A Simple Proof for the Existence of “Good” Pairs of Nested Lattices Or Ordentlich Joint work with Uri Erez November 15th, IEEEI 2012 Eilat, Israel Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
What are lattices? A lattice Λ is a discrete subgroup of R n closed under addition and reflection. Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
Lattice definitions The Voronoi region V of a lattice point is the set of all points in R n which are closest to it. Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
Lattice definitions The Voronoi region V of a lattice point is the set of all points in R n which are closest to it. Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
Lattice definitions A sequence of lattices is good for AWGN coding if the probability that an AWGN (with appropriate variance) is not contained in V vanishes with the dimension. Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
Lattice definitions r cov The covering radius is the radius of the smallest ball that contains the Voronoi region V . Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
Lattice definitions r eff r cov The effective radius is the radius of a ball with the same volume as the Voronoi region V . Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
Lattice definitions r eff r cov A sequence of lattices is good for covering (“Rogers-good”) if r eff r cov → 1 as the lattice dimension grows. Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
Lattice definitions r eff r cov A sequence of lattices is good for MSE quantization if for a random dither U uniformly distributed over V we have E � U � 2 → r 2 eff as the lattice dimension grows. Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
Lattice definitions A lattice Λ c is nested in Λ if Λ c ⊂ Λ. Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
The mod-Λ transmission scheme [Erez-Zamir IT04] N α � Y eff mod-Λ ˆ X Y t � � Q Λ f ( · ) t mod-Λ − U The scheme uses a pair of nested lattice Λ ⊂ Λ f and a dither U . Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
The mod-Λ transmission scheme [Erez-Zamir IT04] N α � Y eff mod-Λ ˆ X Y t � � Q Λ f ( · ) t mod-Λ − U Tx: The transmitted signal is X = [ t − U ] mod Λ X is uniformly distributed over the Voronoi region of Λ. The average transmission power is the second moment of Λ. Λ needs to be good for MSE quantization Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
The mod-Λ transmission scheme [Erez-Zamir IT04] N α � Y eff mod-Λ ˆ X Y t � � Q Λ f ( · ) t mod-Λ − U AWGN Channel: Y = X + N Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
The mod-Λ transmission scheme [Erez-Zamir IT04] N α � Y eff mod-Λ ˆ X Y t � � Q Λ f ( · ) t mod-Λ − U Rx: Y eff = α Y + U = X + U + ( α − 1) X + α N = t − U + λ + U + ( α − 1) X + α N = t + λ + Z eff , where λ ∈ Λ and Z eff = ( α − 1) X + α N . Decoding is correct if Z eff ∈ V f Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
The mod-Λ transmission scheme [Erez-Zamir IT04] N α � Y eff mod-Λ ˆ X Y t � � Q Λ f ( · ) t mod-Λ − U To approach capacity with nearest neighbor decoding we need: The coarse lattice Λ to be good for MSE quantization. The fine lattice Λ f to be good for coding (with NN decoding) in the presence of effective noise Z eff = ( α − 1) X + α N . Note that Z eff depends on the coarse lattice. Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
The mod-Λ transmission scheme [Erez-Zamir IT04] N α � Y eff mod-Λ ˆ X Y t � � Q Λ f ( · ) t mod-Λ − U To be more formal, we want: n E � U � 2 → 2 π e Vol (Λ) 2 / n (or alternatively G (Λ) → 1 1 1 2 π e ). 2 n > 2 π e 1 n E � Z eff � 2 , If V (Λ f ) Pr ([ Q Λ f ( t + Z eff )] mod Λ � = t mod Λ) → 0 . If the two conditions are satisfied the scheme achieves any rate below � Vol (Λ) R = 1 � = 1 � SNR � n log 2 log 1 / n E � Z eff � 2 Vol (Λ f ) Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
How to construct “good” pairs of nested lattices? Binary linear code Lattice Single Nested Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
How to construct “good” pairs of nested lattices? Binary linear code Lattice Single Draw G ∈ Z K × N with entries 2 i.i.d. and uniform over Z 2 Nested Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
How to construct “good” pairs of nested lattices? Binary linear code Lattice Single Draw G ∈ Z K × N Draw G ∈ Z K × N with entries with entries p 2 i.i.d. and uniform over Z p . i.i.d. and uniform over Z 2 Construct a linear code C with this generating matrix. Set Λ = p − 1 C + Z n Nested Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
How to construct “good” pairs of nested lattices? Binary linear code Lattice Single Draw G ∈ Z K × N Draw G ∈ Z K × N with entries with entries p 2 i.i.d. and uniform over Z p . i.i.d. and uniform over Z 2 Construct a linear code C with this generating matrix. Set Λ = p − 1 C + Z n Nested G with Draw G f = − − − G ′ entries i.i.d. and uniform over Z 2 . G generates the coarse code and G f the fine code Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
How to construct “good” pairs of nested lattices? Binary linear code Lattice Single Draw G ∈ Z K × N Draw G ∈ Z K × N with entries with entries p 2 i.i.d. and uniform over Z p . i.i.d. and uniform over Z 2 Construct a linear code C with this generating matrix. Set Λ = p − 1 C + Z n Nested G ? with Draw G f = − − − G ′ entries i.i.d. and uniform over Z 2 . G generates the coarse code and G f the fine code Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
How to construct “good” pairs of nested lattices? Previous works Find a lattice Λ which is good for covering, and has a generating matrix F . Draw G ∈ Z K × N with entries i.i.d. and uniform over Z p . Construct a p linear code C with this generating matrix. p − 1 C + Z n � = p − 1 F C + F Z n . � Set Λ f = F · This work G ∈ Z p with entries i.i.d. and uniform over Z p . Draw G f = − − − G ′ Construct a linear code C from the generating matrix G , and a linear code C f from the generating matrix G f . Set Λ = p − 1 C + Z n , Λ f = p − 1 C f + Z n . Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
Differences from previous work Current approach vs. previous work A simpler ensemble to analyze. A basic proof that makes no use of previous results from geometry of numbers. The coarse lattice only has to be good for MSE quantization (not necessarily for covering). Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
Differences from previous work Current approach vs. previous work A simpler ensemble to analyze. A basic proof that makes no use of previous results from geometry of numbers. The coarse lattice only has to be good for MSE quantization (not necessarily for covering). Historical note: When Erez and Zamir’s work was published it was known that good covering lattices exist, but it was not known that Construction A lattices are usually good for covering. = ⇒ The two-step construction was needed. Erez and Zamir studied the error exponents the mod-Λ scheme achieves. For that purpose it is important that the coarse lattice will be good for covering. For capacity, goodness for quantization suffices. Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
Proof outline We show that w.h.p. the coarse lattice Λ is good for MSE quantization. We show that w.h.p. the fine lattice Λ f is good for coding in the presence of noise that rarely leaves a ball. = ⇒ Most members of the ensemble are “good” pairs of nested lattices. We show that for a coarse lattice Λ which is good for MSE quantization the effective noise Z eff = ( α − 1) X + α N rarely leaves a ball. Ordentlich and Erez Simple Existence Proof for “Good” Pairs of Nested Lattices
Recommend
More recommend
