根据文件中的要求与内容用2020b版本matlab编写脚本文件以及相关程序

为了完成《Assignment1.pdf》中描述的任务，你需要使用MATLAB 2020b编写一个脚本文件（`.mlx`），并实现以下功能：

### 1. 输入设置
定义一组8个符号及其对应的概率。

% Define symbols and their probabilities
Symbols = {'A', 'B', 'C', 'D', 'E', 'F', 'G', 'H'};
Probabilities = [0.25, 0.2, 0.15, 0.12, 0.1, 0.08, 0.06, 0.04];

### 2. 计算熵
计算给定源的熵，并解释其意义。

function H = calculateEntropy(probabilities)
    % Calculate the entropy
    H = -sum(probabilities .* log2(probabilities));
end

% Call the function
H = calculateEntropy(Probabilities);
disp(['Entropy (H): ', num2str(H)]);

### 3. 实现霍夫曼编码和香农-范诺编码

#### 霍夫曼编码
创建自定义函数来生成霍夫曼树和霍夫曼码。

function [codes, avgLength] = huffmanCoding(symbols, probabilities)
    % Create a priority queue (min heap)
    T = struct('symbol', {}, 'probability', [], 'left', [], 'right', []);
    
    % Initialize the leaves
    for i = 1:length(symbols)
        node = struct('symbol', symbols{i}, 'probability', probabilities(i), 'left', [], 'right', []);
        T = [T; node];
    end
    
    % Build the Huffman tree
    while length(T) > 1
        % Sort nodes by probability
        T = sortrows(T, 'probability');
        
        % Take the two nodes with the smallest probabilities
        leftNode = T(1);
        rightNode = T(2);
        
        % Create a new internal node
        newNode = struct('symbol', [], 'probability', leftNode.probability + rightNode.probability, 'left', leftNode, 'right', rightNode);
        
        % Remove the two nodes with the smallest probabilities
        T(1:2) = [];
        
        % Add the new node to the tree
        T = [T; newNode];
    end
    
    % Generate Huffman codes
    codes = cell(length(symbols), 1);
    generateCodes(T{1}, '', codes);
    
    % Calculate the average code length
    avgLength = sum(cellfun(@length, codes) .* probabilities);
end

function generateCodes(node, code, codes)
    if ~isempty(node.symbol)
        codes{strcmp(codes, '') + 1} = code;
    else
        generateCodes(node.left, [code '0'], codes);
        generateCodes(node.right, [code '1'], codes);
    end
end

#### 香农-范诺编码
创建自定义函数来生成香农-范诺码。

function [codes, avgLength] = shannonFanoCoding(symbols, probabilities)
    % Sort symbols by probability in descending order
    [probabilities, idx] = sort(probabilities, 'descend');
    symbols = symbols(idx);
    
    % Split the list into two parts with approximately equal probability
    splitIndex = find(cumsum(probabilities) >= 0.5, 1);
    
    % Assign binary codes recursively
    codes = cell(size(symbols));
    assignCodes(symbols, probabilities, 1, splitIndex, '0', codes);
    assignCodes(symbols, probabilities, splitIndex+1, length(symbols), '1', codes);
    
    % Reorder codes to match original symbol order
    codes = codes(idx);
    
    % Calculate the average code length
    avgLength = sum(cellfun(@length, codes) .* probabilities(idx));
end

function assignCodes(symbols, probabilities, start, endIdx, prefix, codes)
    if start == endIdx
        codes{start} = prefix;
    else
        mid = find(cumsum(probabilities(start:endIdx)) >= 0.5 * sum(probabilities(start:endIdx)), 1) + start - 1;
        assignCodes(symbols, probabilities, start, mid, [prefix '0'], codes);
        assignCodes(symbols, probabilities, mid+1, endIdx, [prefix '1'], codes);
    end
end

### 4. 字符串编码
对样本字符串进行编码，并计算编码后的总长度。

% Sample string
sampleString = 'AABCCDEFGH';

% Encode the sample string using Huffman and Shannon-Fano codes
[huffmanCodes, huffmanAvgLength] = huffmanCoding(Symbols, Probabilities);
[shannonFanoCodes, shannonFanoAvgLength] = shannonFanoCoding(Symbols, Probabilities);

% Function to encode a string using a given codebook
function encodedString = encodeString(str, codes)
    encodedString = '';
    for i = 1:length(str)
        encodedString = [encodedString codes{strcmp(codes, str(i))}];
    end
end

% Encode the sample string
huffmanEncodedString = encodeString(sampleString, huffmanCodes);
shannonFanoEncodedString = encodeString(sampleString, shannonFanoCodes);

% Calculate the total length of the encoded strings
huffmanTotalLength = length(huffmanEncodedString);
shannonFanoTotalLength = length(shannonFanoEncodedString);

% Display the results
disp(['Huffman Encoded String: ', huffmanEncodedString]);
disp(['Shannon-Fano Encoded String: ', shannonFanoEncodedString]);
disp(['Huffman Total Length: ', num2str(huffmanTotalLength)]);
disp(['Shannon-Fano Total Length: ', num2str(shannonFanoTotalLength)]);

### 5. 效率分析
计算并比较平均码长、压缩比、冗余度和编码效率。

% Original size of the sample string
originalSize = length(sampleString) * ceil(log2(length(Symbols)));

% Compression ratio
huffmanCompressionRatio = originalSize / huffmanTotalLength;
shannonFanoCompressionRatio = originalSize / shannonFanoTotalLength;

% Redundancy
huffmanRedundancy = huffmanAvgLength - H;
shannonFanoRedundancy = shannonFanoAvgLength - H;

% Coding efficiency
huffmanEfficiency = (H / huffmanAvgLength) * 100;
shannonFanoEfficiency = (H / shannonFanoAvgLength) * 100;

% Display the results
disp(['Huffman Average Code Length: ', num2str(huffmanAvgLength)]);
disp(['Shannon-Fano Average Code Length: ', num2str(shannonFanoAvgLength)]);
disp(['Huffman Compression Ratio: ', num2str(huffmanCompressionRatio)]);
disp(['Shannon-Fano Compression Ratio: ', num2str(shannonFanoCompressionRatio)]);
disp(['Huffman Redundancy: ', num2str(huffmanRedundancy)]);
disp(['Shannon-Fano Redundancy: ', num2str(shannonFanoRedundancy)]);
disp(['Huffman Coding Efficiency: ', num2str(huffmanEfficiency), '%']);
disp(['Shannon-Fano Coding Efficiency: ', num2str(shannonFanoEfficiency), '%']);

### 6. 数学证明与分析
提供数学证明和可视化结果。

% Function to plot the results
function plotResults(symbols, huffmanCodes, shannonFanoCodes, H, huffmanAvgLength, shannonFanoAvgLength, huffmanEfficiency, shannonFanoEfficiency)
    figure;
    subplot(2, 2, 1);
    bar([cellfun(@length, huffmanCodes), cellfun(@length, shannonFanoCodes)]);
    legend('Huffman', 'Shannon-Fano');
    title('Code Lengths');
    xlabel('Symbol');
    ylabel('Code Length');
    xticks(1:length(symbols));
    xticklabels(symbols);
    
    subplot(2, 2, 2);
    bar([huffmanAvgLength, shannonFanoAvgLength]);
    legend('Huffman', 'Shannon-Fano');
    title('Average Code Length');
    xlabel('Method');
    ylabel('Average Code Length');
    
    subplot(2, 2, 3);
    bar([huffmanEfficiency, shannonFanoEfficiency]);
    legend('Huffman', 'Shannon-Fano');
    title('Coding Efficiency');
    xlabel('Method');
    ylabel('Efficiency (%)');
    
    subplot(2, 2, 4);
    bar([huffmanRedundancy, shannonFanoRedundancy]);
    legend('Huffman', 'Shannon-Fano');
    title('Redundancy');
    xlabel('Method');
    ylabel('Redundancy');
end

% Plot the results
plotResults(Symbols, huffmanCodes, shannonFanoCodes, H, huffmanAvgLength, shannonFanoAvgLength, huffmanEfficiency, shannonFanoEfficiency);

### 7. 输出呈现
展示生成的霍夫曼和香农-范诺码，以及各种指标的比较表。

% Display the generated codes
disp('Huffman Codes:');
for i = 1:length(Symbols)
    disp([Symbols{i}, ': ', huffmanCodes{i}]);
end

disp('Shannon-Fano Codes:');
for i = 1:length(Symbols)
    disp([Symbols{i}, ': ', shannonFanoCodes{i}]);
end

% Create a table to compare the results
resultsTable = table({'Entropy'; 'Huffman Avg Length'; 'Shannon-Fano Avg Length'; 'Huffman Efficiency'; 'Shannon-Fano Efficiency'; 'Huffman Redundancy'; 'Shannon-Fano Redundancy'}, ...
                     {num2str(H); num2str(huffmanAvgLength); num2str(shannonFanoAvgLength); num2str(huffmanEfficiency); num2str(shannonFanoEfficiency); num2str(huffmanRedundancy); num2str(shannonFanoRedundancy)}, ...
                     'VariableNames', {'Metric', 'Value'});
disp(resultsTable);

### 提交指南
将上述代码保存为 `FirstnameLastname.mlx` 文件，并提交到Brightspace。同时，提交一个详细的PDF报告，总结你的发现、数学证明和结果。

希望这些代码和说明能帮助你顺利完成作业！如果有任何问题或需要进一步的帮助，请随时联系我。