Coding for error detection and correction in .NET Integration barcode code 128 in .NET Coding for error detection and correction

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Coding for error detection and correction use .net code 128 barcode integration todraw code 128c with .net Developing with Visual Studio .NET time (branch of trellis) 0 Code 128 Code Set C for .NET 00 a 11 2 K 1 states 11 00 10 01 c 01 01 11 d 10 01 01 10 11 11 00 10 01 01 10 01 10 11 11 00 10 01 11 11 00 10 01 01 10 01 10 11 11 00 10 01 11 11 00 10 00 1 00 2 00 3 00 4 00 5 00 6 data bits 0 1. Figure 7.10 Trellis representation, encoder of Fig. 7.7 Consider now the trellis re visual .net code-128c presentation of the same K = 3, rate-1/2 encoder, as portrayed in Fig. 7.

10. The trellis representation of any other encoder can then easily be obtained in the same way. The 2K 1 states, the four states of this example, are arrayed along the vertical axis, as shown.

Time, in input-bit intervals, is represented by the horizontal axis. Transitions between states are then shown as occurring from one time interval to the next, a transition due to a 0 data input bit always being graphed as the upper line of the two leaving any state. The output bits are shown in order of occurrence alongside each transition, just as in the state diagram of Fig.

7.9. Note how this trellis representation immediately demonstrates the repetitive or cyclical nature of the encoder nite-state machine: starting at any state, one can see at a glance how one moves between states, eventually returning to the state in question.

This representation is useful in evaluating the performance of a given convolutional encoder. The third, tree, representation of convolutional encoders is again best described using the same K = 3, rate-1/2 example. The tree representation for this example appears in Fig.

7.11. It is effectively the same as the trellis representation with sequences of transitions now individually separated out and identi ed as separate paths.

As in the trellis representation of Fig. 7.10, just described, upward transitions are shown corresponding to a 0 data input bit; downward transitions correspond to a 1 data input bit.

The output bits corresponding to a particular transition along a given path are shown above each transition. This representation too demonstrates the repetitive nature of the convolutional encoder state machine, as is apparent by tracing out the various possible paths, and noting that the 2K 1 = 4 states are entered and reentered as time progresses. The important point to note here is that the number of possible paths, each with a different sequence of output bits, increases exponentially with time.

The Hamming distance, the difference in the number of positions between any pair of paths, increases as well. This observation indicates why convolutional codes provide improved performance. As a given path increases in length, it is more readily distinguishable from other paths.

One or more errors occurring in transmission are more readily detected and corrected. This simple observation demonstrates. Mobile Wireless Communications 00 00 a 11 00 input data bits 0 11 b 01 10 00 11. a b c d a b c d a b c d a b c d time 10 01. 11 00. 01 10. 11 1 10 c 00 11 01 01 d 10 00 11. 10 01. 11 00. 01 10. Figure 7.11 Tree representation, coder of Fig. 7.7 as well why increasing the Code 128 for .NET constraint length K and/or the rate of the encoder improve the system performance. Given a particular binary data sequence passed through a convolutional encoder, how does one recover or decode the sequence at the destination, the receiving system Coding of the information-bearing binary sequence, the subject of this chapter, implies that some transmitted information will have been corrupted during transmission, whether due to interference, fading, or noise.

The whole purpose of the convolutional coding we have been discussing is to recover possibly corrupted information with the smallest probability of error. This de nes the best decoder. With convolutional coding this means, given a received signal sequence, choosing the most probable path that would correspond to that sequence.

Say, for example, that we select a sequence L input binary intervals long. There are then 2L possible paths corresponding to the received sequence. The most probable or most likely one of these is the one to be chosen.

A decoder implementing such a strategy is called a maximum-likelihood decoder (Schwartz, 1990; Proakis, 1995). Consider the following simple version of such a decoder. For each input data binary interval there are bits transmitted and signals received.

Because of fading, noise, and interference, each received signal is essentially analog in form, even though transmitted as a binary 1 or 0. Say, for simplicity, that these signals are each individually converted to the best estimate of a 1 or 0. Such a technique is termed hard limiting or hard-decision decoding.

This procedure simpli es the discussion considerably. (Maximum-likelihood decoding of the actual analog received signals is discussed in Schwartz (1990) and in Proakis.
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