Channel-capacity

# Channel-capacity - ±(Note that P b E ≤ P E 2 If R> C...

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Channel Capacity for bandlimited AWGN channel Shannon’s Channel capacity Theorem We are given a binary source of rate R b bits/sec. that is to be transmit- ted across an AWGN channel with power spectral density of S N W ( f ) = N 0 2 . Suppose a bandwidth of W and power of P are available to the modulator. Deﬁne the channel capacity C = W log 2 ± 1 + P WN 0 ² Then 1. Positive Theorem: If R b Binary data of rate R = 1 T b is to be transmitted across an AWGN channel with noise power spectral density of S N W ( f ) = N 0 2 . The available power is P and the available bandwidth is W . Deﬁne the channel capacity as C = W log 2 (1 + P WN 0 ) bits/sec. 1. If R < C , then for any ± > 0, there exists a suﬃciently large K and 2 K signals of duration T = K R = KT b , each having energy E < PT and bandwidth W , such that when used to transmit the binary data, the resulting error probability P ( E ) is less than

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Unformatted text preview: ± . (Note that P b ( E ) ≤ P ( E ).) 2. If R > C , then no system whatsoever can have arbitrarily small error probability. In fact as K → ∞ , P ( E ) → 1 for any system. Properties of channel capacity: We have C = W log 2 (1 + P WN ) bits/sec. Then R < C = ⇒ R W < log 2 (1 + P RN R W ) 1 or since P RN = E b N , 2 R W < 1 + E b N R W which gives E b N > 2 R W-1 R W This is the R W vs. E b N graph we have plotted in the handouts. Note that 1. As P → ∞ , C → ∞ . 2. As W → ∞ , C → P N Ln (2) . Thus as W → ∞ , the orthogonal signals are optimal. For orthogonal signal set if R < P N Ln (2) , and K → ∞ , we can get arbitrarily small P ( E ). 2...
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Channel-capacity - ±(Note that P b E ≤ P E 2 If R> C...

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