1. Lattices in Communications
Cong Ling
Department of Electrical and Electronic Engineering

2. Outline Lattice coding Lattice decoding
Classic additive-noise channel
Multi-input multi-output (MIMO) channel
Lattice decoding
Sphere decoding (Fincke-Pohst, Schnorr-Euchner)
Lattice-reduction-aided decoding (Babai)
Some recent results and questions
Average complexity of LLL and fixed-complexity LLL
Complex-valued LLL
Gap between sphere decoding and Babai
Summary and other applications

3. Digital Communications
Telephone network
Internet
Radio and TV broadcast
Mobile communications
Wi-Fi
Satellite and space communications
Analogue communications
AM, FM
Digital communications
Transfer of information in digits
Dominant technology today

4. Block Diagram Performance measures Power efficiency
Bit error rate vs. signal-to-noise power ratio (SNR)
Limited power in mobile devices, green radio/network
Strong codes and associated decoding algorithms
Spectral efficiency
Spectrum is expensive and scarce

5. Complex Signals Digits cos(t) sin(t) Quadrature Modulator
Lattice: 16 HEX, 16 QAM
More efficient in power and bandwidth

6. Classic Channel
Additive white Gaussian noise channel y = x + w y: received; x: sent; w: Gaussian noise; all complex-valued
Shannon capacity C = log(1+P/N) P: signal power; N: noise power
Shannon proved error-free communication is possible for any rate below capacity (1948)
Gaussian random codes were used in the proof
Without structures, such codes are not implementable
Was the holy grail of coding theorists for half a century
Has been approached by turbo (1990’s) and LDPC codes (1960’s)
Can also be achieved by lattice codes

7. Lattice Codes Lattice code = a lattice  shaping region
There exist lattice codes achieving capacity
With nearest-codeword decoding: optimum performance
With infinite lattice decoding (Erez-Zamir’04): ignores the shaping region
Need linear minimum mean square error (MMSE) filter (Wiener filter) to deal with boundary errors
Proofs are based on an ensemble of random lattices; not explicit
Modular lattices: lattice mod p = linear code over Zp
Then use random coding
Closely related to Minkowski-Hlawka theorem (a pre-Shannon result)
Needs high dimension
Decoding complexity is high (i.e., mostly theoretic investigation)
Lattice codes are not commonly used to achieve capacity

8. Lattice and Codes
Conway and Sloane, Sphere Packings, Lattices and Groups, 1988
Construction of codes from sphere packings
Denser packing  less power
Construction of lattices from codes
Ebeling, Lattices and Codes, 1994
Practice: combine low-dimensional lattices with conventional codes
Trellis coded modulation (E8, D4, Z16…)
Key to the development of modems

9. MIMO Channel Linear MIMO channel without coding y = Hu + w
u Zn + jZn: data vector; H: n  n channel matrix; w: noise
Covers many point-to-point and multiterminal problems in digital communications
CDMA, OFDM, broadcast…
Might be a lattice encoder too 1 n

10. MIMO Lattice Codes Perfect code (Oggier-Rekaya-Belfiore-Viterbo’06)
Full-rate full-diversity
Encoding n2 symbols to collect the maximum diversity n Y = HX + W X: nn matrix
Let x = Vec(X) = Gu y = (IH)Gu + w G: n2n2 matrix, u: n21 vector
The equivalent lattice is of dimension n2 (in fact 2n2 in real Euclidean space)
Code design is based on cyclic division algebra and algebraic lattices
Shaping
A finite region: cubic, spherical, Voronoi
Proposed for latest WiFi standard IEEE n

11. MIMO Decoding Consider y = Bx + w
Minimum distance decoding (closest vector problem)
A  Z
Closest vector problem in a finite subset of a lattice
Suboptimal decoding
Zero-forcing (ZF, Babai’s rounding off): Q{B-1y}
Successive interference cancellation (SIC, Babai’s nearest plane algorithm)
Vertical Bell Labs Space-Time (V-BLAST) Ordering (Forschini’99)
Starting from the end, successively choose the column corresponding to the longest Gram-Schmidt vector
Both can be enhanced by MMSE filtering

12. Lattice Decoding Firstly proposed by Wai Ho Mow’92 and Viterbo’93
Two stages
Lattice reduction
Enumeration of lattice points
Combinations
Lattice reduction only
Sphere decoding only
Both
Lattice reduction can speed up sphere decoding
Bad basis
Good basis
Received signal

13. Sphere Decoding Search strategies
Kannan: ~ nn/2 complexity (Hanrot, Stehle’07)
Fincke-Pohst: find all lattices points inside a sphere
Schnorr-Euchner: find the closest lattice point more quickly
Both Fincke-Pohst and Schnorr-Euchner can easily be tailored for a finite lattice (Viterbo’93)
Sphere decoding
Limit the upper/lower bounds
Average complexity of the Fincke-Pohst strategy is still exponential (Jalden’05)
Assuming i.i.d. normal distribution
Complexity of Schnorr-Euchner strategy is exponential in the worst case
Even with strong lattice reduction (Stehle’07)
Analysis of average complexity (with or without reduction) is missing

14. Lattice-Reduction-Aided Decoding
Computational burden of sphere decoding is too high for real-time hardware implementation even with reduction
The lattice for coded 4-by-4 MIMO has dimension 32 in real space
Lattice-reduction aided decoding (Mow’92, Yao’02)
Close to optimum performance, lower complexity
Lattice reduction + zero forcing/successive interference cancellation  the same as Babai
Performance
Achieves optimum diversity and multiplexing tradeoff if combined with MMSE (Jalden’09)
Has a widening gap to sphere decoding (Ling’06)
How to narrow the gap?
Klein’s algorithm: randomized Babai (Liu, Ling, Stehle’10)
Augmented lattice reduction (Luzzi, Othman, Belfiore’10)

15. Boundary Errors
With lattice reduction, a finite lattice will undergo a linear transform
It is not easy to control the boundary in decoding
A shortage of lattice reduction; will affect
Lattice-reduction-aided decoding
Sphere decoding aided by lattice reduction
No good solutions to this problem
However, negligible for large constellations (e.g., 64QAM)
Can be tackled by MMSE to some extent
Related to constrained optimization
There are applications where the lattice is indeed infinite (MIMO broadcast, differential lattice decoding)

16. Complexity of LLL Algorithm
Computational complexity
O(n4) for integer bases (Lenstra’82)
O(n4logn) for i.i.d. random bases in unit ball (Daude’94)
O(n3logn) for i.i.d. normal-distribution bases (Ling & Howgrave-Graham’07)
Doesn’t affect Gram-Schmidt orthogonalization

17. Fixed-Complexity LLL(-deep)
Complexity of LLL is variable
Throughput is limited by worse-case complexity
A fixed-complexity LLL algorithm was proposed in Ling’09
Desirable in hardware implementation (e.g., FPGA)
Allows pipeline structure; easy to implement
Throughput is as high as that of V-BLAST
Equivalent to LLL-deep
Related to even-odd parallel LLL (Villard’92)

18. Sorted GSO/QR Sorted GSO
At each stage of GSO, sort the remaining columns according to their lengths so that the projections
= sort QR decomposition (Wubben’01)
cf. LLL potential function
As in coding and optimization, sometimes it is beneficial to look at the dual lattice

19. V-BLAST is a relative of LLL
Sorted
GSO/QR
V-BLAST
LLL-deep
Dual lattice
Size reduction
Important property of dual lattice
The lengths of GS vectors satisfy (Lagarias’90)
V-BLAST is equivalent to successively choosing the i-th Gram-Schmidt vector of the dual basis with the minimum length for i = 1, 2, …, n (Ling’06)

20. Complex Lattices Complex equivalent y = Bx + w
x Zn + jZn
B  Cnn
Most reduction algorithms are developed for real-valued lattices
Standard approach: use the equivalent real channel model
Problem: dimension is doubled
Increased complexity (recall: O(n3logn) for LLL reduction)

21. Complex LLL Reduction Definition (Mow, Ling et al’06)
Algorithm should be modified accordingly
50% saving in computational complexity (the number of floating-point operations)
Close performance despite worse approximation factor 1/(-1/2)(n-1)/4
complex calculation

22. Simulation (4x4)

23. Simulation (10x10)

24. How to Determine the Gap?
Gap to infinite lattice decoding
Approach
Examine decision regions
Compare the distances with that of lattice decoding
Partial answer
Up to a factor of 2 for complex Gauss reduction (Yao and Wornell’02)
Cannot capture boundary effect

25. Proximity Factor Definition A measure of the proximity
Quantify the worst-case loss in power efficiency
Not the same as standard approximation factors
Proximity factors are upper-bounded by a constant that depends on n only (Ling’07)
Gauss reduction
KZ reduction
LLL reduction

26. Worse vs. average-case The gap is widening as n gets larger
Dual reduction sometimes leads to smaller bounds
The worse-case bounds on approximation (proximity) factors are quite pessimistic
Do not completely characterize the performance
Average-case analysis?

27. LLL vs. HKZ (Stehle)

28. Specialities of Lattices in MIMO
The basis B
Complex-valued
i.i.d. normal distribution, or other distributions, or correlated
Represented by floating-point numbers
A fast reduction algorithm for integer lattices is not necessarily useful here
Exact arithmetic is not needed
Floating-point algorithms
Dimension can go from 2, 3, 4, … to hundreds (multiuser coded MIMO)
Pre-processing is allowed in some cases (e.g., when channel is fixed)
Since y=Bx+w, y is more likely to be close to a lattice point

29. Questions Lattice reduction over other number fields
Fast lattice reduction for time-varying channels
Incremental LLL, Segment LLL, pairwise reduction?
More efficient enumeration methods than sphere decoding
Better methods than Babai
Bounded-distance decoding
How to deal with boundary effect
Average-case performance analysis of lattice decoding
Average complexity analysis of sphere decoding with lattice reduction
Not straightforward

30. Questions for reduction
Minimize the Frobenious norm
Fixed-complexity LLL, parallel LLL in hardware implementation
the optimal criteria is to min the error prob, but that is intractable

31. Other Applications Quantization ‘Lattices are everywhere’ (Zamir’09)
Continuous signal  digital signal
‘Lattices are everywhere’ (Zamir’09)
Multi-terminal communication, network information theory
Lattices offer structured codes
Alternative to algebra, graph, trellis, tree…
How to make lattice codes more practical at large dimensions?
Interaction between coding and lattice