基于FMCW雷达的测距、测速与测角原理与实现

基于FMCW雷达的测距、测速与测角原理与实现

摘要

线性调频连续波（FMCW）雷达通过发射线性扫频的上斜（或上/下交替）chirp，并对回波与本振信号拍频解调，得到与目标距离和速度相关的拍频与慢时间多普勒信息；配合多天线阵列的空间处理，实现距离–速度–角度联合估计。本文给出从波形到处理链的关键方程、分辨率与无模糊条件、以及工程设计要点。

1. 波形与回波模型

1.1 线性调频上斜 chirp

$$s_{\mathrm{tx}}(t)=\exp \left\{j2\pi \right(f_{c}t+\frac{1}{2}S{t}^2\left)\right\},0\le t<T_c$$

• $s_{\mathrm{tx}}(t)$：发射信号；

• $f_c$：载频$Hz$；

• $S=\frac{B}{T_c}$：调频斜率（Hz/s），$B$ 为扫频带宽$Hz$，$T_c$ 为单个 chirp 时长$s$；

• $t$：快时间$s$；

• $j=\sqrt{-1}$。

1.2 解调（拍频）信号

距离为 $R$、径向速度为 $v$ 的点目标，其回波延迟 $\tau =2R/c$，多普勒 $f_D=2v/\lambda$。与本振拍频后近似得到
$$x(t)\approx A\exp \left\{j2\pi \right(\underset{f_b}{\underbrace{S\tau }}t+f_{D}t+{\varphi }_0\left)\right\}$$

• $x(t)$：解调（拍频）信号；

• $A$：复幅度（含路径损耗与RCS）；

• $\tau =2R/c$：往返时延$s$，$c$ 为光速$m/s$；

• $f_b=S\tau$：拍频$Hz$，与距离成正比；

• $f_D=2v/\lambda$：多普勒频移$Hz$，$\lambda =c/f_c$为波长$m$；

• ${\varphi }_0$：常数相位；

• 近似假设：单个 chirp 内 $R$ 变化很小（小速度、小加速度）。

2. 测距（Range）

2.1 距离–拍频映射

$$R=\frac{c{f}_b}{2S}.$$

• $R$：目标距离$m$；

• $f_b$：拍频$Hz$；

• $S$：斜率$Hz/s$；

• $c$：光速$m/s$。

2.2 距离分辨率

$$\Delta R=\frac{c}{2B}.$$

• $\Delta R$：距离分辨率$m$；

• $B$：单个 chirp 的扫频带宽$Hz$；

• $c$：光速$m/s$。

2.3 最大无模糊距离（采样导致的）

拍频需小于 Nyquist：$f_b<f_s/2$，得
$$R_{\text{max}}=\frac{c{f}_s}{4S}.$$

• $R_{\text{max}}$：最大无模糊距离$m$；

• $f_s$：ADC 采样率$Hz$；

• 其他符号同前。

注：实际还受 IF 带宽、前端抗混叠滤波器、模拟中频范围等限制。

3. 测速（Velocity）

采用 N 个 chirp 的慢时间序列，对同一距离 bin 沿 chirp 维做 DFT，得到慢时间多普勒谱。

3.1 多普勒–速度映射

$$v=\frac{\lambda }{2}{f}_D.$$

• $v$：径向速度$m/s$；

• $f_D$：慢时间估计出的多普勒频率$Hz$；

• $\lambda =c/f_c$：波长$m$。

3.2 速度分辨率

慢时间 PRF 为 $f_{\text{r}}=1/T_r$（相邻 chirp 起始间隔 $T_r$，通常 $T_r\approx T_c)$，N 点 DFT：
$$\Delta v=\frac{\lambda }{2}\cdot \frac{f_{\text{r}}}{N}.\text{\hspace{1em}(7)}$$

• $\Delta v$：速度分辨率（m/s）；

• $\lambda$：波长（m）；

• $f_{\text{r}}$：慢时间采样率（PRF，Hz）；

• $N$：用于多普勒 FFT 的 chirp 数。

3.3 最大无模糊速度

多普勒落在 $(-\frac{f_{\text{r}}}{2},\frac{f_{\text{r}}}{2})$内，得
$$|v|<\frac{\lambda }{4}{f}_{\text{r}}.$$

• $|v|$：速度绝对值（m/s）；

• $\lambda$：波长（m）；

• $f_{\text{r}}$：PRF（Hz）。

交替上/下扫（triangular FMCW）可分离距离拍频与多普勒，提高抗耦合能力与无模糊范围。

4. 测角（Angle）

采用 M 元均匀线阵（ULA），阵元间距 $d\approx \lambda /2$。到达角 $\theta$ 的阵列流形为
$$\mathbf{a}(\theta )={\left[\begin{array}{c}1,e^{j2\pi \frac{d}{\lambda }\sin \theta },\dots ,e^{j2\pi \frac{(M-1)d}{\lambda }\sin \theta }\end{array}\right]}^T.$$

• $\mathbf{a}(\theta )$：阵列导向向量$(M\times 1)$；

• $\theta$：方位角（deg 或 rad）；

• $d$：阵元间距（m）；

• $M$：阵元数；

• $\lambda$：波长（m）。

4.1 FFT 波束形成角度谱

对同一 $(\text{range},\text{doppler})$ 单元的阵列快照 $\mathbf{x}\in {\mathbb{C}}^{M\times 1}$，
$$P_{\text{FFT}}(\theta )=|\mathbf{w}(\theta {)}^H\mathbf{x}{|}^2,\mathbf{w}(\theta )=\frac{\mathbf{a}(\theta )}{|\mathbf{a}(\theta )|}.$$

• $P_{\text{FFT}}(\theta )$：FFT/波束形成角度谱；

• $\mathbf{w}(\theta )$：方向 $\theta$ 的波束权；

• $\mathbf{x}$：阵列观测向量；

• $(\cdot {)}^H$：共轭转置；

• $|\cdot |$：欧氏范数。

4.2 MUSIC 超分辨角度谱（可选）

估计协方差 ${\mathbf{R}}_{xx}$，分解得到噪声子空间 ${\mathbf{E}}_n$ 后
$$P_{\text{MUSIC}}(\theta )=\frac{1}{\mathbf{a}(\theta {)}^H{\mathbf{E}}_n{\mathbf{E}}_n^H\mathbf{a}(\theta )}.\text{\hspace{1em}(11)}$$

• $P_{\text{MUSIC}}(\theta )$：MUSIC 伪谱；

• $\mathbf{a}(\theta )$：阵列导向向量；

• ${\mathbf{E}}_n$：噪声子空间基向量矩阵；

• $(\cdot {)}^H$：共轭转置。

4.3 角度分辨率（波束主瓣近似）

$$\Delta \theta \approx \frac{0.886\lambda }{Md},(\text{rad})\text{或}\Delta \theta \approx \frac{50.8^{\circ }\lambda }{Md}.$$

• $\Delta \theta$：角度分辨率；

• $\lambda$：波长；

• $M$：阵元数；

• $d$：阵元间距；

• 系数 0.886 为均匀线阵近似主瓣宽度常用经验值。

5. 处理链（2D-FFT + CFAR + 角度谱）

5.1 二维 FFT（距离–多普勒）

对解调数据立方体 $\text{(Rx)}\times \text{(chirp)}\times \text{(sample)}$，先沿快时间做 $N_r$ 点 FFT、再沿慢时间做 $N_d$ 点 FFT：
$$X[k,n]={\mathcal{F}}_{\text{slow}}{\mathcal{F}}_{\text{fast}}x[m],k=0\dots N_d-1,n=0\dots N_r-1.$$

• $x[m]$：第 m 个快时间采样（或向量）；

• $n$：距离 FFT bin 索引；

• $k$：多普勒 FFT bin 索引；

• ${\mathcal{F}}_{\text{fast}},{\mathcal{F}}_{\text{slow}}$：快/慢时间 FFT 运算；

• $N_r,N_d$：距离/多普勒 FFT 大小。

5.2 无检测门限的能量图与非相干合成

多天线非相干融合用于检测（抑制相位不一致影响）：
$$\mathrm{RD}[k,n]=\sum _{m=1}^M|X_m[k,n]{|}^2.\text{\hspace{1em}(14)}$$

• $\mathrm{RD}[k,n]$：距离–多普勒能量图；

• $X_m[k,n]$：第 m 个接收通道在 $(k,n)$ 处的复谱值；

• $M$：接收通道数。

5.3 CA-CFAR 阈值（二维均值估计近似）

令参考单元数为 $N_{\mathrm{ref}}$、期望虚警率 $P_{\mathrm{fa}}$，则放缩系数 $\alpha$ 近似为
$$\alpha =N_{\mathrm{ref}}(P_{\mathrm{fa}}^{-1/N_{\mathrm{ref}}}-1).\text{\hspace{1em}(15)}$$

• $\alpha$：CFAR 阈值放缩系数；

• $N_{\mathrm{ref}}$：参考单元数（不含保护单元）；

• $P_{\mathrm{fa}}$：目标虚警概率。

5.4 角度谱切片（RA 图）

对选定 $(k,n)$ 处取阵列快照 ${\mathbf{x}}_{k,n}$，计算角度谱并在角度–距离平面显示：
$$\mathrm{RA}(\theta ,n)=|\mathbf{w}(\theta {)}^H{\mathbf{x}}_{k^{\ast }(n),n}{|}^2,k^{\ast }(n)=\arg \underset{k}{\max }\mathrm{RD}[k,n]$$

• $\mathrm{RA}(\theta ,n)$：角度–距离谱（每个距离选用能量最大的多普勒切片）；

• ${\mathbf{x}}_{k,n}$：在 $(k,n)$ 处的阵列观测向量；

• $\mathbf{w}(\theta )$：波束形成权；

• $k^{\ast }(n)$：距离 bin (n) 上能量最大的多普勒 bin。

6. 关键约束、分辨率与耦合

6.1 距离–速度耦合（单上斜）

若只用上斜 chirp，拍频含有速度项，近似
$$f_b\approx S\tau +f_D.\text{\hspace{1em}(17)}$$

• $f_b$：拍频；

• $S\tau$：与距离成正比项；

• $f_D$：多普勒项。

采用 上/下扫配对或 多帧慢时间处理可减弱耦合：上/下扫的 $f_b$ 对 $f_D$符号相反，可线性消去 $f_D$ 得距离，再以慢时间多普勒求速度。

6.2 距离窗泄漏与定标

距离 FFT 前加窗（Hann/Hamming）降低旁瓣；定标需考虑 TX/RX 通道幅相一致性、IF 带宽与采样链路、ADC 时钟偏差 等。

7. 参数设计与工程指南

• 距离分辨率：由带宽决定，$\Delta R=c/(2B)$——若需 0.15 m 分辨率，带宽需 $B\approx 1$ GHz。

• 速度分辨率：增大 chirp 数 N 或 PRF 降低 $\Delta v$。

• 无模糊距离/速度：给出上限，工程上还要考虑 IF/模数链路实际范围。

• 测角性能：增大阵元数 $M$ 或孔径 $M_d$ 提升分辨率；阵元间距宜 $\le \lambda /2$ 防止栅瓣。

• 检测门限：根据场景回波统计与 $P_{\mathrm{fa}}$ 选择 $N_{\mathrm{ref}},\alpha$，目标稀疏时可用 OS-CFAR。

• 稳健性：目标加速度导致慢时间相位弯曲，可用 Keystone 校正或更高阶模型；强杂波/多径下建议做 空时自适应 或 MIMO 多通道正交培训。

MATLAB代码

%% ============================================================% FMCW Radar: Range / Velocity / Angle All-in-One Demo% - Angle–Range uses per-range strongest Doppler slice% - Legends + Truth Annotations on RD, RA, 3D% Single-file, one-click runnable% Tested: R2020b+ (no special toolboxes required)% =============================================================clear; clc; close all;%% ----------------------- User Parameters -----------------------% Radar & waveformc = 3e8; % speed of lightfc = 77e9; % carrier frequency (Hz)lambda = c/fc;BW = 1e9; % sweep bandwidth (Hz)Tc = 40e-6; % chirp duration (s)S = BW/Tc; % chirp slope (Hz/s)fs = 20e6; % ADC sampling rate (Hz)Ns = round(fs*Tc); % samples per chirpNchirp = 128; % number of chirps (slow-time)Tx = 1; % 1-Tx (TDM off)Rx = 8; % number of Rx (ULA)d = lambda/2; % element spacing% Scene (multi-target): [R(m), v(m/s), azimuth(deg), RCS/amp]targets = [...40, -10, -10, 1.0; % T155, 8, 20, 0.8; % T285, 0, 5, 0.6 % T3];SNR_dB = 20; % SNR of dechirped baseband% Processing paramsNfft_r = 2^nextpow2(Ns*2); % range FFT size (zero-pad)Nfft_d = 2^nextpow2(Nchirp*2); % doppler FFT sizeNfft_a = 256; % angle FFT size (beamforming)use_MUSIC = true; % also compute MUSIC AoA (optional)MUSIC_K = size(targets,1); % number of sources assumed for MUSICguard_r = 2; train_r = 8; % 2D-CFAR params (range)guard_d = 2; train_d = 8; % 2D-CFAR params (doppler)Pfa = 1e-3; % CFAR false alarm% Derived limitsR_max = fs*c/(2*S); % unambiguous rangeV_max = lambda/(4*Tc); % +/- V_max unambiguousfprintf('Unambiguous Range ~ %.1f m, Unambiguous Velocity ~ +/- %.1f m/s\n',...R_max, V_max);%% ----------------------- Time & Grids -------------------------t_fast = (0:Ns-1)/fs; % fast-time within a chirpn_slow = (0:Nchirp-1).'; % chirp index (slow-time)f_fast = (0:Nfft_r-1)*(fs/Nfft_r); % beat freq axis (0..fs)R_axis = c * f_fast / (2*S); % beat freq -> range% Slow-time (Doppler) axis: PRF = 1/Tc, centeredf_doppler = (-Nfft_d/2:Nfft_d/2-1)*(1/(Tc*Nfft_d));V_axis = (lambda/2) * f_doppler;% Angle axis for Rx-ULA (broadside 0 deg; left negative)mu_axis = linspace(-1,1,Nfft_a);ang_axis = asind(mu_axis);%% ----------------------- Signal Simulation --------------------% Data cube: Rx x Nchirp x Ns (dechirped baseband)X = zeros(Rx, Nchirp, Ns);for k = 1:size(targets,1)R0 = targets(k,1);vk = targets(k,2);thet = targets(k,3) * pi/180;amp = targets(k,4);Rn = R0 + vk * n_slow * Tc; % Nchirp x 1tau = 2 * Rn / c; % Nchirp x 1fd = 2*vk/lambda; % Doppler (Hz)m_idx = (0:Rx-1).';phi_arr_m = 2*pi * (m_idx * d * sin(thet) / lambda); % Rx x 1S_tau = S * tau; % Nchirp x 1[S_tau_mat, t_fast_mat] = ndgrid(S_tau, t_fast);[n_slow_mat, ~] = ndgrid(n_slow, t_fast);phase_nt = 2*pi*((S_tau_mat + fd).*t_fast_mat + fd .* n_slow_mat * Tc); % Nchirp x Nsfor m = 1:RxX(m,:,:) = squeeze(X(m,:,:)) + amp * exp(1j*(phase_nt + phi_arr_m(m)));endend% Add AWGNsig_pow = mean(abs(X(:)).^2);noise_pw = sig_pow / (10^(SNR_dB/10));noise = sqrt(noise_pw/2) * (randn(size(X)) + 1j*randn(size(X)));X = X + noise;%% ----------------------- Range FFT ----------------------------win_r = hann(Ns).';X_r = zeros(Rx, Nchirp, Nfft_r);for m = 1:Rxfor n = 1:Nchirpx = squeeze(X(m,n,:)).' .* win_r;X_r(m,n,:) = fft(x, Nfft_r);endendvalid_r = R_axis <= R_max;R_hat = R_axis(valid_r);Nr = numel(R_hat);%% ----------------------- Doppler FFT (fixed) ------------------win_d = hann(Nchirp);Nd = Nfft_d;X_rd = zeros(Rx, Nd, Nr); % Rx x Nd x Nrfor m = 1:RxXr = squeeze(X_r(m,:,:)); % Nchirp x Nfft_rXr = Xr(:, valid_r); % Nchirp x NrXr = Xr .* repmat(win_d, 1, size(Xr,2));Xrd = fftshift(fft(Xr, Nd, 1), 1); % Nd x NrX_rd(m,:,:) = reshape(Xrd, [1, size(Xrd,1), size(Xrd,2)]);end% Non-coherent sum over Rx for detectionRD = squeeze(sum(abs(X_rd).^2, 1)); % Nd x Nr%% ----------------------- 2D-CFAR ------------------------------mag = RD;Nref = (2*train_r*2*train_d);alpha = ca_cfar_alpha(Nref, Pfa);det_map = false(Nd, Nr);for id = 1+train_d+guard_d : Nd - (train_d+guard_d)d_idx = [id-(train_d+guard_d):id-guard_d-1, id+guard_d+1:id+train_d+guard_d];for ir = 1+train_r+guard_r : Nr - (train_r+guard_r)r_idx = [ir-(train_r+guard_r):ir-guard_r-1, ir+guard_r+1:ir+train_r+guard_r];noise_est = mean(mag(d_idx, r_idx), 'all');if mag(id, ir) > alpha * (noise_est + eps)det_map(id, ir) = true;endendendpeak_idx = nms_peaks(mag, det_map, [3 3]); % [id, ir]%% ----------------------- Angle Estimation ---------------------est_list = []; % [R, V, AoA_FFT, Mag, AoA_MUSIC]for p = 1:size(peak_idx,1)id = peak_idx(p,1); % doppler binir = peak_idx(p,2); % range binfd_hat = f_doppler(id);V_est = (lambda/2) * fd_hat;R_est = R_hat(ir);x_m = squeeze(X_rd(:, id, ir)); % Rx x 1 snapshotang_spectrum = abs(fftshift(fft(x_m, Nfft_a))).^2;[~, ia] = max(ang_spectrum);ang_fft = ang_axis(ia);ang_music = NaN;if use_MUSIC && Rx >= (MUSIC_K+1)ang_music = music_aoa(x_m, d, lambda, Nfft_a, MUSIC_K);endest_list = [est_list; R_est, V_est, ang_fft, norm(x_m), ang_music]; %#ok<AGROW>end%% ----------------------- Truth (for annotations) ---------------R_true = targets(:,1);v_true = targets(:,2);th_true = targets(:,3);fd_true = 2*v_true/lambda; % Hz%% ----------------------- Visualization ------------------------% 1) Range profileRP = squeeze(sum(sum(abs(X_r(:,:,valid_r)).^2, 1), 2)); % 1 x Nrfigure; plot(R_hat, 10*log10(RP/max(RP)+1e-12), 'LineWidth',1.3);grid on; xlabel('Range (m)'); ylabel('Normalized Power (dB)');title('Range Profile');% 2) Range-Doppler map + detections + TRUTH + LEGENDfigure;imagesc(R_hat, f_doppler, 10*log10(mag./max(mag(:))+1e-12));axis xy; colorbar; xlabel('Range (m)'); ylabel('Doppler (Hz)');title('Range-Doppler Map (sum over Rx)');hold on;hDet = [];if ~isempty(peak_idx)hDet = scatter(R_hat(peak_idx(:,2)), f_doppler(peak_idx(:,1)), 40, 'w', 'filled', ...'DisplayName','Detections');endhTruth = plot(R_true, fd_true, 'p', 'MarkerSize', 12, 'MarkerEdgeColor', 'k', ...'MarkerFaceColor', [1 1 0], 'LineWidth', 1.5, 'DisplayName','Truth');for i = 1:numel(R_true)text(R_true(i)+0.5, fd_true(i), sprintf('T%d', i), ...'Color', 'k', 'FontWeight', 'bold', 'VerticalAlignment', 'middle');end% Legendif isempty(hDet)legend(hTruth, 'Location','best');elselegend([hDet, hTruth], {'Detections','Truth'}, 'Location','best');endhold off;% 3) Angle spectrum for top-1 detection (FFT beamforming)if ~isempty(peak_idx)id = peak_idx(1,1); ir = peak_idx(1,2);x_m = squeeze(X_rd(:, id, ir));ang_spec = abs(fftshift(fft(x_m, Nfft_a))).^2;ang_spec = ang_spec / max(ang_spec + 1e-12);figure; plot(ang_axis, 10*log10(ang_spec+1e-12), 'LineWidth',1.3); grid on;xlabel('Azimuth (deg)'); ylabel('dB');title('Angle Spectrum (Top-1, FFT Beamforming)');end% 4) Angle–Range 2D Map using "per-range strongest Doppler slice" + TRUTH + LEGEND[~, id_max_per_r] = max(RD, [], 1); % 1 x Nr (each range bin's strongest Doppler bin index)RA = zeros(Nfft_a, Nr);for ir = 1:Nrx_m = squeeze(X_rd(:, id_max_per_r(ir), ir)); % Rx x 1ang_spec = abs(fftshift(fft(x_m, Nfft_a))).^2; % Nfft_a x 1RA(:, ir) = ang_spec;endRA = RA / (max(RA(:)) + 1e-12);figure;imagesc(R_hat, ang_axis, 10*log10(RA + 1e-12));axis xy; colorbar; xlabel('Range (m)'); ylabel('Azimuth (deg)');title('Angle–Range Map (per-range strongest Doppler slice)');hold on;% truth overlay: only plot if target's fd is close to the chosen slice at its nearest range bindf = abs(f_doppler(2) - f_doppler(1)); % Doppler frequency resolutiontol = 1.5*df; % tolerance: ~1.5 binstruth_mask = false(size(R_true));for i = 1:numel(R_true)[~, ir0] = min(abs(R_hat - R_true(i))); % nearest range-bin to target rangefd_slice = f_doppler(id_max_per_r(ir0)); % doppler slice used for this rangetruth_mask(i) = abs(fd_true(i) - fd_slice) <= tol;endhTruthRA = plot(R_true(truth_mask), th_true(truth_mask), 'p', 'MarkerSize', 11, ...'MarkerEdgeColor','k','MarkerFaceColor',[1 1 0],'LineWidth',1.3, 'DisplayName','Truth near slice');for i = 1:numel(R_true)if truth_mask(i)text(R_true(i)+0.5, th_true(i), sprintf('T%d',i), ...'Color','k','FontWeight','bold','VerticalAlignment','middle');endendlegend(hTruthRA, 'Location','best');hold off;% 5) 3D Scatter of Detections in (Range, Velocity, Angle) + TRUTH + LEGENDif ~isempty(est_list)ang_use = est_list(:,5); % prefer MUSIC if availablenan_idx = isnan(ang_use);ang_use(nan_idx) = est_list(nan_idx,3); % fallback to FFT AoAmag_lin = est_list(:,4);mag_db = 20*log10(mag_lin / (max(mag_lin)+1e-12));sz = 30 + 70*(mag_lin / (max(mag_lin)+1e-12)); % marker size by magnitudefigure;hDet3 = scatter3(est_list(:,1), est_list(:,2), ang_use, sz, mag_db, 'filled', ...'DisplayName','Detections');grid on; xlabel('Range (m)'); ylabel('Velocity (m/s)'); zlabel('Azimuth (deg)');title('3D Detection Cloud: Range–Velocity–Angle');cb = colorbar; cb.Label.String = 'Magnitude (dB, relative)';view(45,25);hold on;hTruth3 = plot3(R_true, v_true, th_true, 'p', 'MarkerSize', 12, ...'MarkerEdgeColor','k','MarkerFaceColor',[1 1 0], 'LineWidth',1.5, ...'DisplayName','Truth');for i = 1:numel(R_true)text(R_true(i), v_true(i), th_true(i)+1.0, sprintf('T%d',i), ...'Color','k','FontWeight','bold');endlegend([hDet3 hTruth3], {'Detections','Truth'}, 'Location','best');hold off;end%% ----------------------- Print Detections ---------------------if isempty(est_list)fprintf('No detections. Try lowering threshold (increase Pfa) or raising SNR.\n');else% Sort by magnitude (desc)[~, ord] = sort(est_list(:,4), 'descend');est_sorted = est_list(ord,:);fprintf('\nDetections (after 2D-CFAR + AoA):\n');if use_MUSICfprintf(' %9s %10s %10s %10s %12s %10s\n','Range(m)','Vel(m/s)','AoA_FFT','Mag','AoA_MUSIC','Mag(dB)');for i=1:size(est_sorted,1)mag_db = 20*log10(est_sorted(i,4)/(max(est_sorted(:,4))+1e-12));fprintf(' %9.2f %10.2f %10.1f %10.2f %12.1f %10.1f\n', ...est_sorted(i,1), est_sorted(i,2), est_sorted(i,3), est_sorted(i,4), est_sorted(i,5), mag_db);endelsefprintf(' %9s %10s %10s %10s %10s\n','Range(m)','Vel(m/s)','AoA_FFT','Mag','Mag(dB)');for i=1:size(est_sorted,1)mag_db = 20*log10(est_sorted(i,4)/(max(est_sorted(:,4))+1e-12));fprintf(' %9.2f %10.2f %10.1f %10.2f %10.1f\n', ...est_sorted(i,1), est_sorted(i,2), est_sorted(i,3), est_sorted(i,4), mag_db);endendend%% ----------------------- Helper Functions ---------------------function alpha = ca_cfar_alpha(Nref, Pfa)% Return CA-CFAR scaling alpha for given reference cells and Pfa% Using approximate formula: Pfa = (1+alpha/Nref)^(-Nref)% => alpha = Nref * (Pfa^(-1/Nref) - 1)alpha = Nref * (Pfa^(-1/Nref) - 1);endfunction peaks = nms_peaks(mag, det_map, nms_sz)% Simple Non-Maximum Suppression on det_map using local neighborhood nms_sz% mag: Nd x Nr, det_map: logical same size, nms_sz = [h w][H, W] = size(mag);hh = floor(nms_sz(1)/2);ww = floor(nms_sz(2)/2);idx = find(det_map);keep = true(size(idx));for k = 1:length(idx)if ~keep(k), continue; end[r, c] = ind2sub([H,W], idx(k));r1 = max(1, r-hh); r2 = min(H, r+hh);c1 = max(1, c-ww); c2 = min(W, c+ww);patch = mag(r1:r2, c1:c2);if mag(r,c) < max(patch(:)) - epskeep(k) = false;continue;end[rr, cc] = find(det_map(r1:r2, c1:c2));rr = rr + (r1-1); cc = cc + (c1-1);nbr_lin = sub2ind([H,W], rr, cc);for t = 1:numel(nbr_lin)if nbr_lin(t) == idx(k), continue; endj = find(idx == nbr_lin(t), 1);if ~isempty(j), keep(j) = false; endendendsel = idx(keep);[rr2, cc2] = ind2sub([H,W], sel);peaks = [rr2, cc2];endfunction ang_hat = music_aoa(x_m, d, lambda, Nfft_a, K)% x_m: Rx x 1 snapshot at a single range-doppler bin% Returns peak of MUSIC spectrum (deg)M = numel(x_m);Rxx = (x_m * x_m')/M;Rxx = Rxx + (1e-3*trace(Rxx)/M)*eye(M);[V, D] = eig((Rxx+Rxx')/2);[~, idx] = sort(real(diag(D)), 'ascend');K = min(K, M-1);En = V(:, idx(1:M-K));mu_axis = linspace(-1,1, Nfft_a);P = zeros(1, Nfft_a);m = (0:M-1).';for i = 1:Nfft_aa = exp(1j*2*pi*(m*d/lambda)*mu_axis(i));denom = real(a'*(En*En')*a);if denom <= 0, denom = eps; endP(i) = 1 / denom;end[~, ia] = max(P);ang_hat = asind(mu_axis(ia));end

8. 结论

FMCW 雷达通过单帧 2D-FFT与阵列角度谱即可获得可靠的 R–V–AoA 估计；带宽、PRF、阵列孔径直接决定分辨率与无模糊范围。工程实现应兼顾前端链路、校准与检测门限设计，以在复杂环境中保持稳健性能。