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Wideband Communications

Wideband Communications . Lecture 6-7: Discrete Channel partitioning Aliazam Abbasfar. Outline. Discrete Channel partitioning Vector coding Discrete modal modulation Discrete multi-tone (DMT)/OFDM. Discrete channel partitioning.

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Wideband Communications

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  1. Wideband Communications Lecture 6-7: Discrete Channel partitioning Aliazam Abbasfar

  2. Outline • Discrete Channel partitioning • Vector coding • Discrete modal modulation • Discrete multi-tone (DMT)/OFDM

  3. 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: • mn is a (N+v)-dimension vector • fn(t) = Smk,nf(t-kT’) • x = SXnmn • x(t) = SXnfn(t-kT’) • p(t) = f(t) * h(t) * f*(-t) • y = P x + n

  4. 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’

  5. 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 : • If complex channel : N+v 2(N+v)

  6. Discrete modal modulation • Transmit basis is zero during prefix period • dimension of x = N • Exact matched filter at the receiver • p(t) = f(t) * h(t) * h*(-t) * f*(-t) • y = P x + n • Eigen-vector coding • P is Hermitian • P = F L F* • F is an NxN unitary matrix • L is an NxN diagonal matrix with singular values ln • x = F XT = SXn fn • y = P x + n = F L X + n • Y = F* y = L X + N • Yn = lnXn + Nn • Output noises are independent • n(t) = n0(t) * h*(-t) * f*(-t) • Rn(t) = s2 p(t) • E[nn*] = Rnn = s2 P • E[NN*]= F* P F = s2L • SNRn = ln/ s2

  7. Discrete multi-tone (DMT) • Cyclic prefix + vector coding • Simplifies processing • Channel independent • Eigen-vector coding • P is circulant • P = Q* L Q • Q is an NxN DFT matrix • L is an NxN diagonal matrix with eigen-values ln • x = Q* XT = SXn fn • y = P x + n = Q* L X + n • Y = Q y = L X + N • Yn = lnXn + Nn • lnare the DFT of channel response • Q P = L Q • ln= Pn

  8. DMT/OFDM • Use IFFT and FFT for IDFT and DFT • complexity : N log2(N) • DMT/OFDM performs as well as VC as N   • Wasted energy : v/(N+v)

  9. Noise in DMT/OFDM • Noise is usually colored • E[nn*] = s2Rnn • N = Q n • E[NN*] = s2QRnnQ* • If Rnn is circulant, E[NN*] = diag( sn2 ) • sn2 = Sn(n/NT’) • gn = |ln|2/ sn2 • SNRn = En |ln|2/ sn2 • Energy should be scaled by N/(N+v)

  10. Examples • Channel • h(t) = 1 + 0.9 D-1 |H(f)|2 = 1.81 + 1.8 cosw • N0/2 = 0.181  g(f) = 10( 1 + cosw ) • G = 1 • MF • SNRMFB= E/s2 = 1.81/0.181 = 10 • MT capacity • Sx(f) = K – 1/g(f) • K = 1.33 fmax = 0.44 • c = 1.55 bits/sec • Multi-channel • N = 8, v=1  T = 9 • E = 9 (Eavg = 1), Pavg = 1 • VC • svd singular values: • Energy allocation : • DMM • The same as VC

  11. Examples • Channel • h(t) = 1 + 0.9 D-1 |H(f)|2 = 1.81 + 1.8 cosw • N0/2 = 0.181  g(f) = 10( 1 + cosw ) • G = 1 • DMT • Eigen values, energy allocations • b = 1.38 bits /sec

  12. ADSL/VDSL • ADSL : The most popular broadband Internet service • Over telephone lines • ITU .G992.1 • DMT : T = 250 usec • Down stream • 256 tones, 4.3125 KHz spacing, real baseband • (ADSL2+ /VDSL -> 512/4096 tones) • 1/T’ = 2.208 MHz ( BW = 1.104 MHz) • N + v = 512 + 40 (Hermitian symetry ) • 2-3 tones are not used (phone line) • Tone 64 is pilot ( known QPSK data), Tone 256 not used • Pmax = 20.5 dBm • Up stream • Upstream transmission uses 32 tones to frequency 138 KHz • 1/T’ = 276 KHz ( BW = 138 KHz) • N + v = 64 + 5 (Hermitian symetry ) • 1st tone not used (phone line) • Pmax = 14.5 dBm • Upto 12/1.5 Mbps down/upstream • Bit loading to optimize data rate • bmax = 15

  13. WiFi • Wireless LAN • 802.11a/g @ 5/2.4 GHz • COFDM : T = 4 usec • 64 tones, 312.5 KHz spacing, complex baseband • 1/T’ = 20 MHz BW = 20 MHz (15.56) • N + v = 64 + 16 • Tones : -31 to 31 (48 tones for data ) • -31 to -27, 0, 27 to 32 are not used • -27, -7, 7, and 21 are pilot • Data rate = k * 48 * 250 KHz = 12k Mbps k = bn: bits/tone • Upto 54 Mbps • Variable coding • No bit loading • bn is constant for all tones • Pmax = 16/23/29 dBm

  14. Digital Video Broadcast (DVB) • Digital TV broadcast • Single frequency network (SFN) • Improves coverage • Creates ISI • COFDM • 2048 or 8192 tones, 4.464/1.116 KHz spacing • complex baseband • 1/T’ = 9.142 MHz BW = 20 MHz (15.56) • N T’ = 8192 T’ = 896 usec (1/1.116 KHz) • (N+v) T’ = 924/952/1008/1120 usec • N T’ = 2046 T’ = 224 usec (1/4.464 KHz) • (N+v) T’ = 231/238/252/280 usec • 4/16/64 QAM • Coding : 172/204 * 1/2, 2/3, 3/4, 5/6, or 7/8 • Data rates : 4.98  31.67 Mbps • Carries 2-8 TV channels

  15. Reading • Cioffi Ch. 4.6

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