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De nition A code is called a pre x code or an instantaneous code if no codeword is a pre x of any other codeword. An instantaneous code can be decoded without reference to future codewords since the end of a codeword is immediately recognizable. Hence, for an instantaneous code, the symbol xi can be decoded as soon as we come to the end of the codeword corresponding to it. We need not wait to see the codewords that come later. An instantaneous code is a selfpunctuating code; we can look down the sequence of code symbols and add the commas to separate the codewords without looking at later symbols. For example, the binary string 01011111010 produced by the code of Example 5.1.1 is parsed as 0,10,111,110,10. The nesting of these de nitions is shown in Figure 5.1. To illustrate the differences between the various kinds of codes, consider the examples of codeword assignments C(x) to x X in Table 5.1. For the nonsingular code, the code string 010 has three possible source sequences: 2 or 14 or 31, and hence the code is not uniquely decodable. The uniquely decodable code is not pre x-free and hence is not instantaneous. To see that it is uniquely decodable, take any code string and start from the beginning. If the rst two bits are 00 or 10, they can be decoded immediately. If

for n suf ciently large and R < I (X; Y ) 3 . This proves the existence of a good (2nR , n) code. Now choosing a good codebook and deleting the worst half of the codewords, we obtain a code with low maximal probability of error. In particular, the power constraint is satis ed by each of the remaining codewords (since the codewords that do not satisfy the power constraint have probability of error 1 and must belong to the worst half of the codewords). Hence we have constructed a code that achieves a rate arbitrarily close to capacity. The forward part of the theorem is proved. In the next section we show that the achievable rate cannot exceed the capacity. 9.2 CONVERSE TO THE CODING THEOREM FOR GAUSSIAN CHANNELS In this section we complete the proof that the capacity of a Gaussian P channel is C = 1 log(1 + N ) by proving that rates R > C are not achiev2 able. The proof parallels the proof for the discrete channel. The main new ingredient is the power constraint. Proof: (Converse to Theorem 9.1.1). We must show that if Pe(n) 0 for a sequence of (2nR , n) codes for a Gaussian channel with power constraint P , then R C= 1 P log 1 + 2 N . (9.35)

Possible ET Flavors Strength Strong Weak Description Identi cation is completely determined by the ET itself. Identi cation is partially determined by the ET and partially determined by at least one other ET connected to this one via an Identifying RT. Same as a Strong ET that does not participate in an IT. Same as a Weak ET that does not participate in an IT. Identi cation is completely determined by the Super ET. This is not possible.

avec(ZY I) ax'

Figure 14.7 Block diagram of the digital terrestrial television broadcasting model. (Based on the ITU-R Task Group 11.3 model. From Ref. 18, Figure 4.1. Reprinted with permission.)

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