Approaching Channel Capacity without Error Correction Coding

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Mar 9, 2017 - requiring any additional forward error correction coding. The proposed ... Index Terms—Channel capacity, random-like codes, OFDM,.
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Approaching Channel Capacity without Error Correction Coding through Nonlinear Transformation of OFDM Signals

arXiv:1703.03141v1 [cs.IT] 9 Mar 2017

Sergey V. Zhidkov, Member, IEEE

Abstract—In the context of orthogonal frequency division multiplexing (OFDM), nonlinearity is usually considered a highly undesirable phenomenon. In this letter, we demonstrate that if a suitable memoryless nonlinearity is applied to the uncoded OFDM signal on the transmitter side, the performance of the communication system may be similar to that of state-of-theart, capacity-approaching coded modulation schemes without requiring any additional forward error correction coding. The proposed modulation technique could be used directly or as a precoder for a conventional OFDM modulator, resulting in a system possessing all benefits of OFDM along with reduced crest-factor (CF).

nonlinearly distorted OFDM signals, showing that the BER performance of hard-clipped OFDM signals can be up to 2 dB better than linear OFDM transmission. In this letter, we go one step further to demonstrate that if the uncoded OFDM signal is distorted by memoryless nonlinearity with certain properties, the resultant signal waveform may achieve BER performance close to the best known capacity–approaching coded modulation techniques.

Index Terms—Channel capacity, random-like codes, OFDM, nonlinearity

A. Achieving capacity by nonlinear transformation of uncoded OFDM

I. I NTRODUCTION traditional approach for transmitting digital information over a communication link with high spectral efficiency (η ≥ 2 bit/s/Hz) is to combine good error-correcting code with high-order modulation. Over last two decades, several capacity-approaching coded modulation techniques have been proposed, including low-density parity check (LDPC) coded modulation [1, 2], turbo-coded modulation and turbo trelliscoded modulation [3]. In this letter, we propose an alternative bandwidth-efficient modulation technique, which does not rely on any error correction coding in the traditional sense, while still achieving error rates close to the best known coded modulation schemes. Our method is based on serial concatenation of an orthogonal linear transform, such as the discrete Fourier transform (DFT), with memoryless nonlinearity. We demonstrate that such a simple signal construction may exhibit properties of random code ensemble, as a result approaching channel capacity if decoded using a maximum-likelihood (ML) decoder. Notably, the proposed modulation scheme can be viewed as an OFDM signal distorted by very strong memoryless nonlinearity. In the context of OFDM, any nonlinearity is usually considered a highly undesirable phenomenon. In fact, the sensitivity of OFDM technology to nonlinear distortions is usually cited as a major drawback of OFDM. Nonetheless, in [4], the authors theoretically proved that the bit error rate (BER) performance of OFDM transmission subjected to strong memoryless nonlinearity can outperform linear transmission, provided that an optimal ML-receiver is used. Moreover, in [5], we proposed a practical message-passing receiver for

A

S. V. Zhidkov is with Cifrasoft Ltd., Izhevsk, Russia, e-mail: [email protected].

II. P ROPOSED

MODULATION TECHNIQUE

Consider the modulation scheme illustrated in Figure 1. In the transmitter, K = N/2 − 1 complex, M -ary quadrature amplitude–modulated (M -QAM) baseband symbols {ξk } are transformed into the time domain via N -point, real inverse DFT operation, and passed through a memoryless nonlinearity block. The signal at the modulator output can be expressed as: sn = f (zn ) ,

n = 0, 1, ..., N − 1

(1)

where zn = ℜ

(r

)  K 2πnk 2 X ξk exp j N N

(2)

k=1

and f (z) is a memoryless nonlinear function. Let us now assume that the memoryless nonlinear function f (z) can be represented as f (z) = g (...g (g (z))), where g(z) | {z } l−times

is a deterministic chaotic map. Thus, f (z) has certain properties of chaotic iterated functions, namely, sensitive dependence on initial conditions [6]. Under such assumptions, and if K → ∞ and l → ∞, the ensemble of waveforms generated by (1) possesses all major properties of a random code ensemble, because even small modifications in the message sequence {ξk } (e.g., in a single bit) lead to: a) small modifications, at least, for all samples of the intermediate signal zn due to the spreading properties of the (I)DFT operation, and b) large (and pseudo-random) modifications for all samples of waveform sn due to the properties of the nonlinear function f (z). Therefore, we conjecture that where f (z) has the abovementioned properties if the waveforms generated by (1)

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are demodulated using a ML decoder, system performance may approach channel capacity. Unfortunately, brute-force, maximum-likelihood decoding of (1) is prohibitively complex, K generally of the order O(M 2 ), where M is the modulation order. However, as in many practical applications, we can rely on belief-propagation techniques to approximate ML-decoding. A perfect candidate for decoding the signals generated by (1) is the generalized approximate message passing (GAMP) algorithm [7]. Having already been applied to the decoding of clipped OFDM signals in [5], it has demonstrated good performance and reasonably good convergence behavior.

IDFT

ze



(yn −f (z))2 2µw



(p ˆn (t)−z)2 p 2µn (t)

dz, ∀n

(8)

−∞

µzn

1 (t) = C

Z∞

z 2e



(yn −f (z))2 2µw



(p ˆn (t)−z)2 p 2µn (t)

dz − (ˆ zn (t))2 , ∀n

−∞

(9)

where

{ sn }

{ zn }

{ξ n }

Z∞

1 zˆn (t) = C



(yn −f (z))2 2µw



(p ˆn (t)−z)2 p 2µn (t)

dz, ∀n

(10)

sˆn (t) = (ˆ zn (t) − pˆn (t)) /µpn (t) , ∀n

(11)

C=

f(z)

Z∞

e

−∞

Fig. 1. Proposed modulation scheme

µsn (t) =

1 µpn (t)

B. GAMP algorithm In the additive white Gaussian noise (AWGN) channel, the model of the received signal can be expressed as yn = f (zn ) + wn ,

n = 0, 1, ..., N − 1

Fn,k and

xn =



ℜ (ξn ) , 0 ≤  n ≤ N /2 ℑ ξn−N /2 , N /2 < n < N

µrk (t) =

(3)

N −1 X

n=0

N −1 X k=0

2

|Fn,k | µsn (t)

rˆk (t) = x ˆk (t) + µrk (t)

N −1 X n=0

xˆk (t + 1) =

(4)

!−1

, ∀k

Fn,k sˆk (t) , ∀k

(13)

(14)

M X

m=1

dm Pm,k , ∀k

(15)



µxk (t + 1) = (5)

|Fn,k | µxk (t) , ∀n

(6)

Fn,k x ˆk (t) − µpn (t) sˆn (t − 1) , ∀n

(7)

k=0

pˆn (t) =

2

N −1 X



Model (3) is equivalent to a general problem formulation for the GAMP algorithm [7], which belongs to a class of Gaussian approximations of loopy belief propagation for dense graphs. In our decoder implementation, we use the sumproduct variant of the GAMP algorithm, which approximates minimum mean-squared error estimates of x and z. The GAMP algorithm adapted to our problem is summarized below. The algorithm generates a sequence of estimates x ˆ (t), ˆ z (t), for t = 1, 2, ... through the following recursions: Step 1) Initialization: t = 1, x ˆ (1) = 0, µx (1) = 1, ˆ s (0) = 0 Step 2) Estimation of output nodes: µpn (t) =

(12)

Step 3) Estimation of input nodes:

where wn is the AWGN term with zero mean and variance µw . We can also represent {zn } as z = Fx, where F is a N × N real IDFT (unitary) matrix with elements:  q  2 2πnk  , 0 ≤ k < N /2 N cos N q =  2πnk 2  − , N /2 ≤ k < N N sin N

  µz (t) 1 − pn , ∀n µn (t)

M X

m=1

2

(dm − x ˆk (t + 1)) Pm,k , ∀k

(16)

where

Pm,k =

e



√ M P

rk (t))2 (dm −ˆ

e

2µr (t) k



rk (t))2 (dl −ˆ

,

(17)

2µr (t) k

l=1

M is the number of points in the signal constellation, and {dm } is the vector of constellation points per real/imaginary component. For example, for 4-QAM modulation, {d } = [−1 √ +1], and√ for 16-QAM √ modulation  m√ {dm } = −3 5 − 1 5 +1 5 +3 5 . Steps (6)–(16) are repeated with t → t + 1 until tmax iterations have been performed. At each iteration of the algorithm, we calculate the Euclidean distance E(t) between the received vector {yn } andthe reconstructed time-domain NP −1 Fn,k x ˆk (t) . The final decision is based on waveform f k=0

the vector {ˆ xk (t)} that corresponds to the minimum Euclidean distance E(t).

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C. Choosing optimal nonlinearity f (z)

 sgn (z) (a0 |z| + b0 ) , if 0 ≤ |z| < T1      sgn (z) (a1 |z| + b1 ) , if T1 ≤ |z| < T2 ... f (z) =   sgn (z) (ai−1 |z| + bi−1 ) , if Ti−1 ≤ |z| < Ti    sgn (z) (ai |z| + bi ) , if |z| ≥ Ti (18) with some parameters a = G0 {a0 , a1 , ..., ai }, b = {b0 , b1 , ..., bi }, and T = G−1 0 {0, T1 , T2 , ..., Ti }, where G0 is a scaling factor that does not affect the “shape” of nonlinearity f (z). Several examples of parameter vectors a, b, and T that satisfy our heuristic requirements and that were used in our simulation are given in Table 1 and illustrated in Figure 2. TABLE I E XAMPLE PARAMETERS OF NONLINEARITY f (z) No. 1

2

3

Parameters aG−1 = {1 -3 3 3 -3 3 -3 -3 3 -3 3 3 -3 3 -3 -3 3 -3 3 -0.5} 0 b = {0 4 -3 -4 5 -5 6 7 -6 8 -7 -8 9 -9 10 11 -10 12 -11 3} T G0 = {0 1 1.2222 1.3333 1.4444 1.6666 1.8888 2 2.1111 2.3333 2.5555 2.6666 2.7777 3 3.2222 3.3333 3.4444 3.6666 3.8888 4}; G0 = 0.67 aG−1 0 = {1 2 2 -2 -2 2 2 -2 -2 -0.5} b = {0 -2 -2.5 4 4.5 -4 -4.5 6 6.5 2.5} T G0 = {0 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3}; G0 = 0.515 aG−1 0 = {1 2 2 -2 -2 2 2 -2 -2 -0.5} b = {0 -2 -3.5 4 3.5 -4 -4.5 6 6.5 2.5} T G0 = {0 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3}; G0 = 0.51

III. S IMULATION

f(z)

1

0

−1 −5

0

5

0

5

0 z

5

f(z)

1

0

−1 −5 1

f(z)

Choosing the optimal shape of nonlinearity f (z) is not easy, requiring a balance between two conflicting requirements. Firstly, the nonlinear function should be reasonably "chaotic" to guarantee sensitivity to initial conditions and, therefore, random-like properties of the coded waveforms. On the other hand, our experimental study implies that a "truly chaotic" shape of nonlinearity f (z) precludes GAMP algorithm from converging to the ML-solution. Therefore, we adopted an ad hoc procedure to select and optimize the shape of the memoryless nonlinearity f (z). Our heuristic choice of nonlinearity f (z) relies on two key ideas: • f (z) should contain some linear or almost linear region to allow the message passing decoder to converge. • f (z) should resemble a chaotic iterated function in other regions (in terms of sensitivity to initial conditions). For a general representation of f (z), we suggest using the flexible, piece-wise linear model:

0

−1 −5

Fig. 2. Three examples of nonlinear functions f (z): nonlinearity 1 (top), nonlinearity 2 (middle), and nonlinearity 3 (bottom).

presence of a waterfall region and error floor, and performance improves for larger frame sizes (or number of subcarriers, in our case). In particular, the proposed modulation scheme with nonlinearity 3 and K=8191 achieves target BER=10−5 at Eb /N0 = 3.4 dB, which represents 6.2 dB gain over uncoded 4-QAM modulation and is less than 1.7 dB away from the unconstrained AWGN channel capacity. These results are better or within 0.1 ÷ 0.4 dB of the performance of the capacityapproaching, bandwidth-efficient modulation schemes with η = 2 bit/s/Hz reported in [1-3, 8, 9]. A considerable performance improvement over uncoded modulation was also observed for 16-QAM (η = 4 bit/s/Hz). For instance, with K=4096, nonlinearity 1 and G0 = 0.51, a target BER= 10−4 is achieved at Eb /N0 = 9.2 dB, which represents 3 dB gain over uncoded 16-QAM (graphs are not shown here due to space limitations). However, in case of 16-QAM, the sum-product variant of the GAMP algorithm is sub-optimal in terms of BER performance, and the example nonlinearities that we use for 4-QAM (Table I) are not optimal either. Therefore, application of the proposed technique to high-order modulation formats (η ≥ 4 bit/s/Hz) is an open research topic.

RESULTS

The performance of the proposed modulation scheme with the GAMP decoder was studied by means of Monte-Carlo simulation. Figure 3 illustrates the BER vs Eb /N0 curves for 4-QAM modulation with three nonlinearities (shown in Figure 2) and different values of K. In all simulations, the variance of {zn } was normalized to 1, the maximum number of iterations (tmax ) was set to 50, and integrals (8)–(10) were approximated using numerical summation. For some parameters G0 , a, b, T the proposed modulation scheme exhibits behavior similar to random-like codes with the

IV. T RANSMISSION

VIA FREQUENCY- SELECTIVE MULTIPATH CHANNELS

If signal (1) is transmitted over a frequency-selective multipath channel, the GAMP algorithm no longer directly applies. Convolution with a multipath channel introduces an additional sub-graph between output nodes and observable nodes, and such a model of the received signal requires modification of the message-passing receiver to perform optimally. This same modification, however, adds unpredictable complexity, since the channel matrix (unlike the DFT matrix) is unstructured

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signal. Figure 5 compares the complementary cumulative distribution functions (CCDF) of the direct waveform (1) with nonlinearity 1, the signal generated by the transmission system illustrated in Figure 4 with nonlinearity 2, and a conventional OFDM. As may be seen, the CF of the signal generated by the transmission system illustrated in Figure 4 is somewhat higher than that of the direct waveform (1), but it is still approximately 4 dB lower than the CF of a conventional OFDM signal.

0

10

Conventional OFDM Direct−waveform (nonlinearity 1) Pre−coded OFDM (nonlinearity 2) −1

10

−2

CCDF

10

−3

10

{ξ n }

−4

10

IDFT

f(z)

DFT

Cyclic prefix

Pre-coder 0

2

4

6 8 Crest factor (dB)

10

12

14

Fig. 4. Proposed modulation scheme as a pre-coder for a conventional OFDM transmitter Fig. 5. CCDF of the proposed direct modulation scheme, the proposed precoded OFDM (Figure 4), and a conventional OFDM

V. C ONCLUSIONS −1

10

Uncoded 4−QAM Nonlinearity 1 (K=1023) Nonlinearity 1 (K=2047) Nonlinearity 2 (K=2047) Nonlinearity 2 (K=4095) Nonlinearity 2 (K=8191) Nonlinearity 3 (K=4095) Nonlinearity 3 (K=8191)

−2

10

−3

BER

10

−4

10

−5

10

−6

10

2.5

3

3.5

4

4.5 5 EbNo (dB)

5.5

6

6.5

7

In this letter, we proposed a capacity-approaching modulation technique based on serial concatenation of orthogonal linear transformation (e.g., DFT) with memoryless nonlinearity. We demonstrated that such a simple signal construction may exhibit properties of a random code ensemble, as a result approaching channel capacity. Our computer simulations confirmed that if the decoder relies on the approximate message passing algorithm, the proposed modulation technique exhibits performance on par with state-of-the-art coded modulation schemes that use capacity-approaching component codes. The proposed technique could be extended to modulation formats with higher spectral efficiency (η ≥ 4 bit/s/Hz) and other types of orthogonal transformations (e.g. Walsh–Hadamard transform), offering one possible direction for our future research. R EFERENCES

Fig. 3. BER vs Eb /N0 for the proposed modulation scheme in AWGN channel (η = 2 bit/s/Hz)

and may be time-varying, which precludes using the computationally efficient fast-transform algorithms. In other words, even though the proposed modulation scheme (1) formally resembles conventional OFDM transmission, most advantages of OFDM are lost in practice. An interesting alternative to direct transmission of the signal (1) over a communication channel is to use it instead as a pre-coder for a conventional, linear OFDM transmitter, as illustrated in Figure 4. This approach provides the benefits of a conventional OFDM (namely, simplicity of channel equalization) with lower CF of the transmitted waveform. In particular, the distribution of signal samples at the output of the modulator illustrated in Figure 4 resembles the distribution of M -QAM signal affected by additive Gaussian noise. If the nonlinearity f (z) has a relatively large linear region (as in the case of our example nonlinearity 2) the CF of such a signal will be much lower than the CF of a conventional OFDM

[1] J. Hou, et al., “Capacity-approaching bandwidth-efficient coded modulation schemes based on low-density parity-check codes,” IEEE Trans. Inf. Theory, vol. 49, no. 9, pp. 2141–2155, Sep. 2003. [2] D. Divsalar and C. Jones, “Protograph based low error floor LDPC coded modulation,” in Proc. 2005 IEEE Military Communications Conf., Atlantic City, NJ, Oct. 2005, pp. 378–385. [3] P. Robertson and T. Worz, "Bandwidth-efficient turbo trellis-coded modulation using punctured component codes," IEEE J. Sel. Areas Commun., vol. 16, no. 2, pp. 206-218, Feb. 1998. [4] J. Guerreiro, R. Dinis and P. Montezuma, “On the optimum multicarrier performance with memoryless nonlinearities,” IEEE Trans. Commun., vol. 63, no. 2, pp. 498-509, Feb. 2015. [5] S. Zhidkov, "Detection of nonlinearly distorted OFDM signals via generalized approximate message passing," arXiv:1703.01562 [cs.IT], Mar 2017. [6] J. C. Sprott, Chaos and Time-Series Analysis. Oxford University Press, 2003. [7] S. Rangan, “Generalized approximate message passing for estimation with random linear mixing,” in Proc. IEEE Int. Symp. Information Theory, Saint-Petersburg, Russia, 2011, pp. 2174–2178. [8] J. Hamkins, "Performance of low-density parity-check coded modulation," in Proc. 2010 IEEE Aerospace Conference, Big Sky, MT, 2010, pp. 1-14. [9] G. Durisi, L. Dinoi, and S. Benedetto, “eIRA codes for coded modulation systems,” in Proc. IEEE International Conf. on Comm., Istanbul, Turkey, Jun. 2006, vol. 3, pp. 1125–1130.