1. 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 : T H = (v+1)T’ Discrete basis functions: m n is a (N+v)-dimension vector  n (t) =  m k,n (t-kT’) x =  X n m n x(t) =  X n  n (t-kT’) p(t) = (t) * h(t) *   (-t) y = P x + n

4. Vector coding Singular value decomposition P = F [  ] M* M is an (N+v)x(N+v) unitary matrix F is an NxN unitary matrix  is an NxN diagonal matrix with singular values n 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 [X N-1 … X 1 X 0 | 0 … 0] T =  X n m n y = P x + n = F X + n Y = F* y = X + N Y n = n X n + N n Noises are independent with the same variance Colored noise E[nn * ] = R nn = L L * Whitening y’ = L -1/2 y = L -1/2 P x + L -1/2 n = F’ X + n’ Y’ = F’* y’ = F’* L -1/2 y = X + N’

5. Vector coding (2) SNR n for sub-channels g n = | n | 2 /(N 0 /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) = (t) * h(t) * h  (-t) *   (-t) y = P x + n Eigen-vector coding P is Hermitian P = F  F* F is an NxN unitary matrix  is an NxN diagonal matrix with singular values n x = F X T =  X n f n y = P x + n = F X + n Y = F* y = X + N Y n = n X n + N n Output noises are independent n(t) = n 0 (t) * h  (-t) *   (-t) R n () =  2 p() E[nn * ] = R nn =  2 P E[NN*]= F* P F =  2  SNR n = n / 2

7. Discrete multi-tone (DMT) Cyclic prefix + vector coding Simplifies processing Channel independent Eigen-vector coding P is circulant P = Q*  Q Q is an NxN DFT matrix  is an NxN diagonal matrix with eigen-values n x = Q* X T =  X n f n y = P x + n = Q* X + n Y = Q y = X + N Y n = n X n + N n n are the DFT of channel response Q P = Q n = P n

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 * ] =  2 R nn N = Q n E[NN * ] =  2 QR nn Q* If R nn is circulant, E[NN * ] = diag(  n 2 )  n 2 = S n (n/NT’) g n = | n | 2 / n 2 SNR n = E n | n | 2 / n 2 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 cos  N 0 /2 = 0.181  g(f) = 10( 1 + cos  ) = 1 MF SNR MFB = E/ 2 = 1.81/0.181 = 10 MT capacity S x (f) = K – 1/g(f) K = 1.33 f max = 0.44 c = 1.55 bits/sec Multi-channel N = 8, v=1  T = 9 E = 9 (E avg = 1), P avg = 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 cos  N 0 /2 = 0.181  g(f) = 10( 1 + cos  ) = 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 P max = 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 ) 1 st tone not used (phone line) P max = 14.5 dBm Upto 12/1.5 Mbps down/upstream Bit loading to optimize data rate b max = 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 = b n : bits/tone Upto 54 Mbps Variable coding No bit loading b n is constant for all tones P max = 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