1. EE360 – Lecture 3 Outline Announcements: Classroom Gesb131 is available, move on Monday? Broadcast Channels with ISI DFT Decomposition Optimal Power and Rate Allocation Fading Broadcast Channels

2. Broadcast Channels with ISI ISI introduces memory into the channel The optimal coding strategy decomposes the channel into parallel broadcast channels Superposition coding is applied to each subchannel. Power must be optimized across subchannels and between users in each subchannel.

3. Broadcast Channel Model Both H 1 and H 2 are finite IR filters of length m. The w 1k and w 2k are correlated noise samples. For 1<k<n, we call this channel the n-block discrete Gaussian broadcast channel (n-DGBC). The channel capacity region is C=(R 1,R 2 ). w 1k H1()H1()H2()H2() w 2k xkxk

4. Circular Channel Model Define the zero padded filters as: The n-Block Circular Gaussian Broadcast Channel (n-CGBC) is defined based on circular convolution: 0<k<n where ((. )) denotes addition modulo n.

5. Equivalent Channel Model Taking DFTs of both sides yields Dividing by H and using additional properties of the DFT yields 0<j<n ~ where {V 1j } and {V 2j } are independent zero-mean Gaussian random variables with 0<j<n

6. Parallel Channel Model + + X1X1 V 11 V 21 Y 11 Y 21 + + XnXn V 1n V 2n Y 1n Y 2n N i (f)/H i (f) f

7. Channel Decomposition The n-CGBC thus decomposes to a set of n parallel discrete memoryless degraded broadcast channels with AWGN. Can show that as n goes to infinity, the circular and original channel have the same capacity region The capacity region of parallel degraded broadcast channels was obtained by El-Gamal (1980) Optimal power allocation obtained by Hughes-Hartogs(’75). The power constraint on the original channel is converted by Parseval’s theorem to on the equivalent channel.

8. Capacity Region of Parallel Set Achievable Rates (no common information) Capacity Region For 0<   find {  j }, {P j } to maximize R 1 +  R 2 +  P j. Let (R 1 *,R 2 * ) n,  denote the corresponding rate pair. C n ={(R 1 *,R 2 * ) n,  : 0<   }, C =liminf n C n.  R1R1 R2R2

9. Limiting Capacity Region

10. Optimal Power Allocation: Two Level Water Filling

11. Capacity vs. Frequency

12. Capacity Region

13. Fading Broadcast Channels Broadcast channel with ISI optimally allocates power and rate over frequency spectrum. In a fading broadcast channel the effective noise of each user varies over time. If TX and all RXs know the channel, can optimally adapt to channel variations. Fading broadcast channel capacity region obtained via optimal allocation of power and rate over time Consider CD, TD, and FD.

14. Two-User Channel Model + + X[i] 1 [i] 2 [i] Y 1 [i] Y 2 [i] x x  g 1 [i]  g 2 [i] + + X[i] 1 [i]/  g 1 [i] 2 [i]/  g 2 [i] Y 1 [i] Y 2 [i] At each time i: n={n 1 [i],n 2 [i]}

15. CD with successive decoding M-user capacity region under CD with successive decoding and an average power constraint is: The power constraint implies

16. Proof Achievability is obvious Converse Exploit stationarity and ergodicity Reduces channel to parallel degraded broadcast channel Capacity known (El-Gamal’80) Optimal power allocation known (Hughes- Hartogs’75, Tse’97)

17. Capacity Region Boundary By convexity,  R M +, boundary vectors satisfy: Lagrangian method: Must optimize power between users and over time

18. Water Filling Power Allocation Procedure For each state n, define  (i):{n  (1)  n  (2)  …  n  (M) } If set P  (i) =0 (remove some users) Set power for cloud centers Stop if,otherwise remove n  (i), increase noises n  (i) by P  (i), and return to beginning

19. Time Division For each fading state n, allocate power P j (n) and fraction of time  j (n) to user j. Achievable rate region: Subject to Frequency division equivalent to time-division

20. Optimization Use convexity of region: boundary vectors satisfy Lagrangian method used for power constraint Four step iterative procedure used to find optimal power allocation For each n the channel is shared by at most 2 users Suboptimal strategy: best user per channel state is assigned power – has near optimal TD performance

21. CD without successive decoding M-user capacity region under CD with successive decoding and an average power constraint is: The best strategy for CDWO is time-division