1 Lecture 4. Capacity of Fading Channels Capacity of AWGN Channels Capacity of Fading Channels Ergodic Capacity Outage Capacity
Shannon and Information Theory Claude Elwood Shannon (April 3, 1916 February 4, 1)
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 1 1 1111111 Codebook 1111 111 11111111111 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
Channel Capacity: Binary Symmetric Channel 4 At the transmitter 11 11 1111 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)
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)
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
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
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 Capacity of Fading Channels I: Ergodic Capacity Capacity without CSIT Capacity with CSIT
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
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.)
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?
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 N h l N, 1,...,. optimal Pl l L hl * P 1 l
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 *
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!
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?
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
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
Capacity of Fading Channels II: Outage Capacity Capacity without CSIT Capacity with CSIT
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!
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
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?
Outage Capacity of Rayleigh Fading Channel 4 1%
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
Outage Capacity with L-fold Receive Diversity 6 (=.1)
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)
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.