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Digital Communications 1. Safwan El AssadЧитать онлайн книгу.

Digital Communications 1 - Safwan El Assad


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if the symbol yj appears at the output, there is an uncertainty on the symbol xi, j = 1, ... ,,n which has been sent.

Schematic illustration of ambiguity on the symbol at the input when j is received.

      Figure 2.3. Ambiguity on the symbol at the input when yj is received

      The average value of this uncertainty, or the entropy associated with the receipt of the symbol yj, is:

      [2.26] images

      The mean value of this entropy for all the possible symbols yj received is:

      [2.27] images

      Which can be written as:

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      or:

      [2.29] images

      The entropy H(X/Y) is called equivocation (ambiguity) and corresponds to the loss of information due to disturbances (as I(X, Y) = H(X)− H(X/Y)). This will be specified a little further.

Schematic illustration of uncertainty on the output when we know the input.

      Figure 2.4. Uncertainty on the output when we know the input

      The entropy of the random variable Y at the output knowing the X at the input is:

      [2.30] images

      This entropy is a measure of the uncertainty on the output variable when that of the input is known.

      The matrix P(Y/X) is called the channel noise matrix:

      [2.31] images

      A fundamental property of this matrix is:

      [2.32] images

      Where: p(yj/xi) is the probability of receiving the symbol yj when the symbol xi has been emitted.

      In addition, one has:

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      p(yj) is the probability of receiving the symbol yjwhatever the symbol xi emitted, and:

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      p(xi/yj) is the probability that the symbol xi was issued when the symbol yj is received.

      2.5.2. Relations between the various entropies

      We can write:

      In the same way, as one has: H(Y, X) = H(X, Y), therefore:

      In addition, one has the following inequalities:

      [2.38] images

      and similarly:

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      – Noiseless channel: in this case, on receipt of yj, there is certainty about the symbol actually transmitted, called xi (one-to-one correspondence), therefore:

      [2.40] images

      Consequently:

      [2.41] images

      and:

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      – Channel with maximum power noise: in this case, the variable at the input is independent of that of the output, i.e.:

      [2.43] images

      We then have:

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      Note.– In information security, if xi is the plaintext, and yj is the corresponding ciphertext, then p(xi/yi) = p(xi) is the condition of the perfect secret of a cryptosystem.

      The mutual information obtained on the symbol xi when the symbol yj is received is given by:

      [2.47] images

      The average value of the mutual information, or the amount of information I(X, Y) transmitted through the channel is:

      [2.48] images

      or:

      [2.49] images

      Hence:

      [2.50] images

      [2.51] Скачать книгу

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