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EE360 – Lecture 3 Outline

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. Broadcast Channels with ISI. ISI introduces memory into the channel

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EE360 – Lecture 3 Outline

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  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. H1(w) H2(w) Broadcast Channel Model w1k • Both H1 and H2are finite IR filters of length m. • The w1kand w2k 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=(R1,R2). xk w2k

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

  5. 0<j<n 0<j<n Equivalent Channel Model • Taking DFTs of both sides yields • Dividing by H and using additional properties of the DFT yields ~ where {V1j} and {V2j} are independent zero-mean Gaussian random variables with

  6. Parallel Channel Model V11 Y11 + X1 Y21 + V21 Ni(f)/Hi(f) V1n f Y1n + Xn Y2n + V2n

  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<b find {aj}, {Pj} to maximize R1+bR2+lSPj. • Let (R1*,R2*)n,b denote the corresponding rate pair. • Cn={(R1*,R2*)n,b : 0<b  }, C=liminfnCn . b R2 R1

  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 g1[i] n1[i] Y1[i] x X[i] Y2[i] x n2[i] g2[i] At each time i: n={n1[i],n2[i]} n1[i]/g1[i] Y1[i] X[i] Y2[i] n2[i]/g2[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, mRM+, boundary vectors satisfy: • Lagrangian method: • Must optimize power between users and over time

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

  19. Time Division • For each fading state n, allocate power Pj(n) and fraction of time tj(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

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