二进制相移键控

基础·

BPSK 的时域表达式是

$$e_{BPSK}(t) = \sum_{n = -\infty}^{+\infty} a_nh(t-nT_S)\cos(\omega_ct+\varphi)$$

• $a_n$：待发送的二进制信息
• $T_S$：符号周期
• $h(t)$：成型滤波器的冲激响应
• $\omega_c$：载波中心频率

$T_S\sim N\dfrac{2\pi}{\omega_c}$：未必整数倍

MATLAB 仿真·

按照上图流程进行 MATLAB 仿真

调制与解调·

设定参数：系统时钟频率为 $160MHz$，根升余弦滤波器滚降系数 $\alpha=0.35$，其它参数可修改

sys_clk = 160e6;
Rb = 5e6;                  % //FIXME
Rs = Rb; Ts = 1 / Rs;

usmp_rate = sys_clk / Rs;  % //FIXME
fc = 20e6;                 % //FIXME
hrc = 'rrc';               % //FIXME

1. 随机生成 num 个二进制数，并对极化处理：

num = round(100000 * 10 ^ (EbNo / 10));
b = randi([0 1], 1, num);
b_sign = 1 - 2 * b;
% b_sign = exp(1j * pi * b)  % 1: cos(-\pi)=-1 / 0: cos(\pi)=1

2. 成型滤波，并去除延迟：

switch hrc
    case 'rect'
        h = ones(1, usmp_rate + 1);
    case 'sinc'
        t = -3 * pi : pi / usmp_rate : 3 * pi;
        h = sinc(t);
    case 'rrc'
        beta = 0.35;
        span = 6;
        sps = usmp_rate;
        h = rcosdesign(beta, span, sps, 'sqrt');
end

baseband = conv(upsample(b_sign, usmp_rate), h);
delay = (length(h) - 1 ) / 2;
baseband = baseband(delay + 1:end - delay);

基带信号的时域波形与功率谱密度

3. 载波调制

phase = 2 * pi * rand;
total_t = Ts * num;
t = 0 : 1/sys_clk : (total_t - 1/sys_clk);
carrier = cos(2 * pi * fc * t + phase);
modulated_signal = baseband .* carrier;

调制信号的时域波形与功率谱密度

---

0. 经过高斯白噪声信道

$SNR$ 与 $E_b/n_0$ 的关系：

$$SNR = \dfrac{S}{N} = \dfrac{E_b}{n_0}\dfrac{R_b}{B}$$

SNR = EbNo - 10 * log10(usmp_rate / 2);
noised_signal = awgn(modulated_signal, SNR, 'measured');

1. 相干解调

local_carrier = 2 * carrier;
received_signal = noised_signal .* local_carrier;

相干解调信号的时域波形与功率谱密度

2. 匹配滤波与抽样判决

mf_dout = conv(received_signal, fliplr(h));
delay = (length(h) - 1) / 2;
decision_result = downsample(mf_dout(delay+1:end), usmp_rate);
decision_result = decision_result(1:num);

匹配滤波后信号的时域波形与功率谱密度

仿真结果·

将仿真得到的误比特率曲线与理论误比特率曲线比较，基本重合

理论分析·

载波频率对误比特率的影响·

带通采样定理：

频带信号如图所示，为了保证被采样后不会发生混叠，需要满足条件：

$$\left\{ \begin{aligned} &-f_L + mf_s \leq f_L\\ &-f_H + (m+1)f_s \geq f_H\\ \end{aligned} \right.$$

则

$$\dfrac{2f_H}{m+1} \leq f_s \leq \dfrac{2f_L}{m}$$

在采样频率、码元速率一定时，由带通采样定理可以推导出载波频率 $f_c$ 的范围：

$$\begin{aligned} &\dfrac{2f_H}{m+1} \leq f_s \leq \dfrac{2f_L}{m}\\ &(f_H = f_c + \dfrac{B}{2}, f_L = f_c - \dfrac{B}{2})\\ &\dfrac{m}{2}f_s + \dfrac{B}{2}\leq f_c \leq \dfrac{m+1}{2}f_s - \dfrac{B}{2}\\ &(B = \dfrac{R_S}{\frac{2}{1+\alpha}}\times 2)\\ &\dfrac{m}{2}f_s + \dfrac{R_S(1+\alpha)}{2}\leq f_c \leq \dfrac{m+1}{2}f_s - \dfrac{R_S(1+\alpha)}{2}\\ \end{aligned}$$

当 $f_s = 160MHz, \alpha=0.35, R_S = 5MHz$ 时，可以得到 载波频率的取值范围 是

$$\begin{aligned} &m = 0, \quad 3.375 < f_c < 76.625\\ &m = 1, \quad 83.375 < f_c < 156.625\\ &m = 2, \quad 163.375 < f_c < 236.625\\ &\cdots \end{aligned}$$

内插系数对误比特率的影响·

$$\begin{aligned} &R_S \leq \dfrac{2f_c - mf_s}{1+\alpha}\\ &R_S \leq \dfrac{(m+1)f_s - 2f_c}{1+\alpha}\\ \end{aligned}$$

当 $f_s = 160, \alpha =0.35, f_c = 20$ 时，可以得到内插系数的范围是

$$\left\{ \begin{aligned} &2f_c - m f_s > 0\\ &(m+1)f_s - 2f_c > 0\\ \end{aligned} \right. \Rightarrow m = 0$$

$$R_S \leq \dfrac{40}{1.35}\Rightarrow UsmpRate\geq 5.4$$

接收机采样点位置对误比特率的影响·

由仿真结果可以看出，在最佳的接收机采样点处采样，误比特率最低，距离该点越远，误比特率越高。

成型滤波器对误比特率的影响·

几乎无影响。

FPGA 仿真（调制）·

设定参数：系统时钟频率为 $160MHz$，传输速率 $R_b=5MHz$，发送比特数为 $1024$ 位。

reg [14:0] counter = 15'd0;
always @(posedge clk) begin
    if (!rst_n) begin
        counter[14:0] <= 13'd0;
    end
    else begin
        counter[14:0] <= counter[14:0] + 13'd1;
    end
end

wire [9:0] address;
// sys_clk = 160M / Rb = 5M = 32
assign address[9:0] = counter[14:5];

由系统时钟频率和传输速率可以得到内插系数为 $32$，需要 $5$ 位计数器；发送 $1024$ 位数据，需要 $10$ 位地址，于是定义计数器变量为 reg [14:0] counter，每计数 $32$，地址增 $1$，将 counter 的高 $10$ 位赋值给 $address$.

存储数据、成型滤波、载波调制等过程通过 IP 核实现：

功能	IP 核
存储数据	Block Memory Generator
成型滤波	FIR Compiler
生成载波	DDS Compiler
载波调制	Multiplier
生成系统时钟	Clock Wizard

仿真波形如下图

附录·

噪声建模·

• SNR 与 Eb/n0 的关系

下文用到的符号表示：

• $E_b$：比特能量，单位 $J$
• $n_0$：噪声的功率谱密度，单位 $W/Hz$
• $E_b/n_0$：无量纲
• $S$：信号功率，单位 $W$
• $N$：噪声功率，单位 $W$
• $B$ 带宽
• $SNR$：信噪比，无量纲
• $R_b$：比特速率，单位 $bps$；$T_b$：传输每比特所需的时间
• $R_S$：符号速率，单位 $Baud$
• $R_C$：码片速率，单位 $chip/s$
• $M$：调制星座点个数
• $SSR$：扩频比
• $\alpha$：根升余弦成型滤波器的滚降因子
• $usmp\_rate$：内插系数

SNR（Signal Noise Radio）表示信噪比，$E_b/n_0$ 表示传输 $1bit$ 信息所需要的能量与噪声功率谱密度的比值。对于数字信号来说，用时间长度为 $T_s$ 的波形表示码元，每个码元的平均功率为 $0$，因此不能用功率描述数字信号，因此采用码元能量来描述数字信号波形。

$$SNR = \dfrac{S}{N} = \dfrac{E_bR_b}{n_0 B} = \dfrac{E_sR_s}{n_0 B} = \dfrac{E_cR_c}{n_0 B}$$

其中带宽 $B = f_{smp} / 2$，比特能量与符号能量满足关系 $E_s=E_b \log_2 M$，则

$$SNR = \dfrac{E_sR_s}{n_0 B} = \dfrac{(E_b\log_2{M})(R_c/SSR)}{n_0 \tfrac{f_{smp}}{2}} = \dfrac{E_b}{n_0} \cdot\log_2M \cdot \dfrac{2R_c}{f_{smp}} \cdot \dfrac{1}{SSR}$$

最终得到，

$$\begin{aligned} [SNR] _{dB} &= [E_b/n_0]_{dB} + [\log_2{M}] _{dB} - [\dfrac{usmp\_rate}{2}]_{dB} - [SSR]_{dB}\\ usmp\_rate &= f_{smp} / R_c \end{aligned}$$

注：

• 在常规通信系统中，$usmp\_rate$ 是仿真中的采样速率与 符号速率 之比；
• 在扩频通信系统中，$usmp\_rate$ 是仿真中的采样速率与 码片速率 之比；

BPSK 的 MATLAB 仿真·

• MATLAB 仿真代码（CPU 版）

close all; clear; clc;

tic;

% FIXME System Parameters
fs = 122.88e6;
Rs = 3.072e6;
Rb = Rs; Ts = 1 / Rs;
fc = 2e9;

usmp_rate = floor(fs / Rs);

num = 5000;
b = 1 - 2 * randi([0 1], 1, num);

%% Send
beta = 0.1; span = 12;
sps = usmp_rate;
h = rcosdesign(beta, span, sps, "sqrt");

baseband_tx = conv(upsample(b, usmp_rate), h, "same");

toc;

t = (1 : length(baseband_tx)) / fs;
carrier = cos(2 * pi * fc * t);
tx_wave = baseband_tx .* carrier;

toc;

%% Channel
Ebn0_dB = 10;
SNR_dB = Ebn0_dB - 10 * log10(usmp_rate / 2);
rx_wave = awgn(tx_wave, SNR_dB, 'measured');

toc;

%% Recv
% Coherent Demodulation
local_carrier = 2 * carrier;
demod_wave = rx_wave .* local_carrier;

% Matched Filtering
baseband_rx = conv(demod_wave, h, "same");

% Sampling and Decision
samples = baseband_rx(1 : sps : end);
bits_recovered = double(samples < 0);

toc;

original_bits = (b < 0);
err = sum(original_bits(1:num) ~= bits_recovered(1:num));
fprintf('Bit error rate: %d\n', err / num);

%% Figure
N = 65536;
t = (1 : length(baseband_tx)) / fs;
% Send
figure;
subplot(2, 1, 1);
plot(t, baseband_tx);
xlabel("Time (s)"); ylabel("Amplitude");
title("BPSK Baseband Signal");
grid on;

subplot(2, 1, 2);
f = (-fs/2 : fs/N : fs/2-fs/N);
plot(f, 10 * log10(abs(fftshift(fft(baseband_tx, N)))));
xlabel("Frequency (Hz)"); ylabel("Amplitude (dB)");
title("BPSK Baseband Spectrum");
grid on;

% Recv
figure;
subplot(2, 2, 1);
plot(t, demod_wave);
xlabel("Time (s)"); ylabel("Amplitude");
title("Coherent Demodulation Signal");
grid on;

subplot(2, 2, 2);
plot(t, baseband_rx);
xlabel("Time (s)"); ylabel("Amplitude");
title("Matched Filtering Signal");
grid on;

subplot(2, 2, 3);
f = (-fs/2 : fs/N : fs/2-fs/N);
plot(f, 10 * log10(abs(fftshift(fft(demod_wave, N)))));
xlabel("Frequency (Hz)"); ylabel("Amplitude (dB)");
title("Coherent Demodulation Spectrum");
grid on;

subplot(2, 2, 4);
f = (-fs/2 : fs/N : fs/2-fs/N);
plot(f, 10 * log10(abs(fftshift(fft(baseband_rx, N)))));
xlabel("Frequency (Hz)"); ylabel("Amplitude (dB)");
title("Matched Filtering Spectrum");
grid on;

figure;
subplot(2, 1, 1);
plot(original_bits(1:100));
hold on;
stem(original_bits(1:100), "r", "LineStyle", "none", "Marker", "o");
title('Original Bits (First 100)');
subplot(2, 1, 2);
plot(bits_recovered(1:100));
hold on;
stem(bits_recovered(1:100), "r", "LineStyle", "none", "Marker", "o");
title('Recovered Bits (First 100)');

• MATLAB 仿真代码（GPU 版）

close all; clear; clc;

if gpuDeviceCount > 0
    gpuDevice(1);
    fprintf('Using GPU: %s\n', gpuDevice().Name);
else
    error('No GPU detected!');
end

tic;

% FIXME System Parameters
fs = 122.88e6;
Rs = 3.072e6;
Rb = Rs; Ts = 1 / Rs;
% fc = 2e9;

usmp_rate = floor(fs / Rs);

num = 10000000;
b = gpuArray(1 - 2 * randi([0 1], 1, num));

%% Send
beta = 0.1; span = 12;
sps = usmp_rate;
h = gpuArray(rcosdesign(beta, span, sps, "sqrt"));

baseband_tx = conv(upsample(b, usmp_rate), h, "same");

toc;

% t = (1 : length(baseband_tx)) / fs;
% carrier = cos(2 * pi * fc * t);
% tx_wave = baseband_tx .* carrier;
%
% toc;

%% Channel
Ebn0_dB = 10;
SNR_dB = Ebn0_dB - 10 * log10(usmp_rate / 2);
signal_power = mean(abs(baseband_tx).^2);
noise_power = signal_power / (10^(SNR_dB/10));
noise = sqrt(noise_power) * randn(size(baseband_tx), 'gpuArray');
rx_wave = baseband_tx + noise;

toc;

%% Recv
% % Coherent Demodulation
% local_carrier = 2 * carrier;
% demod_wave = rx_wave .* local_carrier;

% Matched Filtering
baseband_rx = conv(rx_wave, h, "same");

% Sampling and Decision
samples = baseband_rx(1 : sps : end);
bits_recovered = double(samples < 0);

toc;

original_bits = (b < 0);
bits_recovered = gather(bits_recovered);
baseband_tx = gather(baseband_tx);
baseband_rx = gather(baseband_rx);
err = sum(original_bits(1:num) ~= bits_recovered(1:num));
fprintf('Bit error rate: %d\n', err / num);

%% Figure
N = 65536;
t = (1 : length(baseband_tx)) / fs;
% Send
figure;
subplot(2, 1, 1);
plot(t, baseband_tx);
xlabel("Time (s)"); ylabel("Amplitude");
title("BPSK Baseband Signal");
grid on;

subplot(2, 1, 2);
f = (-fs/2 : fs/N : fs/2-fs/N);
plot(f, 10 * log10(abs(fftshift(fft(baseband_tx, N)))));
xlabel("Frequency (Hz)"); ylabel("Amplitude (dB)");
title("BPSK Baseband Spectrum");
grid on;

% Recv
figure;
% subplot(2, 2, 1);
% plot(t, demod_wave);
% xlabel("Time (s)"); ylabel("Amplitude");
% title("Coherent Demodulation Signal");
% grid on;

subplot(2, 1, 1);
plot(t, baseband_rx);
xlabel("Time (s)"); ylabel("Amplitude");
title("Matched Filtering Signal");
grid on;

% subplot(2, 2, 3);
% f = (-fs/2 : fs/N : fs/2-fs/N);
% plot(f, 10 * log10(abs(fftshift(fft(demod_wave, N)))));
% xlabel("Frequency (Hz)"); ylabel("Amplitude (dB)");
% title("Coherent Demodulation Spectrum");
% grid on;

subplot(2, 1, 2);
f = (-fs/2 : fs/N : fs/2-fs/N);
plot(f, 10 * log10(abs(fftshift(fft(baseband_rx, N)))));
xlabel("Frequency (Hz)"); ylabel("Amplitude (dB)");
title("Matched Filtering Spectrum");
grid on;

figure;
subplot(2, 1, 1);
plot(original_bits(1:100));
hold on;
stem(original_bits(1:100), "r", "LineStyle", "none", "Marker", "o");
title('Original Bits (First 100)');
subplot(2, 1, 2);
plot(bits_recovered(1:100));
hold on;
stem(bits_recovered(1:100), "r", "LineStyle", "none", "Marker", "o");
title('Recovered Bits (First 100)');

参考资料·

• What are SNR and Eb/No?
• GEL7014 - Week 6e - EBN0 vs SNR
• GitHub | Euler0525/bpsk-gpu