On the ergodic rate for compute-and-forward

Abstract

Abstract-A key issue in compute-and-forward for physical layer network coding scheme is to determine a good function of the received messages to be reliably estimated at the relay nodes. We show that this optimization problem can be viewed as the problem of finding the closest point of Z [i] n to a line in the n-dimensional complex Euclidean space, within a bounded region around the origin. We then use the complex version of the LLL lattice basis reduction (CLLL) algorithm to provide a reduced complexity suboptimal solution as well as an upper bound to the minimum distance of the lattice point from the line. Using this bound we are able to find a lower bound to the ergodic rate and a union bound estimate on the error performance of a lattice constellation used for lattice network coding. We compare performance of the CLLL with a more complex iterative optimization method as well as with a simple quantized search. Simulations show how CLLL can trade some performance for a lower complexity.Index Terms-Ergodic rate, compute-and-forward, CLLL algorithm, quantized error, successive refinement.

Compute-and-forward protocol design based on improved sphere decoding

“…• The branch-and-bound (BnB) algorithm [4]. • The method based on LLL lattice reduction algorithm [5], where δ is set as 0.75 since further increasing δ achieves little gain in rate but will greatly increase the running time. • The quadratic programming relaxation (QPR) approach [10].…”

“…dist(k) = newdist; 14: d k = t kk /r kk ; // see (17), (19) 15: a k = ⌊d k ⌉; 16: s k = sgn(d k − a k ); [5]. The QS method implemented here consists of two phases: 1) selecting an amplification factor α = α 0 ∈ {1, 2, .…”

“…Besides these, the quadratic programming relaxation method in [10] is of relatively low complexity. Although its performance in terms of the computation rate is better than that of the last two methods proposed in [5], the differences between its performance and that of the optimal methods is still large when the dimension is large and the SNR is high.…”

“…There are methods that give suboptimal solutions. Three methods were proposed in [5]: the one based on the complex LLL [6], the simple quantized search method and the iterative MMSE-based quantization method. Although the average complexity of the LLL algorithm [7] is polynomial if the entries of the basis vectors independently follow the normal distribution N (0, 1) (see, e.g., [8], [9]), the cost of the first method maybe too high since it was proved in [8] that in the MIMO context, the worstcase complexity of the LLL algorithm is not even finite.…”

“…: determine β by (20); 2: dist = 0 n ; // dist(k) = n i=k+1 r 2 ii (a i − d i ) 2 for k < n; 3: d = 0 n ; // computing d by(17)4: a = 0 n ; // intermediate solution5: T = 0 n×n ; // see (19) 6: step = 0 n , step(n) = −1, γ = 0, k = n; 7: while k ≥…”