Lecture 18: Gaussian Channel

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1 Lecture 18: Gaussian Channel Gaussian channel Gaussian channel capacity Dr. Yao Xie, ECE587, Information Theory, Duke University

Mona Lisa in AWGN Mona Lisa Noisy Mona Lisa 100 100 200 200 300 300 400 400 500 500 600 600 700 700 800 800

2 Mona Lisa in AWGN Mona Lisa Noisy Mona Lisa Dr. Yao Xie, ECE587, Information Theory, Duke University

3 Gaussian channel the most important continuous alphabet channel: AWGN Y i = X i + Z i, noise Z i N (0, N), independent of X i model for communication channels: satellite links, wireless phone Z i X i Y i Dr. Yao Xie, ECE587, Information Theory, Duke University 2

4 Channel capacity of AWGN intuition: C = log number of distinguishable inputs if N = 0, C = if no power constraint on the input, C = to make it more meaningful, impose average power constraint: for any codewords (x 1,..., x n ) 1 n x 2 i P n i=1 Dr. Yao Xie, ECE587, Information Theory, Duke University 3

5 Naive way of using Gaussian channel Binary phase-shift keying (BPSK) transmit 1 bit over the channel 1 + P, 0 P Y = ± P + Z Probability of error P e = 1 Φ( P/N) normal cumulative probability function (CDF): Φ(x) = x convert Gaussian channel into a discrete BSC with p = P e. Lose information in quantization 1 2π e t2 2 dt Dr. Yao Xie, ECE587, Information Theory, Duke University 4

6 Definition: Gaussian channel capacity C = max I(X; Y ) f(x):ex 2 P we can calculate from here C = 1 ( 2 log 1 + P ) N maximum attained when X N (0, P ) Dr. Yao Xie, ECE587, Information Theory, Duke University 5

7 C as maximum data rate we can also show this C is the supremum of rate achievable for AWGN definition: a rate R is achievable for Gaussian channel with power constraint P : if there exists a (2 nr, n) codes with maximum probability of error λ n = max 2nR i=1 λ i 0 as n. Dr. Yao Xie, ECE587, Information Theory, Duke University 6

8 Sphere packing why we may construct (2 nc, n) codes with a small probability of error? Fix one codeword consider any codeword of length n received vector N (true codeword, N) with high probability, received vector contained in a sphere of radius n(n + ϵ) around true codeword assign each ball a codeword Dr. Yao Xie, ECE587, Information Theory, Duke University 7

9 Consider all codewords with power constraint, with high probability the space of received vector is a sphere with radius n(p + N) volume of n-dimensional sphere = C n r n for constant C n and radius r n how many codewords can we pack in a total power sphere? C n (n(p + N)) n/2 C n (nn) n/2 = ( 1 + P ) n/2 N rate of this codebook = log 2 (size of the codewords)/n: = 1 2 log 2 ( 1 + P ) N Dr. Yao Xie, ECE587, Information Theory, Duke University 8

10 sphere packing Dr. Yao Xie, ECE587, Information Theory, Duke University 9

11 Gaussian channel capacity theorem Theorem. The capacity of a Gaussian channel with power constraint P and noise variance N is C = 1 ( 2 log 1 + P ) N bits per transmission Proof: 1) achievability; 2) converse Dr. Yao Xie, ECE587, Information Theory, Duke University 10

12 Achievability: New stuff in proof codeword elements generated i.i.d. according X j (i) N (0, P ϵ). So 1 n X2 i P ϵ power outage error E 0 = 1 n n Xj 2 (1) > P j=1 small according to law of large number Converse: Gaussian distribution has maximum entropy Dr. Yao Xie, ECE587, Information Theory, Duke University 11

13 Band-limited channels more realistic channel model: band-limited continuous AWGN *: convolution Y (t) = (X(t) + Z(t)) h(t) Nyquist-Shannon sampling theorem: sampling a band-limited signal at sampling rate 1/(2W ) is sufficient to reconstruct the signal from samples two samples per period Dr. Yao Xie, ECE587, Information Theory, Duke University 12

14 Capacity of continuous-time band-limited AWGN noise has power spectral density N 0 /2 watts/hertz, bandwidth W hertz, noise power = N 0 W signal power P watts 2W samples each second channel capacity C = W log ( 1 + P ) N 0 W bits per second when W, C P N 0 log 2 e bits per second Dr. Yao Xie, ECE587, Information Theory, Duke University 13

15 Telephone line telephone signal are band-limited to 3300 Hz SNR = 33 db: P/(N 0 W ) = 2000 capacity = 36 kb/s practical modems achieve transmission rates up to 33.6 kb/s uplink and downlink ADSL achieves 56 kb/s downlink (asymmetric data rate) Dr. Yao Xie, ECE587, Information Theory, Duke University 14

16 Summary additive white Gaussian noise (AWGN) channel noise power: N, signal power constraint P, capacity C = 1 ( 2 log 1 + P ) N band-limited channel with bandwidth = W C = W log ( 1 + P ) N 0 W Dr. Yao Xie, ECE587, Information Theory, Duke University 15

Lecture 2. Capacity of the Gaussian channel

Lecture 2. Capacity of the Gaussian channel Spring, 207 5237S, Wireless Communications II 2. Lecture 2 Capacity of the Gaussian channel Review on basic concepts in inf. theory ( Cover&Thomas: Elements of Inf. Theory, Tse&Viswanath: Appendix B) AWGN