Hamming Codes

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6.8.1 Hamming Codes

Hamming codes are a simple class of block codes characterized by the structure

where m = 2, 3, . . . . These codes have a minimum distance of 3 and thus, from

Equations (6.44) and (6.47), they are capable of correcting all single errors or detecting

all combinations of two or fewer errors within a block. Syndrome decoding

is especially suited for Hamming codes. In fact, the syndrome can be formed to act

as a binary pointer to identify the error location [5]. Although Hamming codes are

not very powerful, they belong to a very limited class of block codes known as

perfect codes, described in Section 6.5.4.

Assuming hard decision decoding, the bit error probability can be written,

from Equation (6.46), as

where p is the channel symbol error probability (transition probability on the

binary symmetric channel). In place of Equation (6.72) we can use the following

equivalent equation. Its identity with Equation (6.72) is proven in Appendix D,

Equation (D.16):

Figure 6.21 is a plot of the decoded P _B versus channel-symbol error probability,

illustrating the comparative performance for different types of block codes. For the

Figure 6.21 Bit error probability versus channel symbol error probability for several block codes.

Hamming codes, the plots are shown for m = 3, 4, and 5, or (n, k) = (7, 4), (15, 11),

and (31, 26). For performance over a Gaussian channel using coherently demodulated

BPSK, we can express the channel symbol error probability in terms of E_c/N₀,

similar to Equation (4.79), as

where E_c/N₀ is the code symbol energy per noise spectral density, and where Q(x)

is as defined in Equation (3.43). To relate E _c / N ₀ to information bit energy per noise

spectral density (E_b/N₀), we use

For Hamming codes, Equation (6.75) becomes

Combining Equation (6.73), (6.74), and (6.76), P_B can be expressed as a function of

E_b/N₀ for coherently demodulated BPSK over a Gaussian channel. The results are

plotted in Figure 6.22 for different types of block codes. For the Hamming codes,

plots are shown for (n, k) = (7, 4), (15, 11), and (31, 26).

Example 6.11 Error Probability for Modulated and Coded Signals

A coded orthogonal BFSK modulated signal is transmitted over a Gaussian channel.

The signal is noncoherently detected and hard-decision decoded. Find the decoded bit

error probability if the coding is a Hamming (7, 4) block code and the received E _b/N₀

is equal to 20.

Solution

First we need to find E _c/N ₀ using Equation (6.75):

Then, for coded noncoherent BFSK, we can relate the probability of a channel symbol

error to E _c/N ₀, similar to Equation (4.96), as follows

Using this result in Equation (6.73), we solve for the probability of a decoded bit

error, as follows:

Figure 6.22  P_B versus E_b/N₀ for coherently demodulated BPSK over a Gaussian channel for several block codes.

Relevant NI products

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• RF and Communication Hardware and Software
  
• Other Modular Instruments (digital multimeters, digitizers, switching, etc...)
• LabVIEW Graphical Programming Environment

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