### 实现16-QAM调制与解调 在Python中实现16-QAM（Quadrature Amplitude Modulation）调制和解调的过程涉及以下几个核心部分：创建星座图、映射比特到星座点以及逆过程。以下是详细的说明： #### 创建16-QAM星座图 16-QAM是一种四相位幅度调制技术，其星座图为4×4网格结构。每个星座点表示两个比特的信息。 ```python import numpy as np import matplotlib.pyplot as plt def generate_16qam_constellation(): constellation_points = np.array([complex(x, y) for x in [-3, -1, 1, 3] for y in [-3, -1, 1, 3]]) return constellation_points / np.sqrt(10) # Normalize the energy of each symbol to ensure average power is 1. constellation = generate_16qam_constellation() plt.scatter(constellation.real, constellation.imag) plt.axhline(0, color='black', linewidth=0.5, linestyle='--') plt.axvline(0, color='black', linewidth=0.5, linestyle='--') plt.grid(True) plt.title('16-QAM Constellation Diagram') plt.xlabel('In-phase Component') plt.ylabel('Quadrature Component') plt.show() ``` 上述代码生成了一个标准化的16-QAM星座图并绘制出来[^1]。 #### 映射比特流至星座点 为了将输入比特流转换成对应的星座点位置，可以采用灰度编码方式来减少误码率的影响。 ```python def bits_to_symbols(bits): mapping_table = { '0000': (-3-3j), '0001': (-3-1j), '0010': (-3+1j), '0011': (-3+3j), '0100': (-1-3j), '0101': (-1-1j), '0110': (-1+1j), '0111': (-1+3j), '1000': (+1-3j), '1001': (+1-1j), '1010': (+1+1j), '1011': (+1+3j), '1100': (+3-3j), '1101': (+3-1j), '1110': (+3+1j), '1111': (+3+3j), } symbols = [] bit_string = ''.join(map(str, bits)) for i in range(len(bit_string)//4): key = bit_string[i*4:(i+1)*4] symbols.append(mapping_table[key]) return np.array(symbols) / np.sqrt(10) bits = [int(b) for b in list(bin(np.random.randint(0, 2**8))[2:].zfill(8))] symbols = bits_to_symbols(bits) print(f'Bits: {bits}') print(f'Symbols: {symbols}') ``` 此函数接受一个二进制列表作为输入，并将其转化为相应的复数值符号序列。 #### 解调过程——从接收信号恢复原始比特 假设接收到带有噪声干扰后的信号，则可以通过计算欧几里得距离找到最接近的星座点完成解调操作。 ```python def demodulate(received_signal): inverse_mapping_table = {v:k for k,v in mapping_table.items()} detected_bits = '' for s in received_signal: min_distance = float('inf') closest_symbol = None for sym in mapping_table.values(): distance = abs(s-sym/np.sqrt(10)) ** 2 if distance < min_distance: min_distance = distance closest_symbol = sym detected_bits += inverse_mapping_table[closest_symbol * np.sqrt(10)] return [int(b) for b in detected_bits] received_signal = symbols + 0.5*(np.random.randn(*symbols.shape)+1j*np.random.randn(*symbols.shape)) detected_bits = demodulate(received_signal) print(f'Detected Bits: {detected_bits}') ``` 以上展示了如何基于最小均方误差原则执行硬判决检测从而还原出发送端初始的数据比特串。