# 5. Capacity of Wireless Channels

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1 5. Capacity of Wireless Channels 1 Information Theory So far we have only looked at specific communication schemes. Information theory provides a fundamental limit to (coded) performance. It succinctly identifies the impact of channel resources on performance as well as suggests new and cool ways to communicate over the wireless channel. It provides the basis for the modern development of wireless communication. 2 1

2 Capacity of AWGN Channel Capacity of AWGN channel If average transmit power constraint is noise psd is watts/hz, watts and 3 Power and Bandwidth Limited Regimes Bandwidth limited regime capacity logarithmic in power, approximately linear in bandwidth. Power limited regime insensitive to bandwidth. capacity linear in power, 4 2

3 5 Example 1: Impact of Frequency Reuse Different degree of frequency reuse allows a tradeoff between SINR and degrees of freedom per user. Users in narrowband systems have high link SINR but small fraction of system bandwidth. Users in wideband systems have low link SINR but full system bandwidth. Capacity depends on both SINR and d.o.f. and can provide a guideline for optimal reuse. Optimal reuse depends on how the out-of-cell interference fraction f(ρ) depends on the reuse factor ρ. 6 3

4 Numerical Examples Linear cellular system Hexagonal system d Rate bits / s / Hz Frequency reuse factor 1 1/2 1/ Cell edge SNR (db) Rate bits /s / Hz Frequency reuse factor 1 1/2 1/ Cell edge SNR (db) 7 Example 2: CDMA Uplink Capacity Single cell with K users. Capacity per user Cell capacity (interference-limited) 8 4

5 Example 2 (continued) If out-of-cell interference is a fraction f of in-cell interference: 9 Uplink and Downlink Capacity CDMA and OFDM are specific multiple access schemes. But information theory tells us what is the capacity of the uplink and downlink channels and the optimal multiple access schemes. 10 5

6 Uplink AWGN Capacity log 1 + P 2 N 0 R 2 successive cancellation: cancel 1 before 2 B P log P 1 + N 0 conventional decoding C cancel 2 before 1 A P log P 2 + N 0 P 1 log 1 + N 0 R 1 11 Conventional CDMA vs Capacity R 2 ( bits / s / Hz ) 6.66 B 5.67 C A rate increase to weak user 20 db power difference between 2 users D CDMA R 1 ( bits / s /Hz ) Successive cancellation allows the weak user to have a good rate without lowering the power of the strong user. 12 6

7 Orthogonal vs Capacity R 2 ( bits / s / Hz ) orthogonal B C Sum capacity achieved here A 20 db power difference between 2 users R 1 ( bits / s / Hz ) Orthogonal achieves maximum throughput but may not be fair. 13 Downlink Capacity Rate of user orthogonal superposition coding 20 db gain difference between 2 users Rate of user

8 Frequency-selective Channel 's are time-invariant. OFDM converts it into a parallel channel: where is the waterfilling allocation: with λ chosen to meet the power constraint. Can be achieved with separate coding for each sub-carrier. 15 Waterfilling in Frequency Domain 16 8

9 Slow Fading Channel h random. There is no definite capacity. Outage probability: outage capacity: 17 Outage for Rayleigh Channel Pdf of log(1+ h 2 SNR) Outage cap. as fraction of AWGN cap. 18 9

10 Receive Diversity Diversity plus power gain. 19 Transmit beamforming: Transmit Diversity Alamouti (2 Tx): Diversity but no power gain

11 Repetition vs Alamouti Repetition: Alamouti: Loss in degrees of freedom under repetition. 21 Time Diversity (I) Coding done over L coherence blocks, each of many symbols. This is a parallel channel. If transmitter knows the channel, can do waterfilling. Can achieve: 22 11

12 Time Diversity (II) Without channel knowledge, Rate allocation cannot be done. Coding across sub-channels becomes now necessary. 23 Fast Fading Channel Channel with L-fold time diversity: As Fast fading channel has a definite capacity: Tolerable delay >> coherence time

13 Capacity with Full CSI Suppose now transmitter has full channel knowledge. What is the capacity of the channel? 25 Fading Channel with Full CSI This is a parallel channel, with a sub-channel for each fading state. where is the waterfilling power allocation as a function of the fading state, and λ is chosen to satisfy the average power constraint. Can be achieved with separate coding for each fading state

14 Transmit More when Channel is Good 27 Performance At high SNR, waterfilling does not provide any gain. But transmitter knowledge allows rate adaptation and simplifies coding

15 Performance: Low SNR Waterfilling povides a significant power gain at low SNR. 29 Waterfilling vs Channel Inversion Waterfilling and rate adaptation maximize long-term throughput but incur significant delay. Channel inversion ( perfect power control in CDMA jargon) is power-inefficient but maintains the same data rate at all channel states. Channel inversion achieves a delay-limited capacity

16 Example of Rate Adaptation: 1xEV-DO Downlink Multiple access is TDMA via scheduling. (More on this tomorrow.) Each user is rate-controlled rather than power-controlled. (But no waterfilling.) 31 Rate Control Mobile measures the channel based on the pilot and predicts the SINR to request a rate

17 SINR Prediction Uncertainty SINR SINR SINR prediction prediction lag (a) conservative prediction t lag (b) 3 km/hr 30 km/hr 120 km/hr accurate prediction of instantaneous SINR. conservative prediction of SINR. t lag (c) accurate prediction of average SINR for a fast fading channel t 33 Incremental ARQ A conservative prediction leads to a lower requested rate. At such rates, data is repeated over multiple slots. If channel is better than predicted, the number of repeated slots may be an overkill. This inefficiency can be reduced by an incremental ARQ protocol. The receiver can stop transmission when it has enough information to decode. Incremental ARQ also reduces the power control accuracy requirement in the reverse link in Rev A

18 Summary A slow fading channel is a source of unreliability: very poor outage capacity. Diversity is needed. A fast fading channel with only receiver CSI has a capacity close to that of the AWGN channel. Delay is long compared to channel coherence time. A fast fading channel with full CSI can have a capacity greater than that of the AWGN channel: fading now provides more opportunities for performance boost. The idea of opportunistic communication is even more powerful in multiuser situations, as we will see

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