L2-Orthogonal ST-Code Design for Multi-h CPM with fast Decoding

HAL Id: hal-01251193

https://hal.inria.fr/hal-01251193

Submitted on 5 Jan 2016

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

L2-Orthogonal ST-Code Design for Multi-h CPM with fast Decoding

Miguel Angel Hisojo, Jerome Lebrun, Luc Deneire

To cite this version:

Miguel Angel Hisojo, Jerome Lebrun, Luc Deneire. L2-Orthogonal ST-Code Design for Multi-h CPM with fast Decoding. 2013 IEEE European School of Information Theory, Apr 2013, Ohrid, Macedonia. 2013. �hal-01251193�

L2-Orthogonal ST-Code Design for Multi-h CPM with fast Decoding

Miguel Angel Hisojo, J´erˆome Lebrun, Luc Deneire

I3S Lab. CNRS University of Nice Sophia Antipolis, France

Abstract

⊕ CPM: favorable trade-off between power and bandwidth efficiency.

⊕ Multi-h CPM: generalization to further decrease the need for bandwidth. ⊖ Difficult decoding in multi-path environments with no diversity.

How to overcome these limitations:

⊕ IDEA: To combine CPM with Space-Time Block Coding (STBC).

→ Non trivial extension to L2-orthogonal Space-Time codes provides full diversity and better spectral compactness.[1][2]

→ Decoding complexity greatly decreased.[1][2]

The System Model

M-ary multi-h MIMO system of L_{t × Lr} antennas. r_{(t, d) = Hs(t, d) + n(t)} r_{(t, d)(L} r×1), s(t, d)(Lt×1) H_(L r×Lt) with elements hn,m n_{(t) AWGN.}

The vector of transmitted signals: s_{(t, d) =} s

l(t, d) . . . sL_t(t, d)T

The baseband general form [3] s(t, d) = r Es

T exp(jφ(t, d)) The information-carrying phase function

φ(t, d) = 2π

K−1

X

k=0

h_[k]d_{kq(t − (k − 1)T )}

modulation indices h₁, . . . , h_H, cycle in time with period H as:

[k] = mod(k, H) + 1

h_[k] quotient between two relative prime integers.

H_k = (h₁, h₂, . . . , h_k) =  p1 q , p₂ q , . . . , p_k q 

The phase continuity is ensured by the phase pulse q(t), q(t) = ( 0 t < 0 ¯ q(t) elsewhere 1/2 _{t ≥ γT} ¯ q(0) = 0, ¯q(γT ) = 1/2 and lim t→τ q(t) = ¯¯ q(τ ) for 0 _{≤ τ < γT .} γ is the overlapping factor and

the symbol index is given by k = _{⌈t/T ⌉}

L2 Orthogonality

Based on _{k s k}2_L2= R s(t)sH(t)dt, each block fulfills the condition

l_FL_BT

Z

(l_f−1)LBT

s_{(t, d)s}H_{(t, d)dt = I}

The phase continuity is ensured by an antenna dependent phase memory: θ_m,k = θ_m,k−1 + h[k−γ]

2 dm,k−γ + cm,k(T ) − cm,k(0).

Auto-correlation coefficients cancel and cross-correlation coefficients between antennas put to 0 L_B P k=1 kT R (k−1)T exp(j2π[θm,k + k P i=k−γ+1 h_[i]dm,iq(t − (i − 1)T ) + cm,k(t) − θm′,k − k P i=k−γ+1 h_[i]d_m′_,iq(t − (i − 1)T ) − c_m′_,k(t)])dt = 0

To ease the design, two assumptions are introduced, 1. c_m,k(t) = c_m,k′(t)

2. d_m,k = d_m′_,k

For any arbitrary number of transmit antennas, we introduce correction functions c_m,k as: clin_m (t) = m − 1

L_BT t for (k − 1)T < t < kT For m = 1, . . . , L_t and the transmitted signal takes the form of

s_m(t, d) = s(t, d) exp(j2clin_m (t)) L2 Decoding 0 1 2 3 −3π/2 −πι/2 −πι 0 πι πι/2 3πι/2 −3π/2 time (t) φ (t,d) h 2 h 1 h 1

Fig. _{1: Inner-block phase transition for h1 =} 1

4 and h₂ = 3₄, · · · Θ(1) Θ(pMγ−1₎ Θ(2) L_BT 2LBT 3LBT inter-block inner-block MLt pps trellis trellis Θ(1)_Θ(2) Θ(pMγ−1₎ D(d, ˜d₎ M pps, D(d, ˜d₎ L_BT 2LBT

Fig. _{2: Simplified detection with k = 1 (} pps-paths per state).

Classical correlation based multi-h detector expressed blockwise:

D(d, ˜d_{|Θ(k)) =} L_F X k=1 Nsk X n=1 (k+1)T L_B Z kT L_B Re ( r(t, d1_[n], . . . , dL_B[n]) · Lt X m=1 h∗_ms∗_m(t, θ, ˜d₁ [n], . . . , ˜dLB[n]) !) dt ⊖ Impractical:

pM(LtLBNs+γ−1) matched filters of length N_sL

BT employed pMγ−1 times.

pMLB−1 states and MLBLT symbols to decode with N_s samples each.

⊕ Orthogonality: the cross correlation terms are canceled out with blockwise decoding.

DB(d, ˜d|Θ(k)) = L_T X m=1 lT Z (l−1)T Re{r(t, d)h∗mc∗m(t)s∗(t, ˜d)}dt.

To get decoding with complexity growing linearly w.r.t number of transmit antennas, we introduce

x(t, d) = r(t, d)

L_T

X

m=1

h∗_mc∗_m(t).

A simplified expression for a classical multi-h decoder [4] then is given by

D(d, ˜d_{|Θ(k)) =} lT Z (l−1)T Re{x(t, d) · s(t,d˜₎ |φl }dt

Hence our detector based on the correlations gives us an expression which maximizes the minimum distance as Dr(d, ˜d|Θ(k)) = arg max Θ(1)→Θ(pMγ−1)      lT Z (l−1)T Re{x(t, d)s(t,d˜_)}dt      .

The simplified detector can now be represented in a similar form as in [5],

˜ d D ec o d er d CPM r(t, d) s¯ ∗_{(t, ˜d)} h∗ 1¯c∗1(t) pseudo channel ¯H(t) h∗ Ltc¯ ∗ Lt(t) ¯ c_Lt(t) ¯ c₁(t) Simulation Results 5 10 15 20 25 10−5 10−4 10−3 10−2 10−1 100 E b/N0 in dB

Bit Error Rate − (BER)

Tx 1 h={4,6}/16 Tx 2 h={4,6}/16 Tx 3 h={4,6}/16 Tx 1 h={4,2}/12 Tx 2 h={4,5}/12 Tx 3 h={4,5}/12 Tx 1 h={4,6}/15 Tx 2 h={4,6}/15 Tx 3 h={4,6}/15 Tx=3 Tx=1 Tx=2

Fig. _{3: BER for Tx = 1, 2, 3, LB = 2 and} h_i = _{{4, 5}/12, hi} = _{{4, 6}/16, hi} = _{{4, 6}/15.}

Conclusions √

L2-orthogonal STBC provide full diversity by the construction of orthogonal waveforms. √

We have shown that the inner trellis is equivalent to the inter-trellis for multi-h CPM. √

By choosing H _{≤ LB} we get the largest minimum distance in a block. √

L2-orthogonal STBC satisfies the needs for energy and spectral efficiency. √

Decoding complexity increases as P M instead of P M MLT .

References

[1] M. Hisojo, J. Lebrun, and L. Deneire. Wireless robotics: Generalization of an efficient approach with multi-h CPM signaling and L2-Orthogonal space-time coding. Wireless Personal Communications, 2013.

[2] M. Hisojo, J. Lebrun, and L. Deneire. L2-Orthogonal ST-Code design for multi-h CPM with fast decoding. ICC proceedings, 2013.

[3] J.B. Anderson T. Aulin and C.E. Sundenber. Digital Phase Modulation. 1986.

[4] J. B. Anderson and D. P. Taylor. A bandwidth-efficient class of signal-space codes. IEEE. Trans. Inf. Theory, 1978.

[5] M. Hesse, J. Lebrun, and L. Deneire. L2-Orthogonal ST-Code design for CPM. IEEE Trans. Commun., 2011.