is based on the single transmit and receive antennas case for simplicity and the extension to multiple antennas is straight forward. A. SCMA Transceiver Structure In this part, we briefly introduce the transceiver structure of SCMA using the traditional mathematical representation. Denote B = [b1 b2 . . . bK ] to be the information bits transmitted by K uplink users and xk = [xk1 xk2 . . . xkN ]T to be the transmitted symbols at the k th user1 . An SCMA encoder at the k th user is thus defined to be a one-to-one mapping f k : B k → X k with bk ∈ B k and xk ∈ X k and the cardinalities of B k and X k are given by 2NB , where NB is the number of information bits in bk . Note that due to the sparsity property of SCMA scheme, xk may contain zero symbols. The received signal at the base station y, after passing through a block fading multiple access channel, can be expressed as, y=

K X

Hk xk + z,

(2)

B. Capacity Region and Per-user Outage Definition

Time

MPA Receiver 11 0 e*

p(X|y).

In general, to solve the above problem requires a global search over the joint space of K uplink users X 1 × · · · × X K . Due to the sparsity property of SCMA transmission scheme, the MPA detector [6] can be applied to reduce the decoding complexity, which iteratively updates the belief associated with ˆ has been estimated, we the underlying factor graph. Once X can use the inverse mapping function {(f k )−1 } to recover the original information bits B.

Function Node

h4

max

X∈X 1 ×···×X K

(1)

k=1

where Hk = diag[hk1 hk2 . . . hkN ] is the channel condition between the base station and the k th user, z = [z1 z2 . . . zN ] is the additive white Gaussian noise with zero mean and normalized variance, and H = {Hk } is the collection of channel conditions from all uplink users. Given the received signal y = [y1 y2 . . . yN ] and the channel knowledge H, the joint maximum-a-posteriori (MAP) 1 (·)∗ , (·)T , (·)H denote the matrix/vector conjugate, transpose and Hermitian operations respectively. diag[a] denotes a diagonal matrix with the diagonal elements given by a.

|Hnk |2 Pnk , ∀S ⊆ U0

(3)

k∈S n=1

where Pnk = E[xkn xk,∗ n ] is the average transmission power, U0 = {1, . . . , K} is the collection of all uplink users and we adopt the normalized noise variance assumption for simplicity. Due to the multiple access environment, the traditional point-to-point based outage definition may not be applicable since we need to deal with the capacity region rather than a simple capacity formula. To facilitate the following analysis, we define per-user outage event as follows. Definition 1 (Per-user Outage): Given the channel realization H, per-user outage of the k th user in the multiple access channel I k , is defined as, [n o I R({k} ∪ S k ) > C({k} ∪ S k ), R(S k ) ∈ C(S k ) , Sk

∀S k ⊆ U k , where R(A) denotes the rate requirement vector of user set A, U k = U0 − {k} is the uplink user set excluding user k and I(·) is the indicator function, which equals to one when the inner event is true and zero otherwise. The following assumptions are adopted through the rest of the paper. Firstly, we omitted the channel condition H and the time index in the per-user outage expression and the following analysis whenever appropriate for illustration purpose. Secondly, no power control scheme is applied, i.e., Pnk is not a function of time. Thirdly, we assume block fading environment, i.e. the channel condition H remains static with in a fading block but varies between different fading blocks. III. E NERGY E FFICIENCY A NALYSIS In this section, we extend the traditional definition of energy efficiency (EE) given in [8] to the multiple access scenarios and propose a unified framework thereafter to analyze the uplink SCMA transmission scheme under symmetric conditions.

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Definition 2 (Aggregate EE): Given the channel realization H and the rate requirements (R1 , . . . , RK ), we define the aggregate energy efficiency EE(H) to be the sum throughput of all uplink users over the total power consumption in the multiple access systems, which can be expressed as, P k k k∈U0 R (1 − I ) P (4) EE(H) = P N k )/αk + P k ( P + P BS sta n k∈U0 n=1 k where αk and Psta are the power amplifier efficiency and the static power consumption of user k respectively, and PBS is the power consumption at the base station (receiver) side. By averaging over all the possible realization of fading blocks, the average aggregate EE is given by, EE = EH EE(H) (5) P k k EH R (1 − I ) P k∈U0 = P (6) N k k + Pk ( P )/α sta + PBS k∈U0 n=1 n P k k k∈U0 R (1 − Pout ) = P (7) PN k )/αk + P k ( P sta + PBS k∈U0 n=1 n

where we have applied the assumption that Pnk is not a k is the per-user function of time in the second equality and Pout outage probability of user k. Lemma 1 (Per-user Outage Probability): The per-user outk age probability of user k, Pout , can be upper bounded by, hY P X 0 j0 k k Pout ≤ Pr k < 2R − 1 + Pr j ≥ 2 j 0 R k0

j0

S P X i j k − 1 Pr (k ≥ 2R − 1)Pr j ≤ 2 j∈U0 R

(8)

j∈U0

PN i 2 i where i = n=1 |Hn | Pn for all i ∈ U0 , Pr (·) is the probability of the inner event. The summations of j 0 is over all the possible elements S k0 and the production of S k0 is over all the possible S k0 ⊆ U k . Proof: Please refer to Appendix A for the proof. The first term in the right hand side (RHS) of equation (8) represents the event when the rate requirement of user k is beyond the corresponding single user capacity bound and the second term is contributed by the conflict transmission with other multiple access users. Apply Lemma 1, we are able to derive the lower bound of the average aggregate EE by substituting (8) into (7), which facilitates the EE analysis for different multiple access schemes. In the following, we apply the above EE analytical framework to the uplink SCMA transmission scheme. Without loss of generality, the symmetric channel conditions2 kwe consider 2 2 = σH for all resource element n and with EH |Hn | user k. The power model, the transmit power and the rate requirement for different users assumed to be identical, Pare N k k k i.e. αk = α, Psta = Psta , n=1 Pn = P , R = R for 2 In the practical systems, the channel statistics for different uplink users may not be identical. However, we can use the similar approach to derive the EE performance with more complicated notations.

all user k. In addition, each information will be spread into two resource elements as shown in Fig. 1 with equal power such that Pnk = P/2 when xkn 6= 0. With all the above simplified notations, we can derive the average aggregate EE performance for the uplink SCMA scheme and summarize the main results by the following theorem. Theorem 1 (Average Aggregate EE): The average aggregate EE for the uplink SCMA scheme under the symmetric channel condition and identical user assumption is given by3 , i K−1 h η2K (KR) R 1 − η2 (R) 1 − 1 − η2 (R) EE ≥ (9) P/α + Psta + PBS /K where Cnk denotes n choose k operation and ηs (r) = R v u−1 r t e−t dt denoting the incomγ(s, 2P σ−1 2 ) with γ(u, v) = 0 H plete Gamma function. Proof: Please refer to Appendix B for the proof. Theorem 1 establishes the first order relationship between the EE performance of the uplink SCMA scheme and the corresponding network parameters, such as the channel condition and the transmit power. From Theorem 1, we show that the average aggregate EE can be improved by sharing the common base station power consumption PBS while maintaining the similar system throughput when η2K (KR) is small. IV. L OW C OMPLEXITY I MPLEMENTATION To verify the above analytical EE results, we shall implement and test the uplink SCMA transmission scheme in the prototype systems. In this section, we briefly introduce the original MPA algorithms and point out the implementation issues for prototype. A low complexity decoding algorithm is then proposed with significant complexity reduction. In the MPA representation, the observed signals are denoted by function nodes (FNs) and the candidate symbols are denoted by variable nodes (VNs). The basic process of the MPA algorithm is to change the probability distribution of the candidate symbols (VNs) iteratively by using the observed information (FNs). Mathematically, the MPA decoding process can be written by, XX k i j (l) (l−1) i (l−1) j e−d(H,ˆxn ,ˆxn ,ˆxn ) PV i Fn (ˆ PFn V k (ˆ xkn ) = xn )PV j Fn (ˆ xn ), x în

(l)

PV k Fn (ˆ xkn ) =

x ˆjn (l) PFn V k (ˆ xkn ) , P (l) PFn V k (ˆ xkn ) x ˆk n

where l is the iteration index, d(H, x ˆkn , x în , x ˆjn ) = |yn − k k i i j j 2 (hn x ˆn + hn x ˆn + hn x ˆn )| is the Euclidean distance between the observed signal yn and the estimated transmit symbols x ˆkn , x în , x ˆjn given the channel condition H, and the summation k of x ˆn , x în , x ˆjn is over all the possible candidate symbols in k i Xn , Xn and Xnj respectively. Note that we can set the 3 Note that the lower bound of average aggregate EE in Theorem 1 is the theoretical results with ideal assumptions only. In the practical system, the actual value of average aggregate EE may even be smaller than the analytical lower bound due to the throughput loss caused by other factors (such as imperfect channel estimation, time/frequency offset and RF imperfectness).

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initial condition for the iteration process with equal proba(0) (0) bilities assumption, e.g., PV k Fn (ˆ xkn ) = 1/|Xnk |, PV i Fn (ˆ xin ) = (0)

1/|Xni |, PV j Fn (ˆ xjn ) = 1/|Xnj |. Although the MPA algorithm mentioned above is sufficient for software demonstration, the prototype system is still suffering from the following issue: 1) The distance calculation of d(·) involves exponential operations, which require huge size of look-up table in the prototype implementation for efficient computation; 2) The dynamic range of the term k i j e−d(H,ˆxn ,ˆxn ,ˆxn ) can be quite high, which requires high resolution hardware implementation; 3) Each massage passing process contains significant number of multiplications, which is computational time consuming as well. To solve the above issues, we propose to use the logarithm domain message passing algorithm (Log-MPA) and the iteration process becomes, log,(l) k log,(l−1) i PFn V k (ˆ xn ) = maxj − d(H, x ˆkn , x în , x ˆjn ) + PV i Fn (ˆ xn ) x în ,ˆ xn

log,(l−1)

+PV j Fn log,(l) k PV k Fn (ˆ xn )

=

(ˆ xjn ) ,

log,(l) k xn ), PFn V k (ˆ

(10) (11)

where we have applied the Jacobian logarithm log(ea + eb ) ≈ max(a, b) in the simplificationand eliminate the normalization operation from VN to FN4 . Fig. 2 shows the complexity comparison between the traditional MPA and the proposed Log-MPA algorithms when the number of iterations l = 5. The codebook size |X k | is equal to 4 for all k and other parameters are chosen to be the same as described in Section III. The upper half compares the decoding complexities in terms of operation numbers, where Log-MPA algorithm saves more than 90% multiplication and completely eliminates the exponent calculations. The lower half is the tested running time results of two algorithms. Although Log-MPA algorithm increases the number of addition operations, the running time saving from the reduction of multiplication is much more significant. Moreover, due to the hardware efficiency difference in calculating the exponent and the maximum operations, Log-MPA algorithm saves extra 20% of running time. V. N UMERICAL R ESULTS In this section, we verify the EE performance of the uplink SCMA scheme via the software simulation (MATLAB) and the prototype measurement. The prototype system is built based on the software-defined radio (SDR) concept. For the base station side, the baseband signal processing is supported by programming the Huawei Tecal RH2288 server and the radio frequency (RF) part is utilized by the Huawei commercial product RRU3232, while for the user side, we use the MacBook Pro ME294CH/A to model the baseband which connects to a mobile RF module. They are all connected to the 4 The aim of the normalization operation from VN to FN is to control log,(l) the dynamic range of {PV k F } due to the existance of the exponential term k

i

j

Fig. 2. Complexity comparison between the traditional MPA algorithm and the proposed Log-MPA algorithm. In the upper half, we compare the complexity of two algorithms by the number of operations and in the lower half, we use the running time in the prototype systems to compare the complexity of two algorithms. TABLE I S IMULATION AND P ROTOTYPING PARAMETERS

Modulation Codebook Size No. of Users (K) No. of Spreading REs (N) Channel Coding Coding Rate System Bandwidth FFT/IFFT Size Channel Model Power Model

LTE-A QPSK N.A. 4 N.A.

SCMA N.A. 4 6 4 Turbo Code 1/2 10 MHz 1024 Jakes Pedestrian-B Psta = 44w, PBS = 300w

channel emulator (Anite EB Propsim@F32) directly to model the uplink environment via RF cables as shown in Fig. 3. Key parameters for simulation and prototyping are listed in Table I. Average BLER Performance comparison between SCMA and LTE-A systems5 is shown in Fig. 4, where the red curves corresponds to the baseline system (LTE-A) and the blue curves corresponds to the uplink SCMA scheme. The dotted curves are the theoretical bounds for LTE-A and SCMA schemes, and the solid curves are BLER performance under

n

xn ,ˆ xn ,ˆ xn ) . However, this requirement is much relaxed if we compute e−d(H,ˆ the probability in the logarithm domain and the normalization operation can be eliminated.

5 For fair comparison, we choose to maintain the same throughput requirement and the same output power for each uplink user in the simulation as well as the prototyping systems.

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Average BLER Performance Comparison between SCMA and LTE−A Systems

0

10

−1

BLER

10

−2

10

−3

10

Single−user Outage Probability (Theoretical) LTE−A (Matlab, Float, Ideal) LTE−A (Matlab, Float, Practical) LTE−A (Matlab, Fixed, Practical) LTE−A (Measurement) Per−user Outage Probability (Theoretical, Lemma 1) SCMA (Matlab, MPA, Float, Ideal) SCMA (Matlab, MPA, Float, Practical) SCMA (Matlab, Log−MPA, Float, Practical) SCMA (Matlab, Log−MPA, Fixed, Practical) SCMA (Measurement)

−4

10

−2

0

2

4 Es/No (dB)

6

8

10

Fig. 4. Average BLER performance comparison between SCMA and LTE-A systems. The per-user outage probability bound given by Lemma 1 (dotted blue) provides accurate BLER approximation of the ideal uplink SCMA schemes (solid blue). Practical system considerations including channel estimation and time/frequency offest reflect 2dB loss in BLER performance for both LTE-A (dashed red) and SCMA systems (dashed blue). Curves with stars are the measurement results from the prototype system.

Fig. 3. Hardware environment illustration of the prototype system. The upper half is the base station with RRU3232 as the RF part and RH2288 server as the baseband part. The lower half is the channel emulator and 6 users with Macbook as the baseband part and the mini broad as the RF part.

the ideal system assumptions, including the ideal channel estimation and perfect time/frequency synchronization. Simulation results show that the per-user outage probability bound as derived in Lemma 1 provides accurate BLER approximation of the ideal uplink SCMA schemes. In Fig. 4, BLER performance under the practical system considerations, including the imperfect channel estimation and time/frequency offset, are labeled with dashed curves, which is around 2dB worse than the corresponding ideal case. From those curves, we show the negligible performance loss caused by 16-bit fixed-point quantization (circle and lower triangle) and by replacing MPA with low complexity LogMPA algorithm (circle and upper triangle). Curves with stars are the BLER measurement results from the prototype system, which shows less than 1dB performance gap if compared with the fixed-point simulation results6 . Note that for the SCMA systems, we can actually aggregate 50% more users 6 In Fig. 4, we can see the error floor of BLER curve at around 10−3 for prototyping results. This is actually due to the joint effect from the insufficient quantization bits, the imperfect channel estimation error and the background noise caused by the practical hardware systems.

in the uplink directions with marginal per-user performance loss compared with LTE-A systems. Fig. 5 provides the average aggregate EE performance comparison for SCMA and LTE-A systems, where the dashed curves are the analytical results and the solid curves are the measurement results. From simulation as well as measurement results, we show that: 1) the proposed analytical framework is able to predict the average aggregate EE behavior for the uplink SCMA and LTE-A systems (especially in the high SNR regime); 2) when the numbers of users in SCMA and LTEA systems are equal, the average aggregate EE of LTE-A system is relatively better due to the orthogonal transmission nature; 3) SCMA scheme can support extra number of users with the help of the non-orthogonal transmission and the average aggregate EE can therefore be improved by sharing the common BS power consumption PBS , which matches well with the analytical results of Theorem 1. VI. C ONCLUSION In summary, we have proposed an analytical framework for the energy efficiency analysis of the non-orthogonal multiple access systems, which is then utilized to show the energy effectiveness of the uplink SCMA scheme. We propose a low complexity algorithm to deal with the message passing decoding and make the hardware implementation much more friendly. To verify the analytical results, we provide some numerical results with MATLAB simulation and establish a SDR-based prototype system using the proposed low complexity Log-MPA decoding algorithm. Simulation and measurement results show that SCMA scheme provides extra multiple access capability with reasonable complexity and

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Combining the above cases, we have the expression of the per-user outage probability as shown in Lemma 1.

Average Aggregate EE Comparison between SCMA and LTE−A Systems 55

50

A PPENDIX B P ROOF OF T HEOREM 1

45

Due to the page limit, we provide the sketched proof as follows. Under the symmetric conditions, we can calculate the per-user outage probability term by term as follows. The first term Pr k < 2R − 1 is equal to η2 (R), where η2 (R) k the cumulative distribution function of theQrandom variable P j0 R ≥ evaluated at 2 − 1. The product term S k0 Pr j0 P Q j0 R P (j ≥ 2 j0 − 1 can be upper bounded by j∈U k r

40

EE (KBit/Joule)

35

30

25

20 LTE−A (Theoretical, K=4) LTE−A (Measurement, K=4) SCMA (Theorem 1, K=6) SCMA (Measurement, K=6) SCMA (Theorem 1, K=4) SCMA (Measurement, K=4)

15

10

5 −2

0

2

4 Es/No (dB)

6

8

j

10

Fig. 5. Average aggregate EE performance comparison between SCMA and LTE-A systems.The dashed curves are the analytical results and the solid curves are the measurement results. With the special structure of SCMA scheme, we can aggregate more users in the uplink direction to support the future massive connectivity requirements for 5G systems.

energy consumption, and hence, can be regarded as an energy efficient approach for 5G wireless communication systems. A PPENDIX A P ROOF OF L EMMA 1

k

2R − 1), which is equal to (1 − η2 (R))|U | since j and k follow the independent and identical χ2 distribution with degrees of freedom P equal j to 4 for all k and j. The last term P j Pr ( j∈U0 ≤ 2 j∈U0 R ) is equal to η2K (KR). Hence, the upper bound of the per-user outage probability is given by, |U k | k 1 − η2 (R) η2K (KR) Pout ≤ η2 (R) + 1 − η2 (R) Substitute the above result into (7), we have k |U0 |R 1 − Pout EE = |U0 |(P/α + Psta ) + PBS h i K−1 R (1 − η2 (R)) 1 − 1 − η2 (R) η2K (KR) ≥ . P/α + Psta + PBS /K ACKNOWLEDGEMENT

To prove Lemma 1, we first separate the original outage event into different cases according to the cardinality of S k . Case 1: |S k | = 0 ⇒ R({k}) > C({k}). The outage condition becomes log(1 + k ) < Rk , and the outage probability k can be described by P r(k < 2R − 1). Case 2 (m ≥ 2): |S k | = m−1 ⇒ R({k}∪S k ) > C({k}∪ k S ), R({k}) ≤ C({k}), R(U k ) ∈ C(U k ). To calculate the probability case by case is quite tedious and we consider the union event for all S k ⊆ U k and |S k | ≥ 1, where the mathematical representation is ∪S k ⊆U k ,|S k |≥1 {R({k}∪S k ) C({k} ∪ S k )}, R({k}) ≤ C({k}), R(U k ) ∈ C(U k ). The corresponding outage probability is thus given by Pr R({k}) ≤ C({k}) Pr R(U k ) ∈ C(U k ) 1 − Pr ∪ {R({k} ∪ S k ) ≤ C({k} ∪ S k )}|R({k}) ≤ C({k}), R(U k ) ∈ C(U k ) . We now calculate the outage probability term by term as follows. The first term Pr R({k}) ≤ C({k}) is equal to Pr (k ≥ k 2R − 1), and the second term Pr R(U k ) ∈ C(U k ) is equal P 0 Q P 0 Rj to S k0 ⊆U k Pr ( j 0 ∈S k0 j ≥ 2 j0 ∈Sk0 − 1). Regarding S the third term, we note that the event {R({k} ∪ S k ) ≤ C({k} ∪ S k )}|R({k}) ≤ C({k}), R(U k ) ∈ C(U k ) is equivalent to R(U0 ) ∈ C(U0 )|R({k}) ≤ C({k}), R(U k ) ∈ C(U k ) and the corresponding probability can be upper bounded by 1 − Pr (R(U0 ) ∈PC(U0 )) ≤ 1 − Pr (R(U0 ) ≤ C(U0 )) = j P Pr ( j∈U0 j ≤ 2 j∈U0 R ).

This paper is in part supported by the National High Technology Research and Development Program of China (863 Program No. 2012AA011400) and the National Basic Research Program of China (973 Program No. 2012CB316000). R EFERENCES [1] A. Osseiran and et al., “The METIS 2020 Project – Laying the foundation of 5G”, METIS 2020, Nov. 2012. [Online]. Available: https://www.metis2020.com/ [2] G. Anthes, “NSF Grant to Help Point Way to 5G Wireless”, ACM News, Nov. 2013. [Online]. Available: http://cacm.acm.org/news/169528nsf-grant-to-help-point-way-to-5g-wireless/fulltext [3] R. Razavi, M. AL-Imari, M. A. Imran, R. Hoshyar, and D. Chen, “On Receiver Design for Uplink Low Density Signature OFDM (LDSOFDM),” IEEE Trans. Commun., vol. 60, no. 11, pp. 3499 – 3508, Nov. 2012. [4] H. Nikopour and H. Baligh, “Sparse Code Multiple Access,” in IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), London, Sep. 2013, pp. 332 – 336. [5] “5G Radio Access – Research and Vision”, Ericsson White Paper, June 2013. [Online]. Available: http://www.ericsson.com/news/1306255g-radio-access-research-and-vision 244129228 c [6] F. R. Kschischang and B. J. Frey, “Iterative Decoding of Compound Codes by Probability Propagation in Graphical Models,” vol. 16, no. 2, pp. 219 – 230, Feb. 1998. [7] R. Razavi, R. Hoshyar, M. A. Imran, and Y. Wang, “Information Theoretic Analysis of LDS Scheme,” IEEE Commun. Lett., vol. 15, no. 8, pp. 798 – 800, Aug. 2011. [8] S. Zhang, Y. Chen, and S. Xu, “Improving Energy Efficiency through Bandwidth, Power, and Adaptive Modulation,” in IEEE Vehicular Technology Conference Fall (VTC-Fall), Ottawa, Sep. 2010, pp. 1 – 5.

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