On Error Exponents of Nested Lattice Codes for the AWGN Channel

On Error Exponents of Nested Lattice Codes for the AWGN Channel Tie Liu

Pierre Moulin

Ralf Koetter

University of Illinois 405 North Mathews Avenue Urbana, IL 61801, USA email: [email protected]

University of Illinois 405 North Mathews Avenue Urbana, IL 61801, USA email: [email protected]

University of Illinois 1308 West Main Street Urbana, IL 61801, USA email: [email protected]

Abstract — We present a new lower bound for the error exponents of nested lattice codes for the additive white Gaussian noise (AWGN) channel. The exponents are closely related to those of an unconstrained additive noise channel where the noise is a weighted sum of a white Gaussian and a spherically uniform random vector. The new lower bound improves the previous result derived by Erez and Zamir and stated in terms of the Poltyrev exponents. More surprisingly, the new lower bound coincides with the random coding error exponents of the optimal Gaussian codes for the AWGN channel in the nonexpurgated regime. One implication of this result is that minimum mean squared error (MMSE) scaling, despite its key role in achieving capacity of the AWGN channel, is no longer fundamental in achieving the best error exponents for rates below channel capacity. These exponents are achieved using a lattice inflation parameter derived from a large-deviation analysis.

I. Introduction Recently, Erez and Zamir [1] showed that using nested lattice codes in conjunction with an MMSE-scaled transformation of the AWGN channel into a modulo lattice additive noise (MLAN) channel, lattice decoding can achieve capacity of the AWGN channel. Later in [2], they studied the error exponents of the MLAN channel and showed that the proposed lattice encoding and decoding scheme achieves the Poltyrev exponents which were previously derived in the context of coding without restrictions for the AWGN channel [4]. Both [1] and [2] assumed a noise-matched lattice decoder which performs maximum-likelihood (ML) decoding. In a more recent paper [3], these results were further extended by considering the more practical Euclidean lattice decoder which performs minimum-Euclidean-distance (MED) decoding. Although MED lattice decoding is strictly inferior to ML lattice decoding (for reasons which will be clear shortly), they showed that MED lattice decoded nested codes can also achieve the Poltyrev exponents. They even conjectured that the MLAN channel is asymptotically equivalent to Poltyrev’s unconstrained AWGN channel [3]. In this extended summary, we present a new lower bound for the error exponents of MED lattice decoded nested codes for the AWGN channel. It is, of course, also a lower bound for the error exponents of ML lattice decoded nested codes for the AWGN channel. The new lower bound is better than the Poltyrev exponents for all rates up to channel capacity, which contradicts Erez and Zamir’s conjecture on the asymptotic optimality of the Poltyrev exponents for the MLAN channel. Moreover, the new lower bound coincides with the random 1 This research was supported by NSF under ITR grants CCR 00-81268 and CCR 03-25924.

α

N

c

+

−

mod - Λ

X

+

Y

+

Y’’

Q Ω(Λ ) 1

^c

mod - Λ

U

Figure 1: Erez and Zamir’s lattice encoding and decoding scheme for the AWGN channel Y = X + N. coding error exponents of the optimal Gaussian codes for the AWGN channel in the nonexpurgated regime. This is quite surprising considering that transformation of channels usually incurs loss of information which might manifest itself in the error exponents. An ingredient inherent in Erez and Zamir’s lattice encoding and decoding scheme is the self noise [6]. Self noise refers to the uniform component of the MLAN which is a weighted sum of a white Gaussian and a uniform random vector, aliased into the fundamental Voronoi region of the coarse lattice. Self noise comes from the quantization process during encoding and enters the picture of decoding because of the use of MMSE scaling to reduce the power of the effective noise and to achieve capacity of the original power-constrained AWGN channel. See [7] for an exposition of the role played by MMSE estimation in this problem. Due to the self noise and the aliasing, the MLAN is not strictly Gaussian (not even spherically symmetric). However, with a proper choice of a sequence of coarse lattices, it approaches a Gaussian distribution as the dimension of the lattices tends to infinity. This fact raises the question of how self noise affects the error exponents: it is the large deviation rather than the limiting behavior of the sequence of channel law that determines the error exponents. One essential step in deriving the lower bound in [3] is to “bound” the self noise by a deliberately chosen Gaussian random vector (needs to be “blown up” appropriately) and then relate the error probability of the MLAN channel to that of an unconstrained AWGN channel. Elegant as it is, the Gaussian bound turns out to be loose for deriving the best error exponents. Notation. We use N (x, σ 2 In ) to denote the distribution of an n-dimensional Gaussian random vector with mean x and covariance matrix σ 2 In (In is the n × n identity matrix) and U(A) to denote the distribution of an n-dimensional random vector uniformly distributed over the subset A of Rn . Finally, we use Bn (x0 , r) to denote the n-dimensional ball centered at x0 and of a radius r.

II. Statement of the Problem Referring to Figure 1, we consider Erez and Zamir’s lattice encoding and decoding scheme for the AWGN channel Y = X+N

(1)

where the channel £ ¤input X satisfies an average power constraint EX n1 kXk2 ≤ PX and N ∼ N (0, PN In ) is the

AWGN. Let {Λn ⊆ Λ1,n } be a sequence of nested lattices of increasing dimension n such that:

We always have Pe,MLAN ≤ Pe,MLAN−MED

1. The order of the quotient group Λ1,n /Λn satisfies 1 log |Λ1,n /Λn | → R, n

as

n → ∞.

(2)

2. The coarse lattice sequence {Λn } is Rogers-good [8]. 3. For each n, the fine lattice Λ1,n satisfies the MinkowskiHlawka theorem [5]. The existence of such “good” nested lattices are proved in [3]. The points of the set C = {Λ1 ∩ V(Λ)}

(3)

where V(Λ) is the fundamental Voronoi region of Λ are coset leaders of the fine lattice Λ1 relative to the coarse lattice Λ; for each c ∈ C the shifted lattice Λc = c + Λ is a coset of Λ in Λ1 . The coding rate of the nested lattice code C is defined as R , n1 log |C|. It follows that V (Λ) 1 1 log |Λ1 /Λ| = log . n n V (Λ1 )

R =

(4)

We choose Λ such that the second moment σ 2 (Λ) = PX to satisfy the power constraint on the channel input. Let mod-Λ denote modulo lattice operation with respect to the Voronoi region V of the coarse lattice Λ. Let Ω denote some fundamental region of the fine lattice Λ1 to be specified later, and let QΩ(Λ1 ) denote the corresponding lattice quantizer. For encoding, we associate each message with each of the cosets of Λ1 relative to Λ, as represented by the coset leaders C = {c}. Let U ∼ U (V(Λ)) be a dither vector shared between the encoder and the decoder as a source of common randomness. Given the message c ∈ C, the encoder sends X

=

(c − U)

mod Λ.

(5)

Due to the use of dither, for any c the channel input X is also uniform over V(Λ) and the average transmitted power is PX [7]. Furthermore, X and c are statistically independent [3]. Let α ∈ (0, 1] be some scaling parameter to be optimized later. The decoder computes ˆ c

=

QΩ(Λ1 ) (αY + U)

mod Λ.

because ML decoding is optimal in minimizing the error probability. For asymptotically infinite block length, define the error exponents of the MLAN channel as EMLAN (R) = lim sup − n→∞

EMLAN−MED (R) = lim sup − n→∞

Definition 1 Assume x > 0, a > 0, and 0 < b ≤ 1. We define p λ1 (x; a, b)

=

β(a, b)

=

λ2 (x; a, b)

=

=

Eex (x; a, b)

=

C(a, b)

=

(10)

Rcr (a, b)

=

For MED lattice decoding, let Ω(Λ1 ) be the fundamental Voronoi region V(Λ1 ) of the fine lattice Λ1 . The error probability of the MED lattice decoded MLAN channel is

Rex (a, b)

=

where the MLAN N0 is independent of c and is distributed as N0 = ((1 − α)U + αN)

mod Λ.

(8)

The probability of decoding error for any codeword c is given by Pe = Pr{N0 ∈ / Ω(Λ1 )}. (9) For ML lattice decoding, let Ω(Λ1 ) be the ML decoding region Ω∗ (Λ1 ) with respect to N0 of the zero codeword. The error probability of the MLAN channel is Pe,MLAN = Pr{N0 ∈ / Ω∗ (Λ1 )}.

Pe,MLAN−MED = Pr{N0 ∈ / V(Λ1 )}.

(11)

(15)

We first state the following definitions which will be used to simplify the statement of our results.

(6)

(7)

∀ R > 0.

III. Main Results

Esl (x; a, b)

mod Λ.

1 ∗ log Pe,MLAN−MED (R, n) (14) n

EMLAN (R) ≥ EMLAN−MED (R),

=

QΩ(Λ1 ) (c + N0 )

(13)

∗ where Pe,MLAN−MED (R, n) is the minimum value of Pe,MLAN−MED over all nested lattices in Rn with fixed rate R and all α ∈ (0, 1]. Clearly, by (12), we have

Esp (x; a, b)

=

1 ∗ log Pe,MLAN (R, n) n

∗ where Pe,MLAN (R, n) is the minimum value of Pe,MLAN over all nested lattices in Rn with fixed rate R and all α ∈ (0, 1]. Similarly, define the error exponent of the MED lattice decoded MLAN channel as

Using the inflation lattice lemma proved in [3], we obtain ˆ c

(12)

b4 + 4a2 (1 − b)2 x − b2 , 2a(1 − b)2 q a(1 − b)2 + 3b2 + (a(1 − b)2 + 3b2 )2 − 8b4 2a ¡ ¤ the unique solution for y ∈ x4 , x2 , assuming x ≥ 2β(a, b), to the equation p 2b2 x 2ay − b2 − b4 + 4a2 (1 − b)2 y = , 4y − x µ ¶ ax a(1 − b)2 + b2 2 +1− 2b2 ax λ1 (x; a, b) 1 − log λ1 (x; a, b), 2 β(a, b) 1 , Esp (β(a, b); a, b) − log 2 x Esp (λ2 (x; a, b); a, b) µ ¶ 1 x − log 1 − , 2 4λ2 (x; a, b) 1 a log , 2 a(1 − b)2 + b2 1 1 log , 2 β(a, b) 1 1 log . 2 2β(a, b)

Theorem 1 Let {Λn ⊆ Λ1,n } be a sequence of “good” nested lattices of increasing dimension n. The error exponents of the

lattice decoded MLAN channel satisfy the following asymptotic inequalities: EMLAN (R)

≥

EMLAN−MED (R)

≥

E(R − on (1); SNR, α) − on (1)

(16)

Pr{Z1 ∈ / V(Λ1 )} n

≤

for any α ∈ (0, 1] where

 for [Rex (SNR, α)]+ < R ≤ [Rcr (SNR, α)]+ ,       Esp (e−2R ; SNR, α),      for [Rcr (SNR, α)]+ < R ≤ C(SNR, α) (17) and [x]+ , max{x, 0}. In particular, let α∗ (R; SNR) = argmaxα∈(0,1] E(R; SNR, α).

(18)

Lattice encoding and decoding can achieve the error exponent E ∗ (R − on (1); SNR) − on (1) where E ∗ (R; SNR) = E(R; SNR, α∗ (R; SNR))

(19)

for the AWGN channel. Sketch of the Proof. Following the footsteps of Erez and Zamir, we first use the following lemmas to upper bound the error probability of the MED lattice decoded MLAN channel. Lemma 1 Let N00 = (1 − α)U + αN; due to (8), we have mod Λ.

(20)

The error probability of the MED lattice decoded MLAN channel can be upper bounded as Pr{N0 ∈ / V(Λ1 )} ≤ Pr{N00 ∈ / V(Λ1 )}.

(21)

Lemma 2 Let B1 ∼ U (Bn (0, rcov (Λ))) (rcov (Λ) is the covering radius of Λ) and B1 is independent of N. Let Z1 = (1 − α)B1 + αN. We have Pr{N00 ∈ / V(Λ1 )} ≤ enε1 (Λ) Pr{Z1 ∈ / V(Λ1 )}

enδ nπ 2 Γ( n +1) 2

R 2d 0

wn−1 Pr{Z1 ∈ Dn (d, w)}dw

+ Pr{kZ1 k ≥ d},

E(R; SNR, α) =  Eex (e−2R ; SNR, α),       for 0 < R ≤ [Rex (SNR, α)]+ ,       Esl (e−2R ; SNR, α),

N0 = N00

In [4], Poltyrev showed that for any unconstrained additive noise channel where the noise Z1 is spherically distributed, the exists a lattice Λ1 such that the decoding error probability

(22)

where ε1 (Λ) → 0 in the limit n → ∞ if Λ is chosen to be Rogers-good. In [3], Erez and Zamir further upper bound the error probability Pr{Z1 ∈ / V(Λ1 )} by Pr{N1 ∈ / V(Λ1 )} (with appropriate “blown up”) where N1 = (1−α)G+αN and G ∼ N (0, PX In ) and is independent of N. This links the error probability of the MED lattice decoded MLAN channel to that of Poltyrev’s unconstrained AWGN channel with noise N1 . This is a Gaussian type bound in the sense that, in contrast to N0 , N1 is strictly Gaussian. However, the Gaussian bound turns out to be loose for deriving the best error exponents. Instead, we focus on the upper bound (22) which connects the error probability of the MED lattice decoded MLAN channel to that of the unconstrained additive noise channel with the noise Z1 being a weighted sum of a white Gaussian and a spherically uniform random vector.

(23)

∀d>0

1 where δ = n1 log V (Λ is the normalized logarithmic density 1) of Λ1 and Dn (d, w) is the section of the n-dimensional ball Bn (0, d) which is cut off by the hyperplane that orthogonally intersects Bn (0, d) at a distance w2 from the center. This upper bound can be improved for small values of δ by means of expurgation as

Pr{Z1 ∈ / V(Λ1 )} n

≤

16enδ nπ 2 Γ( n +1) 2

R 2d

√ nρδ

wn−1 Pr{Z1 ∈ Dn (d, w)}dw

+ Pr{kZ1 k ≥ d},

∀d>0

(24) 1 where ρδ = 2πe2δ+1 . Let {Λn ⊆ Λ1,n } be a sequence of rate-R nested lattices such that {Λn } is Rogers-good and {Λ1,n } satisfies (23) and (24), and note that Z1 is a weighted sum of two independent spherical random vectors and hence is also spherical distributed. The desired error exponents follow from an investigation on the asymptotic behavior of (23) and (24) for the non-Gaussian noise Z1 . See [12] for the details of the proof. Remark. Whereas Erez and Zamir derived the Poltyrev exponents (as a lower bound) from an analysis of the performance of their own scheme [3], we derive the error exponents for Erez and Zamir’s scheme from the Poltyrev exponents.

IV. Numerical Examples and Discussions In Figures 2-4, we have plotted E(R; SNR, αMMSE ) defined in (17), E ∗ (R; SNR) defined in (19), EP (R; SNR), the Poltyrev exponents [3, Eqn (63)], and EG (R; SNR), the random coding error exponents of the optimal Gaussian codes for the power-constrained AWGN channel [9, Chapter 7.4], as a function of R for SNR = −10dB, 0dB, and 10dB which are typical of low-SNR, medium-SNR, and high-SNR regime, respectively. In the same figures, we have also plotted α∗ (R; SNR) as a function of R for the same SNRs. Based on the results of Theorem 1 and the numerical examples illustrated in Figure 2-4, we make the following remarks: SNR 1. Fix α = αMMSE = 1+SNR and we have EP (R; SNR) < E(R; SNR, αMMSE ) for any 0 < R < C = 12 log(1+SNR) and for any finite SNR, which implies that Erez and Zamir’s conjecture on the asymptotic optimality of the Poltyrev exponents for the MLAN channel is not true. The gap between EP (R; SNR) and E(R; SNR, αMMSE ) (and hence E ∗ (R; SNR)) is particularly large in the lowSNR low-rate regime.

2. Comparison of E ∗ (R; SNR) with EG (R; SNR) reveals a surprising fact: E ∗ (R; SNR) = EG (R; SNR) 2 , i.e., the transformation from the power-constrained AWGN channel to the MLAN channel suffers no loss in the error exponents, for rates in the nonexpurgated regime 2 It can be verified that this equality holds analytically for rates in the nonexpurgated regime.

0.25

0.25

E*(R;SNR) E (R;SNR,αMMSE) EP(R;SNR) E (R;SNR)

E*(R;SNR) E (R;SNR,αMMSE) EP(R;SNR) E (R;SNR)

G

G

0.2

Normalized Error Exponent

Normalized Error Exponent

0.2

0.15

0.1

0.1

0.05

0.05

0

0.15

0 0

0.1

0.2

0.3

0.4

0.5 R/C

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5 R/C

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

(a)

(a) 1

1

0.95

Normalized Scaling Parameter

Normalized Scaling Parameter

0.9

0.85

0.8

0.75

0.7

0.99

0.98

0.65

0.6

0.55

0.5

0.97 0

0.1

0.2

0.3

0.4

0.5 R/C

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5 R/C

(b)

(b)

Figure 2: Plots of (a) the error exponents (normalized by SNR) (b) the optimal scaling parameter α∗ (R; SNR) 1 (normalized by αMMSE = 11 ) as a function of rate R (normalized by C) for SNR = −10 dB, which is typical of the low-SNR regime.

Figure 4: Plots of (a) the error exponents (normalized by SNR) (b) the optimal scaling parameter α∗ (R; SNR) (normalized by αMMSE = 10 11 ) as a function of rate R (normalized by C) for SNR = 10 dB, which is typical of the high-SNR regime. µ

0.25 E*(R;SNR) E (R;SNR,αMMSE) EP(R;SNR) EG(R;SNR)

1 2

log

¶

q 1 2

+

SNR 4

+

1 2

1+

SNR2 4

< R ≤ C.

How-

Normalized Error Exponent

0.2

0.15

0.1

0.05

0

0

0.1

0.2

0.3

0.4

0.5 R/C

0.6

0.7

0.8

0.9

1

(a) 1

Normalized Scaling Parameter

0.95

0.9

0.85

0.8

0.75

0

0.1

0.2

0.3

0.4

0.5 R/C

0.6

0.7

0.8

0.9

1

(b) Figure 3: Plots of (a) the error exponents (normalized by SNR) (b) the optimal scaling parameter α∗ (R; SNR) (normalized by αMMSE = 12 ) as a function of rate R (normalized by C) for SNR = 0 dB, which is typical of the medium-SNR regime.

ever, the way we derive this intriguing result is essentially through “brute-force” calculations; the problem whether there is a “high-level” way of seeing it remains open. 3. In the high-SNR regime, we observe that E ∗ (R; SNR) ∼ = EP (R; SNR) for any 0 < R ≤ C (see, for example, Figure 4), which suggests that the MLAN channel is asymptotically equivalent to Poltyrev’s unconstrained AWGN channel in the limit SNR → ∞. This is because in the high-SNR regime the scaling parameter α∗ goes to 1 fast enough and hence the effective noise becomes Gaussian, before aliasing. 4. αMMSE is strictly suboptimal in maximizing E(R; SNR, α) except for R = C. The reason is that for the Gaussian bound α affects the Poltyrev exponents only through the variance of the Gaussian noise N1 , where αMMSE is the unique minimizer. However, our new lower bound takes into account the heaviness of the tail of the MLAN which is also controlled by α. Recall that whereas a Gaussian distribution has a “normal tail”, a uniform distribution over a ball has “no tail”; one thus would expect the optimal scaling parameter to become smaller to favor the latter to balance large-deviation exponents. The deviation of α∗ (R; SNR) from αMMSE shows that there is a tradeoff between the tail heaviness and the variance of the noise in optimizing the error exponents of the MLAN channel. Since αMMSE is optimal only for

mutual information (which boils down to the second moment for linear Gaussian channels), it is not so surprising that a different α may be optimal for other properties. ∗

5. The fact that α (R; SNR) < αMMSE for any 0 < R < C reminds us the Gaussian arbitrarily varying channel (GAVC) where the worst-case noise is equivalent in distribution (induced by the random encoder/decoder) to the sum of a white Gaussian and a uniform (over the surface of a ball) random vector. It is shown in [11] that the error exponents of the GAVC are greater than those of the AWGN channel with the same noise variance.

V. Conclusions We have derived a new lower bound for the error exponents of nested lattice codes for the AWGN channel. The key here is a detailed analysis on the effects of the self noise upon the proposed lattice encoding and decoding scheme. The new lower bound is obtained by pursuing the optimal tradeoff between the tail heaviness and the variance of the MLAN which manifests itself in the rate-adaptive scaling parameter α∗ (R; SNR). We show that the error exponents of the MLAN channel coincide with those of the optimal Gaussian codes for the AWGN channel in the nonexpurgated random-coding regime. Therefore, the recent conjecture on the asymptotic optimality of the Poltyrev exponents for the MLAN channel [3] is pessimistic, especially in the low-SNR low-rate regime. Recent research in information theory and communications has shown that MMSE estimation is a key component in many digital transmission problems of achieving channel capacity, for example, Costa’s “dirty-paper channel” [10]. One purpose of this paper is to provide an explanation of why MMSE estimation plays such a key role in the context of lattice encoding and decoding for the AWGN channel. Instead of focusing on the problem of achieving channel capacity, we have considered the general problem of transmission at arbitrary rates below channel capacity and used error exponents as performance criteria. Our results show that in terms of achieving the best error exponents, MMSE estimation no longer plays a fundamental role. MMSE estimation becomes possibly optimal as we exit the large-deviation regime and optimization of the error exponents reduces simply to maximization of mutual information.

References [1] U. Erez and R. Zamir, “Lattice decoding can achieve 12 log(1 + SNR) on the AWGN channel using nested codes,” in Proc. Int. Symp. Inform. Theory, Washington, DC, June 24 - 29, 2001. [2] U. Erez and R. Zamir, “Lattice decoded nested codes achieve the Poltyrev exponent,” in Proc. Int. Symp. Inform. Theory, Lausanne, Switzerland, June 30 - July 5, 2002. [3] U. Erez and R. Zamir, “Achieving 12 log(1+SNR) on the AWGN channel with lattice encoding and decoding,” IEEE Trans. Inform. Theory, to appear October 2004. [Online] Available: http://www.eng.tau.ac.il/∼zamir/papers/lat dec.pdf [4] G. Poltyrev, “On coding without restrictions for the AWGN channel,” IEEE Trans. Inform. Theory, vol. 40, no. 2, pp. 409417, March 1994. [5] H. A. Loeliger, “Averaging bounds for lattice and linear codes,” IEEE Trans. Inform. Theory, vol. 43, No. 6, pp. 1767-1773, November 1997.

[6] R. Zamir, S. Shamai (Shitz), and U. Erez, “Nested linear/lattice codes for structured multiterminal binning,” IEEE Trans. Inform. Theory, vol. 48, no. 6, pp. 1250-1276, June 2002. [7] G. D. Forney Jr., “On the role of MMSE estimation in approaching the information-theoretic limits of linear Gaussian channels: Shannon meets Wiener,” in Proc. 41th Annal Allerton Conf. on Communication, Control, and Computing, Montecello, IL, October 2003. [8] R. Zamir, “On lattice quantization noise,” IEEE Trans. Inform. Theory, vol. 42, no. 4, pp. 1152-1159, July 1996. [9] R. G. Gallager, Information Theory and Reliable Communication. New York: Wiley, 1968. [10] M. H. M. Costa, “Writing on dirty paper,” IEEE Trans. Inform. Theory, vol. 29, no. 3, pp. 439-441, May 1983. [11] T. G. Thomas and B. Hughes, “Exponential error bounds for random codes on Gaussian arbitrarily varying channels,” IEEE Trans. Inform. Theory, vol. 37, no. 3, pp. 643-649, May 1991. [12] T. Liu, P. Moulin, and R. Koetter, “On error exponents of nested lattice codes for the AWGN channel,” IEEE Trans. Inform. Theory, submitted June 2004. [Online] Available: http://www.ifp.uiuc.edu/∼tieliu/Liu Moulin Koetter IT04.pdf