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Chapter 6: Problem 23

A digital source produces sequences of nine letters with the following probabilities. $$ \begin{array}{|l|c|c|c|c|c|c|c|c|c|} \hline \text { letter } & \mathrm{a} & \mathrm{b} & \mathrm{c} & \mathrm{d} & \mathrm{e} & \mathrm{f} & \mathrm{g} & \mathrm{h} & \mathrm{i} \\ \hline \text { probability } & \frac{1}{4} & \frac{1}{8} & \frac{1}{8} & \frac{1}{8} & \frac{1}{8} & \frac{1}{16} & \frac{1}{16} & \frac{1}{16} & \frac{1}{16} \\ \hline \end{array} $$ a) Find a Huffman code that compresses this source. How does the resulting code compare with the best possible code? b) A clever engineer proposes the following (6,3) code to correct errors after transmission through a digital channel. $$ \begin{array}{|l|l|} c_{1}=d_{1} & c_{4}=d_{1} \oplus d_{2} \oplus d_{3} \\ \hline c_{2}=d_{2} & c_{5}=d_{2} \oplus d_{3} \\ \hline c_{3}=d_{3} & c_{6}=d_{1} \end{array} $$ What is the error correction capability of this code? c) The channel's bit error probability is \(1 / 8\). What kind of code should be used to transmit data over this channel?

Short Answer

Expert verified
Use Huffman coding for minimal average code length, (6,3) code can correct one error, and use BCH or Reed-Solomon for high error probability.

Step by step solution

01

Understand the Huffman Coding Problem

The digital source produces sequences of nine letters with their respective probabilities given in the table. Our task is to create a Huffman code, which is an optimal, prefix-free binary code. The goal is to compress the data represented by these letters in a way that minimizes the overall average code length.
02

Construct the Huffman Tree

To construct the Huffman tree, start by listing all the probabilities in increasing order. Then, iteratively combine the two symbols with the lowest probabilities into a single new node and update the probability list. Repeat this process until only one node remains. This node becomes the root of the Huffman tree. Assign binary digits (0 and 1) to the branches connecting the nodes.
03

Assign Huffman Codes to Each Letter

Once the tree is constructed, each letter is represented by a path from the root to the leaf node corresponding to that letter. Write down the sequence of 0s and 1s encountered on this path; this sequence is the Huffman code for that letter. Make sure all letters have a unique binary sequence.
04

Calculate the Average Code Length

To analyze the efficiency of your Huffman code, calculate the average code length by multiplying the probability of each letter with its code length, and summing the results. Use the formula:\[ L = \sum_{i=1}^{n} P(x_i) \cdot l(x_i) \] where \(P(x_i)\) is the probability of each letter and \(l(x_i)\) is the length of the code for each letter.
05

Compare with the Best Possible Code

Calculate the entropy \(H\) of the source using the formula:\[ H = -\sum_{i=1}^{n} P(x_i) \log_2 P(x_i) \]The entropy gives the theoretical lower limit of the average code length. Compare your calculated average code length from Step 4 with this entropy to determine the efficiency of the Huffman code.
06

Analyze Error Correction Capability of the Code

The proposed (6,3) code involves redundancy, which is used for error detection and correction. The code introduces 3 parity bits, and the configuration suggests Hamming code features, allowing for single error correction (SEC) and double error detection (DED).
07

Determine the Suitable Error Correction Code for the Channel

Given a bit error probability of \(\frac{1}{8}\) (12.5%), this is a relatively high error rate. A simple single error correction code may not be sufficient. Therefore, a more robust error-correcting code, such as a BCH or Reed-Solomon code, which can handle multiple errors, would be more suitable for reliable data transmission over this channel.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Code Compression
In the world of digital communication, data often needs to be compressed to save space and increase efficiency. Huffman Coding is a popular method used to achieve this. It is an optimal technique for lossless data compression, meaning the original data can be perfectly reconstructed from the compressed data.
Huffman Coding works by assigning shorter codes to more frequently occurring symbols and longer codes to less frequent ones. This is achieved by building a binary tree, known as a Huffman tree, from the probabilities of the symbols. Shorter paths in the tree correspond to shorter codes, leading to a reduction in the overall length of the data when encoded.
The primary advantage of Huffman Coding is its ability to provide the most efficient prefix-free binary codes for a given set of symbol probabilities, thereby minimizing the average code length. Despite its efficiency, there are situations where Huffman Coding might not be suitable, such as when the distribution of probabilities changes frequently, as it requires updating the tree and codebook accordingly.
Error Correction Codes
Error correction codes are essential in digital communication as they help detect and correct errors in transmitted data. One type of error correction code, illustrated in the exercise, is the (6,3) code. This code uses a combination of data and parity bits to identify and correct errors.
The code works by adding three parity bits to every three data bits. These parity bits are calculated using specified operations, such as XOR (exclusive OR), to create a redundancy that allows single error correction and double error detection. This process of adding redundancy can greatly improve the reliability of data transmission, especially when errors are likely to occur.
Understanding how error correction codes function is key to optimizing communication systems. The simpler codes, like the one described, can handle certain types of errors efficiently. However, as error rates increase, more complex codes, such as Reed-Solomon or Turbo Codes, may be needed to maintain data integrity.
Digital Communication Channels
Digital communication channels are the pathways through which data is transmitted from one device to another. The quality of these channels can vary, and factors such as noise and interference can introduce errors during transmission.
Channels with high error rates, like the one with a bit error probability of 1/8 in the exercise, require sophisticated error correction mechanisms to ensure reliable data transfer. In such environments, choosing the right type of error correction code is crucial. Traditional single error correction codes might not suffice due to the high chance of multiple errors occurring simultaneously.
Advanced codes, such as BCH and Reed-Solomon, are often recommended for channels with significant error probabilities. These codes can handle multiple error corrections, offering a more robust solution for preserving the integrity of the data. By applying the appropriate error correction strategies, digital communication systems can achieve reliable transmission even under challenging conditions.

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Most popular questions from this chapter

In a so-called multi-tone system, several bits are gathered together and transmitted simultaneously on different carrier frequencies during a \(T\) second interval. For example, \(B\) bits would be transmitted according to $$ x(t)=A \sum_{k=0}^{B-1} b_{k} \sin \left(2 \pi(k+1) f_{0} t\right) \quad, \quad 0 \leq t

We want to examine both analog and digital communication alternatives for a dedicated speech transmission system. Assume the speech signal has a \(5 \mathrm{kHz}\) bandwidth. The wireless link between transmitter and receiver is such that 200 watts of power can be received at a pre-assigned carrier frequency. We have some latitude in choosing the transmission bandwidth, but the noise power added by the channel increases with bandwidth with a proportionality constant of 0.1 watt \(/ \mathrm{kHz}\) a) Design an analog system for sending speech under this scenario. What is the received signal-to-noise ratio under these design constraints? b) How many bits must be used in the \(\mathrm{A} / \mathrm{D}\) converter to achieve the same signal-to-noise ratio? c) Is the bandwidth required by the digital channel to send the samples without error greater or smaller than the analog bandwidth?

Stereophonic radio transmits two signals simultaneously that correspond to what comes out of the left and right speakers of the receiving radio. While FM stereo is commonplace, AM stereo is not, but is much simpler to understand and analyze. An amazing aspect of AM stereo is that both signals are transmitted within the same bandwidth as used to transmit just one. Assume the left and right signals are bandlimited to \(W \mathrm{~Hz}\). $$ x(t)=A\left(1+m_{l}(t)\right) \cos \left(2 \pi f_{c} t\right)+A m_{r}(t) \sin \left(2 \pi f_{c} t\right) $$ a) Find the Fourier transform of \(x(t)\). What is the transmission bandwidth and how does it compare with that of standard AM? b) Let us use a coherent demodulator as the receiver, shown in Figure 6.34 . Show that this receiver indeed works: It produces the left and right signals separately. c) Assume the channel adds white noise to the transmitted signal. Find the signal-to-noise ratio of each signal.

In designing a digital version of a wireless telephone, you must first consider certain fundamentals. First of all, the quality of the received signal, as measured by the signal-to-noise ratio, must be at least as good as that provided by wireline telephones \((30 \mathrm{~dB})\) and the message bandwidth must be the same as wireline telephone. The signal-to-noise ratio of the allocated wirelss channel, which has a \(5 \mathrm{kHz}\) bandwidth, measured 100 meters from the tower is \(70 \mathrm{~dB}\). The desired range for a cell is \(1 \mathrm{~km}\). Can a digital cellphone system be designed according to these criteria?

Consider the following 5 -letter source. $$ \begin{array}{|l|l|} \hline \text { Letter } & \text { Probability } \\ \hline \mathrm{a} & 0.4 \\ \hline \mathrm{b} & 0.2 \\ \hline \mathrm{c} & 0.15 \\ \hline \mathrm{d} & 0.15 \\ \hline \mathrm{e} & 0.1 \\ \hline \end{array} $$ a) Find this source's entropy. b) Show that the simple binary coding is inefficient. c) Find the Huffman code for this source. What is its average code length?
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