1 / 21

Outline

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

lyle-walker
Download Presentation

Outline

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Should Coded Modulation Use Nyquist Pulses?John B AndersonHigh Speed Wireless Center andElectrical & Information Technology Dept.Lund University, Sweden

  2. 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

  3. P/2W -W W 0 Shannon’s Square PSD Capacity bits/s • Consideration of Shannon’s argument shows: • As W grows, P/W is fixed if Es and Eb are fixed (NB: Es=P/2W) . • 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.

  4. Capacity at Different Energies & Bandwidths per Data Bit Physical channel has bandwidth W, power P and Es = P/2W

  5. What is a Faster than Nyquist Signal?

  6. 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:

  7. FTN Example Send 1 -1 1 – 1- 1

  8. Information Rates for Un-Square PSDs Now suppose the signals have a PSD that is not square.

  9. 0.5 0 -1 1 AWGN Information Rate for a PSD PSD = , power P Generalization of Low power example: Low bit density coding Fix W at 1 Hz Square PSD, C=1 b/Hz-s 30% rtRC, C=1.03 b/Hz-s

  10. 500 0 -1 1 Increase the Power ...but not the bandwidth. So, more bits/Hz-s. High bit density coding Square PSD, C=9.97 b/Hz-s 30% rtRC, C=11.81 b/Hz-s In the limit P → ∞, C for 30% rtRC C for square PSD _____________ → 1.3 (i.e., 1 + excess bandwidth factor)

  11. * A 0 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 pulse is T-orthogonal if its PSD is antisymmetrical about (A/2, 1/2T) The effect grows stronger with bit density (noticeable > 3 bits/Hz-s)

  12. More about Bit Densities ... • Binary antipodal signaling: 1 data bit / (T s)(1/2T 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, 64QAM + R=1/2 coding: (1/2) 6 / (T s)(1/T Hz) = 3 bits/Hz-s

  13. Capacity and FTN with Linear Modulation

  14. 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 !!

  15. 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

  16. 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 rtRC 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% rtRC, need • In fact, codes based on binaryFTN can have higher rate than any code based on orthogonal pulses.

  17. FTN Constrained Capacities Under bounds to capacity for binary-data FTN based on 30% rtRC pulse (red). Nyquist C (green) has no input constraint. Top C is capacity for PSD (blue). ( Rusek & Anderson, Trans. IT, Feb. 09)

  18. Shannon BER Bound and 2 Systems at 3 Bits/Hz-s FTN: A. Prlja, 2008 TCM: W. Zhang, 1995

  19. 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)

  20. 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

  21. Tack!

More Related