MIT16_36s09_lec13_14

MIT16_36s09_lec13_14 - MIT OpenCourseWare http:/ocw.mit.edu...

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MIT OpenCourseWare http://ocw.mit.edu 16.36 Communication Systems Engineering Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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16.36: Communication Systems Engineering Lectures 13/14: Channel Capacity and Coding Eytan Modiano Eytan Modiano Slide 1
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Channel Coding 1 0 1-Pe 1-Pe Pe When transmitting over a noisy channel, some of the bits are received with errors Example: Binary Symmetric Channel (BSC) 0 Pe = Probability of error 1 Q: How can these errors be removed? A: Coding: the addition of redundant bits that help us determine what was sent with greater accuracy Eytan Modiano Slide 2
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Example (Repetition code) Repeat each bit n times (n-odd) Input 0 Code 000……. .0 1 11. .……. .1 Decoder: If received sequence contains n/2 or more 1 s decode as a 1 and 0 otherwise Max likelihood decoding P ( error | 1 sent ) = P ( error | 0 sent ) = P[ more than n / 2 bit errors occur ] n n ) n i P i (1 P e e = i i = n /2 Eytan Modiano Slide 3
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Repetition code, cont. For P e < 1/2, P(error) is decreasing in n for any ε , n large enough so that P (error) < ε Code Rate: ratio of data bits to transmitted bits For the repetition code R = 1/n To send one data bit, must transmit n channel bits “bandwidth expansion” In general, an (n,k) code uses n channel bits to transmit k data bits Code rate R = k / n Goal: for a desired error probability, ε , find the highest rate code that can achieve p(error) < ε Eytan Modiano Slide 4
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Channel Capacity (Discrete Memoryless Channel) The capacity of a discrete memoryless channel is given by, Channel Y C = max p ( x ) I ( X ; Y ) X Example: Binary Symmetric Channel (BSC) I(X;Y) = H (Y) - H (Y|X) = H (X) - H (X|Y) H(Y|X) = H b (p e ) (why?) H(Y) 1 (why?) H(Y) = 1 when p 0 = 1/2 P 0 P 1 =1-P 0 1 0 C = 1 - H b (p e ) Try to compute the capacity starting with H(X) - H(X|Y). 1 0 1-Pe 1-Pe Pe Eytan Modiano Slide 5
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Capacity of BSC C = 1 - H b (P e ) C = 0 when P e = 1/2 and C = 1 when P e = 0 or P e =1 Eytan Modiano Slide 6
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Capacity of AWGN channel Additive White Gaussian Noise channel r = S + N N is AWGN with power spectral density N o /2 Transmission over band-limited channel of bandwidth W Average input (transmit) power = P Band-limited equivalent noise power = WN o 1 P C = log(1 + ) bits per transmission 2 WN 0 P R s 2W C = Wlog(1 + ) bits per second WN 0 Notes Rs 2W is implied by sampling theorem (see notes on sampling theorem) Capacity is a function of signal-to-noise ratio (SNR = P/WN o ) Where the signal power is measured at the receiver As W increases capacity approaches a limit: Increasing W increases the symbol rate, but also the noise power Eytan Modiano P P Slide 7 Limit W →∞ Wlog(1 + ) = log(e) 1.44 P WN 0 N 0 N 0
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Capacity of AWGN channel (example) The capacity of a cooper telephone channel
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MIT16_36s09_lec13_14 - MIT OpenCourseWare http:/ocw.mit.edu...

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