Letter sequences, decoding

For equiprobable source letters, the decoding rule of Eq. (4-85) thus chooses the most likely source letter conditional on the received sequence. If no source letter is more probable than all others, we assume that a decoding error is made. 

In general, the decoding reliability can be improved by decoding an entire watermark letter sequence d, where the known encoding of m into d can be exploited to estimate the most likely d, or equivalently, to estimate the most likely watermark message m. The simple codebook structure of SCS can be exploited to efficiently estimate d. First, data y is extracted from the received data r. This extraction process operates sample-wise, where the extraction rule for the nth element is... 

To prove Theorem 4-9 formally, let the sequences u, x, y, and v represent source letters, channel input symbols, channel output symbols, and decoded output letters respectively. For any given coder and decoder, a probability measure will be defined on all these sequences. Then, using Eq. (4-51)... 

When the source letter a = um appears, the channel transmits the code word xm = ( ml, and the decoder receives the sequence... 

The decoder, observing the sequence, y, guesses at which, source letter was put into the coder. We call this decoded letter v, and if v a decoding error has occurred. 

Given a particular set of code words, and given the decoding rule of Eq. (4-85), the probability of decoding error can in principle be calculated. Let Ym be the set of channel output sequences that satisfy Eq. (4-85) for each m, 1 <, m <, M. Then the probability of decoding error for Hie source letter um is... 

Formally, we can think of the combination of source letter, set of code words, and output sequence as an elementary event in a product ensemble. The decoding ride, Eq. (4-85), specifies for each of these elementary events whether or not a decoding error occurs, and thus Pe is a well-defined quantity. 

The simplest approach for the redundant embedding of the information bits b into the host data x is the repeated embedding of each bit. Let p denote the repetition factor, thus the sequence of watermark letters d is p times longer than b. The watermark letters d are embedded, transmitted and demodulated as described in Sec. 2. However, instead of deciding for each demodulated value Un what transmitted watermark letter dn is most likely, the decoder can estimate directly the most likely transmitted watermark information bit bk from p different demodulated values in y. Without loss of generality, we assume that the fcth iirformationbit bk has been embedded into the data elements Xpk, Xpk+i,..., Xpk+p-i. For an AWGN attack, the demodulated... 

Thus, the genetic code stored in DNA is an alphabet with four letters C, T, A, and G. Any combination of three letters—CCT, TAG, CGG, etc.—is a word of code that stands for an amino acid. Sections of DNA may contain redundant codes. If we think of the sequence of words like a sentence, a sequence of three letters could function like a period in a sentence. Also, there are sections of code that appear not to have any meaning, or at least their meaning has not yet been decoded. 

---