Lecture 4. Capacity of Fading Channels

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1 1 Lecture 4. Capacity of Fading Channels Capacity of AWGN Channels Capacity of Fading Channels Ergodic Capacity Outage Capacity

2 Shannon and Information Theory Claude Elwood Shannon (April 3, 1916 February 4, 1)

3 Code Rate and Error Probability 3 Digital System Attenuation, Noise, Distortion, SOURCE Source Info. Transmitter Transmitted signal Channel Received signal Receiver Received info. User Information sequence Codebook What is the maximum rate R for which arbitrarily small P e can be achieved? Error Prob.: P ˆ e Pr{ x x} Size: M= k Code length: n>=k bits Code Rate: R=k/n=log M/n

4 Channel Capacity: Binary Symmetric Channel 4 At the transmitter Code length n Discrete-memoryless Binary Symmetric Channel 1-p p 1 p 1-p 1 At the receiver: For any codeword x i, np bits will be received in error with high probability, if n is large. The number of possible error codewords corresponding to x i is n n! nhb ( p) np ( np)!( n(1 p))! H ( p) plog p(1 p) log (1 p) b Choose a subset of all possible codewords, so that the possible error codewords for each element of this subset is NOT overlapping! n b The maximum size of the subset: M nhb ( p) The maximum rate that can be reliably communicated : n(1 H ( p)) 1 C log M 1 Hb ( p ) n (bit/transmission)

5 Channel Capacity: Discrete-time AWGN Channel 5 x is an input sequence with power constraint: 1 n x n i1 i P y x z Noise z i is a zero-mean Gaussian random variable with variance. 1 n n i 1 1 n n z i1 i 1 n n i y P i for large n ( y ) 1 i xi yx n n-dimensional hyper-sphere with radius n y np ( ) n-dimensional hyper-sphere with radius np ( ) How many input sequences can we transmit over this channel at most such that the hyperspheres do not overlap? The maximum rate that can be reliably communicated : M P n n ( ) /( ) 1 1 P C logm log (1 ) n (bit/transmission)

6 Channel Capacity: Continuous-time AWGN Channel 6 Capacity of discrete-time AWGN channel: 1 1 P C log M log (1 ) bit/transmission n For continuous-time AWGN baseband channel with bandwidth W, power constraint P watts, and two-sided power spectral density of noise N /, What is the average noise power per sampling symbol? N W What is the minimum sampling rate without introducing distortion? W Capacity of continuous-time AWGN channel: 1 P C W log (1 ) NW W log (1 P ) NW bit/s

7 More about Capacity of Continuous-time AWGN Channel 7 Can we increase the capacity by enhancing the transmission power? Yes, but the capacity increases logarithmically with P when P is large. If we can increase the bandwidth without limit, can we get an infinitely large channel capacity? No. lim C P log e W N How to achieve (approach) AWGN channel capacity? Turbo codes, LDPC,? C W log (1 P ) bit/s NW

8 More about Capacity of Continuous-time AWGN Channel 8 C W log (1 P ) bit/s NW Spectral Efficiency: C log (1 SNR) bit/s/hz C (bit/s/hz) C log SNR SNR P NW N P: power per unit bandwidth P C SNRlog e SNR (db)

9 9 Capacity of Fading Channels I: Ergodic Capacity Capacity without CSIT Capacity with CSIT

10 Channel Model 1 Consider a flat fading channel. Suppose each codeword spans L coherence time periods. Without CSIT: the transmission power at each coherence time period is a constant P. P l =P, l=1,, L. With CSIT: different transmission power can be allocated to different coherence time periods according to CSI. The average power is P. P l =f(h l ), l=1,, L. Suppose the receiver has CSI. 1 L L l 1 P P. l

11 Ergodic Capacity without CSIT 11 At each coherence time period, the reliable communication rate is The average rate is 1 L log (1 hl SNR) L log 1 (1 hl SNR) l SNR P N 1 L For ergodic channel: lim log 1 (1 hl SNR) E h[log (1 h SNR)] L l L Ergodic capacity without CSIT: C h SNR wo e E[log(1 h )] How to achieve it? AWGN capacity-achieving codes Coding across channel states (Codeword should be long enough to average out the effects of both noise and fading.)

12 Ergodic Capacity without CSIT 1 C h SNR wo e E[log h (1 )] log (1 E [ ] ) h h SNR log (1 SNR) CAWGN At low SNR, C h SNR e SNR e C wo e E[ h ]log log AWGN At high SNR, wo C E[log ( h SNR)] log SNRE[log h ] C e h h C AWGN AWGN E[log h h ] < What if the transmitter has full CSI?

13 Ergodic Capacity with CSIT 13 For a given realization of the channel gains h 1,, h L at L coherence time periods, the maximum total rate is where satisfies: 1 L Pl h l max log 1 P 1 1,..., P l L L N 1 L Subject to: P. l 1 l P L P optimal l 1 L L N hl l1 N h l P Waterfilling Power Allocation Note: x x if x otherwise

14 14 N h l N, 1,...,. optimal Pl l L hl * P 1 l

15 Ergodic Capacity with CSIT 15 As L, P E h N h X converges to a constant which depends on the channel statistics (not the instantaneous channel realizations). The optimal power allocation is given by: Ergodic capacity with CSIT: P N ( h) h * C w e Eh log 1 P ( h) h N *

16 Ergodic Capacity with/without CSIT 16 With CSIT C Eh log 1 P ( h) h N * No CSIT C h SNR E[log(1 h )] Capacity of fading channel can be higher than that of AWGN channel if CSIT is available!

17 To Overcome Fading or to Exploit Fading? 17 No CSIT: C h SNR E[log(1 h )] Equal power allocation Constant rate Coding across channel states To average out fading! With CSIT: C Eh log 1 P ( h) h N * How to exploit fading?

18 How to Exploit Fading? 18 h l N h l Waterfilling power allocation Variable rate Higher power at better subchannel Higher rate at better subchannel No coding across channel states

19 Summary I: Ergodic Capacity 19 Without CSIT: to average out the fading effect Close to AWGN channel capacity Constant rate, coding according the channel states With CSIT: to exploit the fading effect Higher than AWGN channel capacity at low SNR Waterfilling power allocation + rate allocation

20 Capacity of Fading Channels II: Outage Capacity Capacity without CSIT Capacity with CSIT

21 Outage Event 1 The ergodic capacity is based on the assumption that the channel is ergodic. What if the ergodicity requirement is not satisfied? Suppose that: CSI is available only at the receiver side. The channel gain remains unchanged during the whole transmission. Can we find a non-zero transmission rate R>, below which reliable communication is guaranteed for any channel realization? No. For any R>, there always exists some channel realization h such that log (1 ) h SNR R outage!

22 Outage Probability Outage Probability p out (R): the probability that the system is in outage for given transmission rate R bit/s/hz. Outage Probability without CSIT: p R h SNR R wo out ( ) Pr{log (1 ) } R 1 Pr h SNR F h R 1 SNR F x h x ( ) Pr{ } h wo For given rate R, the outage probability pout ( R) decreases as SNR increases. wo For given SNR, the outage probability p ( R) decreases as rate R decreases. wo pout ( R) > for any SNR< and R>. out

23 Outage Capacity 3 Outage Capacity: C max{ R: p ( R) } out Outage Probability without CSIT: Outage Capacity without CSIT: p R h SNR R wo out ( ) Pr{log (1 ) } 1 ( F SNR ) h wo C log (1 F ( ) SNR ) 1 h high SNR wo 1 C log SNRlog F ( ) h C AWGN log F ( ) 1 h < low SNR wo 1 C F ( ) SNRlog h e F 1 h ( ) C AWGN How to improve the outage capacity?

24 Outage Capacity of Rayleigh Fading Channel 4 1%

25 Outage Capacity with Receive Diversity 5 With L-fold receive diversity: Outage probability without CSIT: p R SNR R worx out ( ) Pr{log (1 h ) } R ( 1) LSNR! Outage Capacity without CSIT: L L F Chi-square with L degrees of freedom: worx 1 C log(1 F () SNR) log(1 ( L!) SNR) h h R 1 SNR 1 L F ( x) h L x L! high SNR: low SNR: worx C log SNR log ( L!) 1 L worx 1/ L C ( L!) SNRlog e

26 Outage Capacity with L-fold Receive Diversity 6 (=.1)

27 Outage Capacity with CSIT 7 Suppose that: CSI is available at both the receiver and the transmitter sides. The channel gain remains unchanged during the whole transmission. With CSIT: w C max{ R: min p ( R, P( h)) } P( h):e [ P( h)] P h out The optimal power allocation strategy: P * ( h) P h E h[1/ h ] Channel inversion Outage capacity with CSIT: C w SNR log 1 E[1/ ] h h Delay-limited capacity (zero-outage capacity)

28 8 Summary II: Outage Capacity Outage capacity is a useful metric for delay-sensitive scenarios. Constant transmission rate Usually much lower than AWGN channel capacity Without CSIT: outage is unavoidable. With CSIT: zero-outage is achieved by adjusting the transmission power inversely proportional to the channel gain.

Lecture 4 Capacity of Wireless Channels

Lecture 4 Capacity of Wireless Channels I-Hsiang Wang ihwang@ntu.edu.tw 3/0, 014 What we have learned So far: looked at specific schemes and techniques Lecture : point-to-point wireless channel - Diversity: