Should Coded Modulation Use Nyquist Pulses John B

Should Coded Modulation Use Nyquist Pulses? John B Anderson High Speed Wireless Center and Electrical & Information Technology Dept. Lund University, Sweden

Outline • • • Faster than Nyquist signaling Capacity of signals with a PSD Linear modulation with orthogonal pulses Extend to FTN signaling Dmin, capacities and performance of FTN The outcome: Coded linear modulation based on nonorthogonal pulses has superior information rates Lund University / Department of Electrical and Information Technology

Shannon’s Square PSD Capacity P/2 W bits/s -W 0 W Consideration of Shannon’s argument shows: • As W grows, P/W is fixed if Es and Eb are fixed (NB: Es=P/2 W). • So log[1+P/WNo] data bits/s are carried in each +Hz of the square PSD. • This is a flow of bits, per Hz-s. • For fixed P/No, flow is maximized by the square PSD. • But the square PSD stems from sinc pulses. Lund University / Department of Electrical and Information Technology

Capacity at Different Energies & Bandwidths per Data Bit Physical channel has bandwidth W, power P and Es = P/2 W Lund University / Department of Electrical and Information Technology

What is a Faster than Nyquist Signal? Lund University / Department of Electrical and Information Technology

Faster-than-Nyquist Signaling (Mazo, 1975) • Baseband signals of form • • • Reduce symbol time to τT ≤ T Euclid. min. distance preserved for Asymptotic error performance thus unaffected Allows higher data bits/Hz More bandwidth efficiency at same energy • The new form: Lund University / Department of Electrical and Information Technology

FTN Example Send 1 -1 1 – 1 - 1 Lund University / Department of Electrical and Information Technology

Information Rates for Un-Square PSDs Now suppose the signals have a PSD that is not square. Lund University / Department of Electrical and Information Technology

AWGN Information Rate for a PSD = , power P Generalization of Low power example: Low bit density coding Fix W at 1 Hz 0. 5 Square PSD, C=1 b/Hz-s 30% rt. RC, C=1. 03 b/Hz-s -1 0 1 Lund University / Department of Electrical and Information Technology

Increase the Power. . . but not the bandwidth. So, more bits/Hz-s. High bit density coding 500 Square PSD, C=9. 97 b/Hz-s 30% rt. RC, C=11. 81 b/Hz-s -1 0 1 In the limit P → ∞, C for 30% rt. RC → 1. 3 _______ C for square PSD (i. e. , 1 + excess bandwidth factor) Lund University / Department of Electrical and Information Technology

Why Does This Happen? Reason 1: C is linear in W, but log in P Reason 2: Moving PSD parts by Nyquist’s reflection principle* can only increase C * A 0 * *A pulse is T-orthogonal if its PSD is antisymmetrical about (A/2, 1/2 T) The effect grows stronger with bit density (noticeable > 3 bits/Hz-s) Lund University / Department of Electrical and Information Technology

More about Bit Densities. . . • Binary antipodal signaling: 1 data bit / (T s)(1/2 T Hz) = 2 bits/Hz-s • QPSK: 2 bits / (T s)(1/T Hz) = 2 bits/Hz-s • Convolutional R=1/2 coding + above: 1 bit/Hz-s • DTV, 64 QAM + R=1/2 coding: (1/2) 6 / (T s)(1/T Hz) = 3 bits/Hz-s Lund University / Department of Electrical and Information Technology

Capacity and FTN with Linear Modulation Lund University / Department of Electrical and Information Technology

Coded Orthogonal Pulse Linear Modulation • Most modulations/coded modulations are this, with h T-orthogonal • Average PSD of s(t) is , even if h(t) not T-orthogonal • Form codes as subsets of these signals, having same PSD • Capacity for s(t) signal set is same for any orthogonal h(t). So it must be the square PSD C. • But !! Lund University / Department of Electrical and Information Technology

What Does Sq PSD C < RC PSD C Mean for Linear Modulation? • Orthogonal h(t) cannot achieve C for PSD shape unless sinc h • Can Faster than Nyquist linear modulation ? ? • Let’s try an FTN h(t) ! • Same h(t), coming faster - h(t) not orthogonal - Higher bits/Hz-s, by - Same average PSD Lund University / Department of Electrical and Information Technology

Dmin and Capacity in FTN Signaling • Fact: Consider set of s(t) signals made from For sinc pulse, (25% more bits/Hz-s, same PSD, Pe ) For rt. RC pulse, (42% more) • Theorem: FTN codewords can achieve the PSD capacity (Ph. D. Thesis, F. Rusek, 2007; Rusek & Anderson, IT Trans. , Feb. 2009) For 30% rt. RC, need • In fact, codes based on binary FTN can have higher rate than any code based on orthogonal pulses. Lund University / Department of Electrical and Information Technology

FTN Constrained Capacities Under bounds to capacity for binary-data FTN based on 30% rt. RC pulse (red). Nyquist C (green) has no input constraint. Top C is capacity for PSD (blue). ( Rusek & Anderson, Trans. IT, Feb. 09) Lund University / Department of Electrical and Information Technology

Shannon BER Bound and 2 Systems at 3 Bits/Hz-s FTN: A. Prlja, 2008 TCM: W. Zhang, 1995 Lund University / Department of Electrical and Information Technology

What about Sinc Pulses? • No gain in C available if h(t) is a sinc ! (but sinc FTN linear modulation is better than sinc without FTN) • But is coding based on sinc of practical interest? • Can show: For finite frames, sinc is not an optimal use of time and bandwidth (does not give the best time - frequency occupancy) Lund University / Department of Electrical and Information Technology

Conclusions • Signal capacity must be computed from true PSD at high bit density • Extra information rate available with non-sinc pulses • It exceeds that available with any use of orthogonal pulses • FTN linear modulation can achieve this capacity Lund University / Department of Electrical and Information Technology

Tack! Lund University / Department of Electrical and Information Technology