1 / 12

Wideband Communications

Wideband Communications . Lecture 4-5: Channel partitioning Aliazam Abbasfar. Outline. Channel partitioning. How to partition channels?. Partition the signal into N dimensions (sub-channels) Signal duration : T Channel Bandwidth/Signal energy limits N Continuous transmission

eagan-kaufman
Download Presentation

Wideband Communications

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. Wideband Communications Lecture 4-5: Channel partitioning Aliazam Abbasfar

  2. Outline • Channel partitioning

  3. How to partition channels? • Partition the signal into N dimensions (sub-channels) • Signal duration : T • Channel Bandwidth/Signal energy limits N • Continuous transmission • Received signal

  4. ISI and ICI • ICI : Inter-Channel Interference • The interference between sub-channels • Intra Symbol Interference • Orthogonal sub-channels = zero ICI • ISI : Inter Symbol Interference • The interference between subsequent symbols (duration T) • Generalize Nyquist criterion: • qn,m(kT) = d(n-m,k) • Q(k) = d(k) I

  5. Modal Modulation • Optimum transmit basis • Eigen-functions of channel autocorrelation function • r(t) = h(t)*h*(-t) • fn(t) * r(t) = rnfn(t) • Orthonormal set exists • The time limit avoids ISI • The noise samples at MFs are independent • E[ninj] = ri N0/2 di,j • SNRn = (rn)2 En /rn N0/2 = rn En / (N0/2) • gn = rn / (N0/2)

  6. Modal Modulation (2) • Each eigen-function is a mode (Modal modulation) • gn = rn /(N0/2) • Operate in N best modes (N largest eigen-values) • Each mode is detected independently (best performance) • Optimality • Eigen-functions maximizes the bit rate for N dimensions (for a given set of {En}) • En is computed by water-filling algorithm • Best average SNR for any N orthonormal basis • Disadvantages: • Channel dependent • Difficult to solve • except for no ISI channels : r(kT) = d(k) • Can be found using sampling

  7. Other transmit basis • Time-division partitioning • Channel causes both ICI and ISI • ICI and ISI is removed by equalization • ISI can be removed by Guard Time • Frequency division partitioning (MT) • No ICI when no overlap in frequency • No ISI if sub-channel BW is small • Practical filters = Excess bandwidth • MT is MM when T is very large • Best energy allocation

  8. OFDM • Orthogonal frequency division multiplexing • fn(t) = rect(t/T) ej2pnf0 t • f0 = 1/T • Guard time removes ISI and ICI • Spectrum • Sinc functions • Cyclic prefix • Periodic channel • Requires multiple OSCs with precise frequency • Analog implementation is hard • Residual ICI add up

  9. Discrete channel partitioning • We look at the discrete-time representation of the system • Channel response (p(t)) is sampled at 2x the highest frequency to be used • A vector of (N+v) samples is considered a symbol • T= (N+v) T’ • Each sample can be one dimension (N+v dimensions) • Guard period of v samples is considered for no ISI • Channel response span of time : TH = (v+1)T’ • Discrete basis functions: • fn(t) = Smk,nf(t-kT’) • mn is a (N+v)-dimension vector • p(t) = f(t) * h(t) * f*(-t) • y = P x + n

  10. Vector coding • Singular value decomposition • P = F [L | 0 N, n] M* • M is an (N+v)x(N+v) unitary matrix • F is an NxN unitary matrix • L is an NxN diagonal matrix with singular values ln • Vector coding • Choose the first N column of M as transmit basis • Choose N column of F to get x • Discrete matched filters • x = M [X | 0 … 0]T = M [XN-1 … X1 X0 | 0 … 0]T = SXnmn • y = P x + n = F L X + n • Y = F* y = L X + N • Yn = lnXn + Nn • Noises are independent with the same variance • Colored noise • E[nn*] = Rnn = L L* • Whitening • y’ = L-1/2 y = L-1/2 P x + L-1/2 n = F’ L X + n’ • Y’ = F’* y’ = F’* L-1/2 y = L X + N’

  11. Vector coding (2) • SNRn for sub-channels • gn = |ln|2 /(N0/2) • Water-filling to allocate energy • Note that there are (N+v) dimensions • Bit rate :

  12. Reading • Cioffi Ch. 4.4-5

More Related