Efficient iterative decoding of serially concatenated ...

S. Srinivasan, L. M. Davis, “Efficient iterative decoding of serially concatenated multiplexed Alamouti codes,” in Proc. IEEE Int. Conference on Communications (ICC), pp. 1-5, doi: 10.1109/icc.2011.5962689, Kyoto, Japan, June 2011.

Efficient iterative decoding of serially concatenated multiplexed Alamouti codes

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Sudharshan Srinivasan and Linda M. Davis Institute for Telecommunications Research University of South Australia {sudharshan.srinivasan, linda.davis }@unisa.edu.au

Abstract—We present a low complexity iterative decoding method for a multiplexed Alamouti space-time code serially concatenated by a convolutional code. The space-time signalling code consists of a Gray coded 16-QAM mapper and two spatially overlaid Alamouti signalling blocks. A linear complexity decoder is used for decoding the inner space-time code. Ordinarily, an iterative decoding between the inner and outer decoders at the receiver does not improve performance when using a Gray coded modulation scheme. However we show that turbo gains are possible when a multiplexed Alamouti code is used. The low complexity inner decoder is based on Bayesian interference cancelling approach. We present a method of using both the extrinsic and full a posteriori values from the outer decoder to facilitate gains from iterations. The modified iterative decoder presented here favourably compares to a space-time bit interleaved coded modulation (ST-BICM) system providing the same spectral efficiency in terms of complexity-performance tradeoff.

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I. I NTRODUCTION Due to the promise of increased capacity with the introduction of multiple input multiple output (MIMO) systems [1], space-time signalling codes have received wide interest. Space-time signalling codes constructed by orthogonal designs [2] are attractive because the individual symbols can be independently decoded, thereby reducing the decoder complexity compared to non-orthogonal designs. A popular orthogonal space-time signalling code is Alamouti code [3] which applies to a 2-transmit antenna 1-receive antenna system, or simply a 2 × 1 system, and is of rate one. It was shown in [2] that complex orthogonal designs of rate one are not possible for more than two antennas. In order to provide higher spectral efficiencies, non-orthogonal space-time codes can be used over higher order MIMO systems, which, in general, have a high decoding complexity. Alternatively, multiplexed orthogonal codes can be used to provide higher rates and have a relatively low decoding complexity. Multiplexed Alamouti codes belong to this category, wherein individual Alamouti signalling blocks are spatially or code multiplexed. The popular Silver code [4], [5] for a 2 × 2 system is of full rate and belongs to the latter category. In this paper, we consider a spatially multiplexed Alamouti code consisting of two Alamouti blocks, applied on a 4×2 system. Four symbols are transmitted using four antennas in two time slots. In order to improve the overall system error performance, a forward error correction (FEC) code is serially concatenated as an outer code with an interleaver before the inner space-time block code (STBC). Performance gains might then be expected

from iterative decoding between the inner and outer decoders. However, it was shown in [6] that when orthogonal STBCs with Gray mapped constellations are used, ‘turbo’ gains cannot be achieved as they have a flat extrinsic information transfer (EXIT) function. Later, [7] showed that optimized non-Gray modulation mappings are needed to get turbo performance gains. We recognize that the multiplexed Alamouti code is not orthogonal, thus giving scope for improving performance by iterative decoding. We report performance improvements by iterating between the inner STBC decoder and the outer decoder, while using a Gray mapped 16-ary quadrature amplitude modulation (16-QAM) scheme. Our contributions in this paper are the following. We use a Gray mapped 16-QAM modulation with a multiplexed Alamouti code concatenated with a 4-state, rate 1/2 convolutional code and show how performance gains can be obtained by a few turbo iterations. We present a suboptimum linear complexity inner decoder based on Bayesian interference cancelling approach. We show how this decoder, by a suitable modification, obtains performance gains very close to an optimum decoder. The presented scheme offers an attractive performance-complexity tradeoff for the targeted spectral efficiency. The paper is organized as follows. Section II provides the preliminaries including details of the multiplexed Alamouti code, soft-in soft-out APP decoding and suboptimum Bayesian decoding of multiplexed Alamouti codes. Section III describes the system considered in this paper and its iterative decoding structure. Section IV presents simulation results and associated discussions on performance and complexity. Finally, Section V concludes the paper. II. P RELIMINARIES A. Notations Upper case notations represent matrices while lower case notations represent vectors or scalars. All vectors and matrices are notated in bold face fonts. An n × n identity matrix is denoted as In . B. Multiplexed Alamouti blocks An Alamouti block CT is given by   c1 −c∗2 , CT = c2 c∗1

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S. Srinivasan, L. M. Davis, “Efficient iterative decoding of serially concatenated multiplexed Alamouti codes,” in Proc. IEEE Int. Conference on Communications (ICC), pp. 1-5, doi: 10.1109/icc.2011.5962689, Kyoto, Japan, June 2011.

C. APP decoding The optimum APP decoder delivers exact reliability values on each bit of all the symbols transmitted. Given   1 2 (4) p(r|s, c) ∝ exp − 2 r − Hc − Gs , 2σ the APP output for a bit cij (i = {1, 2}, 0 ≤ j < log2 (M )) in symbol ci is given by   p(r|c, s)P (c)P (s) , (5) P (cij = b|r) = p(r) 2 2

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where each row represents one antenna and each column represents one signalling time index. The symbols c1 and c2 are taken from a symbol constellation of size M . Let α1 and α2 be the channel gains from each Tx antenna to the Rx antenna, which remain constant over the two symbol periods. Let n1 and n2 be two independent samples of a complex Gaussian random variable with zero-mean and covariance 2σ 2 . The receiver view of the system, which is the expression of the received signals with respect to decoding the transmitted symbol vector, is given by        α1 α2 c1 n1 r1 = + r2∗ α2∗ −α1∗ c2 n2 or r = Hc + n, where H is called the induced channel. In order to provide multiplexing gain, a 2 × 2 system can be used and two Alamouti codes can be code-multiplexed and transmitted. In this case, the signal transmitted per each antenna is a linear combination of symbols from two different users or data sources. As an alternative, a 4 × 2 system can be used and each Alamouti block can independently use two of the transmit antennas. A code-multiplexed 2 × 2 system provides a full diversity and full rate system, where the general form of the transmitted code block is given by     x3 −x∗4 x1 −x∗2 + Λ , (1) X= x2 x∗1 x4 x∗3

where b = {0, 1}. A similar relation holds for P (sij = b|r). In logarithm domain, we have ⎞ ⎛ ⎜   Lb (cij ) = log ⎜ ⎝

This code is also known as Silver code [8]. The general receiver view,  when Kt Alamouti blocks are K multiplexed, is given by r = k t Hk ck + n, where r is the received complex column vector of dimension 2Nr × 1, Hk is the 2Nr × 2 induced complex channel matrix seen by the k th transmitted vector at the receiver, and n is the 2Nr × 1 Gaussian noise vector that adds to the received vector. In this paper, we consider a system which has Kt = 2 multiplexed Alamouti blocks, whose receiver view is given by

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r = Hc + Gs + n,

(3)

where c and s are the symbol vectors to be decoded and H and G are the corresponding induced channels at the receiver. As far as the receiver view is concerned, the 2 × 2 full-rate full-diversity codes proposed by Sezginer and Sari [9] and Rabiei-Al-Dhahir [10] also have the same structure (see [11] for details). A soft-in soft-out (SISO) decoder for the multiplexed Alamouti code takes a priori soft inputs and delivers soft decisions on each of the bits xij , where x = {c, s}, i = {1, 2} and 0 ≤ j < log2 (M ). The focus of this paper is a low complexity SISO decoder based on Bayesian Interference cancelling technique which performs as well as a max-log APP decoder. The following subsections briefly explain the two decoders.

 p(r|c, s)P (c)P (s) ⎟ ⎟. ⎠ p(r) 2

(6)

c∈M2 s∈M cij =b

The a posteriori log likelihood ratio (LLR) or the L value of a bit xij is given by   P (xij = 1|r)  = L1 (xij ) − L0 (xij ), (7) L(xij ) = log P (xij = 0|r) where x refers to either c or s. Thus, we have   ⎞ ⎛ p(r|c, s)P (c)P (s) 2 2 :cij =1 s∈M ⎟ ⎜ c∈M  L(cij ) = log ⎝ ⎠ (8) p(r|c, s)P (c)P (s)

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where the matrix Λ is a unitary matrix. One such code was given by Tirkkonen et al. [4], [5] where Λ is chosen to be   1 1+i −1 + 2i . (2) Λ= √ 7 −1 − 2i −1 + i

c∈M s∈M cij =b

c∈M2 :cij =0 s∈M2

and a similar expression for L(sij ). For the multiplication of two numbers, the equivalent operation in the log domain is addition. For the addition of two numbers, the equivalent log domain operation is a Jacobian, given by log(ea + eb ) = max(a, b) + log(1 + e−|a−b| ). In many implementations, the correction term is omitted, resulting in what is called as the ‘max-log’ approximation, given by log(ea + eb )  max(a, b). In the calculation of APP output for each bit, the metric calculated over all other combination of bits is summed. Therefore, an optimum APP decoder is extremely complex even for MIMO systems of small order using constellation sizes of 16 or above. For example, in a 2 × 2 full-rate, full diversity system using 16QAM, decoding of one code block, without any complexity optimization, requires calculation of metrics for 24·4 = 65536 combinations and 16 × 65536 summations. Due to the impractically large complexity of optimum APP decoding, low complexity soft output decoders are required that are suboptimal, yet appreciably close to the optimum performance. D. Bayesian decoder To decode the multiplexed orthogonal blocks, without any a priori information on the symbols, a Bayesian interference cancelling (BIC) decoder [12] decodes a symbol vector by assuming the other symbol vector as an interference with

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S. Srinivasan, L. M. Davis, “Efficient iterative decoding of serially concatenated multiplexed Alamouti codes,” in Proc. IEEE Int. Conference on Communications (ICC), pp. 1-5, doi: 10.1109/icc.2011.5962689, Kyoto, Japan, June 2011.

where ¯s is the transformed received signal vector given by 1 T ¯s = (¯ (10) s1 , s¯2 ) = 2 G† R−1 r, βs

where (·)† refers to conjugate transpose. In the first iteration, the a priori value P (s) is equiprobable for all s (or La (s) = 0 in the log domain). The parameter βs is given by   λ2 G2 1 − , (11) βs2 = σ2 1 + σ 2 /H2 while R−1 is a quaternion symmetric matrix given by I4 HH† − σ2 σ 2 (H2 + σ 2 )

(12)

and λ = H† G/(HG). The parameter λ is an inner product of two unit quaternion vectors, which measures the angle between the decoded signal channel vector G and the interference signal channel vector H [12]. Since s1 and s2 are independent, we can write   βs2 2 P (si |r) ∝ exp − |si − s¯i | P (si ) ; i = {1, 2}. (13) 2 and

c∈M cij =b

(17) In an iterative decoding setup, from the second iteration onwards, the BIC decoder need not evaluate (10) to (12). Instead, a hard decision interference estimate on the symbols of one of the symbol vectors (say s) is made from the a priori values supplied by the outer decoder, as sˆi = arg max P (si ), i = {1, 2}. si ∈M

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  ⎟ ⎜ βs2 2 ⎟ |s exp − − s ¯ | Lb (sij ) = log ⎜ i i ⎠ + La (s). ⎝ 2 si ∈M sij =b

(14)

The hard decision of each symbol si is given by sˆi = arg max P (si |r). si ∈M

(15)

The hard decision of the symbol vector (ˆs) is then used to cancel out its contribution (interference) from the received vector, to decode the bits in the other symbol vector (c). Given the interference-cancelled received signal vector r = (r − Gˆs), 2 βc2 = H σ 2 and ¯= c

1 H† r , H2

(16)

(18)

This hard decision is used to cancel out the contribution from the original received signal and the bits in the other symbol vector (c) decoded. After this decoding, a hard decision of the symbol vector c is obtained from the refined a posteriori value of P (c|r) to cancel its contribution from the original received vector and then the bits in vector s are decoded. Complexity: The calculations (10) to (12), which involve a few matrix operations, are needed only for the first iteration. Subsequently, the BIC decoder performs interference cancellation and soft-in soft-out demapping. The interference cancellation and projection of the received signal requires two matrix multiplications and one vector addition. Note that if the Gray-coded QAM constellation is constructed as a cartesian product of two pulse amplitude modulation (PAM) constellations, the soft out decoding in (14) and (17) can be independently (and parallely) performed for each coordinate, thus significantly reducing decoding complexity and improving speed. The transformation of the complex system into individual real domains is better explained in [13].

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R−1 =

the bits in the other symbol vector c can be decoded from ⎞ ⎛   2 ⎟ ⎜ β Lb (cij ) = log ⎜ exp − c |ci − c¯i |2 ⎟ ⎠ + La (c). ⎝ 2 2

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known statistics. Specifically, the interference is assumed to be Gaussian distributed with the same mean and variance as the data. The joint posterior distribution p(r|c, s) is marginalized with respect to the symbol vector to be decoded. The received signal is then projected to the appropriate signal space and decoded. While [12] presented maximum likelihood hard decision decoding using the Bayesian interference cancelling approach, we consider SISO decoding here. In the first iteration, after determining the norms of the two induced channels, if G2 > H2 , the vector c is treated as interference and if otherwise, s is treated as interference. Assuming without loss of generality that c is an interference, the a posteriori probability of s given the received signal with interference is approximated by (see [12] for details)    βs2  2 2 |s1 − s¯1 | + |s2 − s¯2 | P (s), (9) P (s|r) ∝ exp − 2

III. S ERIALLY CONCATENATED MULTIPLEXED A LAMOUTI CODE

A. System setup We consider a serially concatenated system with the outer code being a rate 1/2 convolutional code encoded by a 4-state, (5, 7) encoder. The information block length is 400 bits. An interleaver of length 800 bits and spread 20 is placed after the outer convolutional encoder. The output of the interleaver is fed to the inner STBC. The inner STBC consists of a 16-QAM Gray mapper space-time encoded by the spatially multiplexed Alamouti code giving place to a 4 × 2 system. This system transmits at the rate of two 16-QAM symbols in one symbol period. The spectral efficiency of the system is 4 bps/Hz. The transmitter is shown in Fig. 1. This system belongs to the family of bit interleaved space-time block codes (BI-STBC) [14], where the inner STBC code is replaced by spatially multiplexed Alamouti codes to provide higher spectral efficiency. The channel gains between the individual transmit and receive antenna links are uncorrelated. A flat fading Rayleigh channel, modelled as a zero mean circular complex Gaussian

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S. Srinivasan, L. M. Davis, “Efficient iterative decoding of serially concatenated multiplexed Alamouti codes,” in Proc. IEEE Int. Conference on Communications (ICC), pp. 1-5, doi: 10.1109/icc.2011.5962689, Kyoto, Japan, June 2011.

A. Simulation results

Transmitter, Concatenated Multiplexed Alamouti code

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Fig. 1.

Fig. 3 shows the performance of the decoders after one, two and six iterations of the decoders. After one iteration, that is, with no feedback from the outer to inner decoder yet, there is an appreciable difference between the max-log APP decoder and the BIC decoder. At a block error rate (BLER) of 2×10−2 , the Bayesian decoder performs worse by 5.6 dB. The overall performance can be improved by iterating between the inner and outer decoders. Still, after two iterations, at a BLER of 10−2 , the BIC decoder performs worse than the APP decoder by 7.6 dB. After six iterations, the BIC decoder performs worse by 2.1 dB at a BLER of 10−2 . B. Turbo gains using Gray coded modulation

Fig. 2.

Receiver, Concatenated Multiplexed Alamouti code

process with unit variance, is considered. The channel is a fast fading one, remaining constant within one space-time block (two symbol periods) while changing independently from one space-time block to another. The receiver is shown in Fig. 2. The receiver iterates between the inner STBC decoder and the outer convolutional decoder. A low complexity decoder for the inner decoding is the main focus of this paper. B. Iterative decoding

C. Progressive improvement over iterations

When the max-log APP inner decoder is used, the performance improves from the first iteration to the second, but does not improve considerably over subsequent iterations. The max-log APP decoder being close to optimum is not able to progressively gain after the second iteration as spatial diversity provides a small additional benefit in the fast fading scenario compared to time diversity. In the case of the suboptimum BIC decoder, the interference estimates are based on hard decisions, which are error prone in the initial iterations. Thus, due to the

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The extrinsic L value for each bit from one decoder is provided as the a priori input to the other decoder. Let the a priori L value for a particular bit xij to the inner decoder be given by La (xij ). It is assumed that the soft values are rendered independent by the interleaver. Therefore, the a priori L value of a 16-QAM symbol xi = xi0 xi1 xi2 xi3 is given by

The decoder for the inner Gray mapped STBC code is able to make use of a priori inputs from the outer decoder. Orthogonal STBCs using Gray mapped constellation are reported to be unable to gain from iterations [6], [7]. Multiplexed Alamouti codes are not orthogonal although the individual Alamouti blocks are. One can see the similarity of the received signal expression (3) to a single input single output channel with interference. Given the induced channels H and G, a better knowledge on one symbol vector does give some extra information on the other symbol vector. Thus a priori inputs are useful to the inner decoder to produce more reliable soft outputs.

La (xi ) =

3 

βj (xi )La (xij ),

(19)

j=0

rate 1/2 CC, 4X2, fast fading, 16QAM, BLER

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where βj (xi ) ∈ {0, 1} refers to the j th bit in the labelling of the symbol xi . The extrinsic L value for the bit xij , Le (xij ) is given by Le (xij ) = L(xij ) − La (xij ). All the summations are implemented using max-log approximation. The outer decoder is a standard soft-in soft-out log-APP decoder [15] using full Jacobian for summations. IV. R ESULTS AND D ISCUSSION

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We report here the performance of a suboptimum Bayesian decoder and an APP decoder implemented with max-log approximation, for the inner decoder. Due to excessive computational load, it was not feasible to have results for the APP decoder using full Jacobian. To the extent simulated (up to Eb /N0 of 8dB ), the max-log APP decoder performance after two iterations was found to be very close to the APP decoder using full Jacobian. The error rates were calculated after 50 block errors were recorded.

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BIC, 1itr BIC, 2itr BIC, 6itr APP, 1itr APP, 2itr APP, 6itr

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Fig. 3.

Fast fading, APP, BIC decoders, 1, 2, 6 iterations

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S. Srinivasan, L. M. Davis, “Efficient iterative decoding of serially concatenated multiplexed Alamouti codes,” in Proc. IEEE Int. Conference on Communications (ICC), pp. 1-5, doi: 10.1109/icc.2011.5962689, Kyoto, Japan, June 2011.

D. Enhancing BIC decoder performance

V. C ONCLUSIONS A serially concatenated MIMO system employing multiplexed Alamouti STBC was presented along with a low complexity iterative decoding technique that performs very close to the optimum decoder. By exploiting both the extrinsic and APP soft values from the outer decoder, we showed that a low complexity Bayesian decoder can perform similar to a max-log APP decoder. For a spectral efficiency of 4 bps/Hz, this system provides a good performance-complexity tradeoff since the inner STBC decoder is implemented by a linear complexity, fast decoding technique. R EFERENCES

[1] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Europ. Trans. Telecommun., vol. 10, pp. 585-596, Nov./Dec. 1999. [2] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-Time codes from orthogonal designs,” IEEE Trans. Info. Theory., vol. 45, No. 5, pp. 14561467, Jul. 1999. [3] S. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Select. Areas. Commun., Vol. 16, No. 8, pp. 14511458, Oct. 1998. [4] O. Tirkkonen and R. Kashaev, “Performance optimal and information maximal MIMO modulations,” in Proc. IEEE Intl. Symp. Info. Theory (ISIT), Jul. 2002. [5] A. Hottinen and O. Tirkkonen,“Precoder designs for high rate space-time block codes,” in Proc. Conf. Inf. Sci. Syst., Princeton, U.S.A., Mar. 2004. [6] G. Bauch, “Concatenation of space-time block codes and “turbo”-TCM,”, in Proc. Intl. Conf. Comm (ICC), pp. 1202-1206, Vancouver, Canada, Jun. 1999. [7] G. Bauch and F. Schreckenbach, “How to obtain turbo gains in coherent and non-coherent orthogonal transmit diversity,” in Proc. IEEE Intl. Symp. Personal Indoor and Mobile Radio Comm. (PIMRC), pp. 1988-1992, Beijing, China, Sep. 2003. [8] C. Hollanti, J. Lahtonen, K. Ranto, R. Vehkalahti and E. Viterbo, ”On the algebraic structure of the silver code: a 2× 2 perfect space-time block code,” in Proc. IEEE Info. Theory Workshop, pp. 91-94, Porto, Portugal, May 2008. [9] S. Sezginer, H. Sari and E. Biglieri, “On high-rate full-diversity 2× 2 space-time codes with low-complexity optimum detection,” IEEE Trans. Comm., vol. 57, No. 5, pp. 15321541, May. 2009. [10] P. Rabiei and N. Al-Dhahir, “A new information lossless STBC for 2 transmit antennas with reduced-complexity ML decoding,” in Proc. IEEE Veh. Tech. Conf. (VTC) Fall, pp. 773 - 777, Baltimore, U.S.A., Sep. 2007. [11] S. Sirianunpiboon, Y. Wu, A.R. Calderbank, and S.D. Howard, “Fast optimal decoding of multiplexed orthogonal designs by conditional optimization,” Int. Symp. on Inform. Theory, Vol. 56, No. 3, pp. 1106-1113, Mar. 2010. [12] S. Sirianunpiboon, A.R. Calderbank, and S.D. Howard, “Bayesian analysis of interference cancellation for Alamouti multiplexing,” IEEE Trans. on Inform. Theory, vol. 54, No. 10, pp. 4755-4761, October 2008. [13] L. M. Davis, S. Srinivasan and S. Sirianunpiboon, “Flexible complexity fast decoding of multiplexed Alamouti codes in space-time-polarization systems, ” IEEE Vehicular tech. Conf. (VTC) (Spring), Taipei, Taiwan, May 2010. [14] Z. Hong and B. L. Hughes, “Bit-interleaved space-time coded modulation with iterative decoding,” IEEE Trans. Wireless Comm., vol. 3, no. 6, pp. 1912–1917, Nov. 2004. [15] S. Benedetto, D. Divsalar, G. Motorsi and F. Pollara, “A Soft-Input Soft-Output Maximum A posteriori (MAP) Module to Decode Parallel and Serial Concatenated Codes,” TDA Progress Report, nov. 1996. [16] A. M. Tonello, “Space-time bit-interleaved coded modulation with iterative decoding strategy,” in Proc. IEEE Vehicular. Tech. Conf. (Fall), pp. 473 - 478, Boston, U.S.A., Sept. 2000.

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The poor performance of the BIC decoder was suspected to be due to incorrect hard interference estimates in the initial iterations. Therefore, to enhance its performance, a key idea we used is to utilise the full a posteriori soft outputs from the outer decoder as a priori inputs for the initial interference estimate from 2nd iteration onwards (see (18)). The subsequent soft output calculations in the BIC decoder still use the extrinsic outputs from the outer decoder. The full APP output of the outer decoder for each bit is calculated by adding its extrinsic output to the inner decoder’s previous soft output (which was given as a priori input to the outer decoder). Apart from this addition, the enchanced BIC decoder’s complexity is the same as that discussed in Section II-D. The performance of this modified BIC decoder is shown in Fig. 4 with the legend ‘BIC*’. A considerable improvement over the unmodified BIC decoder is clearly seen. While the APP decoder achieves close its best performance in two iterations, the enhanced BIC decoder achieves the same performance in 6 iterations. From a decoding speed point of view, iterating a linear complexity Bayesian decoder more times than an exhaustive search APP decoder is still preferable, given the significant difference in complexity. rate 1/2 CC, 4X2, fast fading, 16QAM, BLER

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it looses by only 0.7 dB at a BLER of 1 × 10−3 from the STBICM scheme. This is outweighed by the significantly lower complexity of the BIC decoder, given that the 4×2 ST-BICM decoder needs to search over four 4-PSK constellations for calculating the soft output for each bit.

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enhancement of noise in detection, the soft outputs for the first and second passes are generally not reliable and there is no appreciable improvement in two iterations. However, with more iterations, the outer decoder’s soft outputs strengthen the reliability of the interference estimates which in turn helps improve the reliability of the inner decoder’s soft outputs.

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Fast fading, APP decoder, modified BIC (BIC*) decoder

A comparable system that provides the same spectral efficiency of 4 bps/Hz is the ST-BICM proposed in [16]. Fig. 4 shows the ST-BICM system performance after 7 iterations under similar channel conditions. The modified suboptimum BIC decoder performs noticeably better than the ST-BICM receiver for low SNRs. Although it worsens at higher SNRs,

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