Multichannel Monostatic Sensing

The BS illuminates the scene with its known downlink OFDM waveform and receives echoes through $R$ synchronized sensing channels. Keeping their relative complex phases allows the system to estimate angle under the ULA model in addition to range and Doppler.

1. Multichannel Time-Frequency Grid

Define the received vector on resource $(n,m,\gamma)$ as

$$\boldsymbol Y_{n,m,\gamma}^\mathrm{sens} = \begin{bmatrix} Y_{n,m,\gamma}^{(0)}&\cdots&Y_{n,m,\gamma}^{(R-1)} \end{bmatrix}^{T}.$$

When the maximum echo delay fits inside the cyclic prefix and intra-symbol Doppler is much smaller than $\Delta f$,

$$\boldsymbol Y_{n,m,\gamma}^\mathrm{sens} =b_{n,m,\gamma}^\mathrm{DL} \sum_{p=1}^{P+C} \beta_p\boldsymbol a(\theta_p) e^{j2\pi\left(f_{D,s,p}t_{m,\gamma} -\kappa_n\Delta f\tau_{s,p}^\mathrm{link}\right)} +\boldsymbol Z_{n,m,\gamma},$$

where $t_{m,\gamma}=(\gamma M+m)T_O$ and $\tau_{s,p}^\mathrm{link}=\tau_{s,p}^\mathrm{prop}+\tau_\mathrm{sens}^\mathrm{RF}$. Since $b_{n,m,\gamma}^\mathrm{DL}$ is known, modulation is removed element-wise:

$$\boldsymbol F_{n,m,\gamma} =\frac{\boldsymbol Y_{n,m,\gamma}^\mathrm{sens}} {b_{n,m,\gamma}^\mathrm{DL}} =\sum_{p=1}^{P+C} \beta_p\boldsymbol a(\theta_p) e^{j2\pi\left(f_{D,s,p}t_{m,\gamma} -\kappa_n\Delta f\tau_{s,p}^\mathrm{link}\right)} +\tilde{\boldsymbol Z}_{n,m,\gamma}.$$

Retaining every channel forms the tensor, where $M_\mathrm{sens}$ is the sensing-symbol count per frame:

$$\mathcal F_\gamma \in\mathbb C^{N\times M_\mathrm{sens}\times R}.$$

2. Array-Channel Calibration

Let fixed relative complex gains be

$$\boldsymbol C=\operatorname{diag}(g_0,g_1,\ldots,g_{R-1}).$$

The calibrated vector is

$$\boldsymbol F_{n,m,\gamma}^{\mathrm{cal}} =\boldsymbol C^{-1}\boldsymbol F_{n,m,\gamma}.$$

Range and Doppler can be processed per channel, but angle estimation must use calibrated inter-channel phase. A common complex gain changes only total amplitude and phase; relative channel-phase error directly biases angle.

3. Slow Time and Clutter Suppression

Select uniformly spaced sensing symbols from continuous frames:

$$\boldsymbol F[n,q] =\boldsymbol F_{n,m_q,\gamma_q}^{\mathrm{cal}}, \qquad t_q=qT_\mathrm{slow}.$$

With one sample every $M_D$ OFDM symbols, $T_\mathrm{slow}=M_DT_O$. TDD uses only active downlink resources. A conventional Doppler FFT requires uniform samples; a nonuniform selection must use the actual $t_q$.

Static and near-static reflections concentrate near zero Doppler. Apply a high-pass MTI filter along $q$ for each subcarrier and array channel:

$$\tilde{\boldsymbol F}[n,q] =\frac{1}{a_0} \left( \sum_{i=0}^{I}b_i\boldsymbol F[n,q-i] -\sum_{j=1}^{J}a_j\tilde{\boldsymbol F}[n,q-j] \right).$$

$\{b_i\}$ and $\{a_j\}$ are the feedforward and feedback coefficients. The filter creates a stopband around zero Doppler, suppressing fixed leakage and static clutter while retaining motion outside the notch. Near-stationary targets require a narrower notch or the unfiltered tensor.

4. Range–Doppler Processing

Split the slow-time stream into coherent intervals of $M_s$ samples. Let $N_\mathrm{Per}$ and $M_\mathrm{Per}$ be the delay-IFFT and Doppler-FFT lengths. With two-dimensional window $w[n,q]$, the array vector in each range–Doppler cell is

$$\boldsymbol z_\gamma[k_\tau,k_f] =\sum_{q=0}^{M_s-1}\sum_{n=0}^{N-1} \tilde{\boldsymbol F}_\gamma[n,q]w[n,q] e^{j2\pi\kappa_n k_\tau/N_\mathrm{Per}} e^{-j2\pi qk_f/M_\mathrm{Per}}.$$

Angle-independent array power is

$$P_\mathrm{RD}[k_\tau,k_f] =\frac{1}{NM_s} \left\|\boldsymbol z_\gamma[k_\tau,k_f]\right\|_2^2.$$

With signed, FFT-shifted Doppler index $k_f$,

$$\hat\tau^\mathrm{link}=\frac{\hat k_\tau}{N_\mathrm{Per}\Delta f}, \qquad \hat f_D=\frac{\hat k_f}{M_\mathrm{Per}T_\mathrm{slow}},$$ $$\hat\tau^\mathrm{prop} =\hat\tau^\mathrm{link}-\tau_\mathrm{sens}^\mathrm{RF}, \qquad \hat r=\frac{c\hat\tau^\mathrm{prop}}{2}, \qquad \hat v=\frac{c\hat f_D}{2f_c}.$$

Zero padding increases display sampling density, but fundamental range resolution remains set by $B$ and Doppler resolution by coherent duration $M_sT_\mathrm{slow}$.

Resolution and Unambiguous Ranges

With contiguous bandwidth $B=N\Delta f$, delay resolution is $\Delta\tau=1/B$, giving monostatic range resolution

$$\Delta r_\mathrm{mono}=\frac{c}{2B}.$$

Uniform subcarrier sampling has circular delay-ambiguity period $1/\Delta f$, corresponding to monostatic range period $c/(2\Delta f)$; the interference-free echo delay should still remain within $T_\mathrm{CP}$. For slow-time interval $T_\mathrm{slow}$ and coherent length $M_s$,

$$\Delta f_D=\frac{1}{M_sT_\mathrm{slow}}, \qquad |f_D|<\frac{1}{2T_\mathrm{slow}}.$$

The velocity resolution is $c\Delta f_D/(2f_c)$. Zero padding increases display sampling density but does not change these fundamental resolution or ambiguity limits.

5. ULA Angle Processing

Scan the ULA steering vector at a fixed $(k_\tau,k_f)$ cell:

$$P_\mathrm{RDA}[k_\tau,k_f,\theta] =\frac{ \left|\boldsymbol a^H(\theta) \boldsymbol z_\gamma[k_\tau,k_f]\right|^2 }{R^2NM_s},$$ $$\hat\theta =\arg\max_\theta P_\mathrm{RDA}[k_\tau,k_f,\theta].$$

For one dominant target, calibrated array phase follows

$$\angle z_r\approx\phi_0+r\mu, \qquad \mu=\frac{2\pi d_a}{\lambda}\sin\theta.$$

After phase unwrapping and slope fitting,

$$\hat\theta =\arcsin\!\left(\frac{\lambda\hat\mu}{2\pi d_a}\right).$$

$d_a\le\lambda/2$ prevents visible-region spatial aliasing. If several targets share a range–Doppler cell, one phase slope no longer represents one angle; beam scanning, a spatial FFT, or a higher-resolution array estimator is then required.

6. Multichannel Micro-Doppler

First transform subcarriers to per-channel range–slow-time data:

$$\boldsymbol r[k_\tau,q] =\frac{1}{N}\sum_{n=0}^{N-1} \tilde{\boldsymbol F}[n,q]e^{j2\pi nk_\tau/N}.$$

At target range cell $k_\tau^\star$, either combine channel powers or beamform toward $\theta_0$:

$$r_{\theta_0}[q] =\frac{1}{R}\boldsymbol a^H(\theta_0) \boldsymbol r[k_\tau^\star,q].$$

The short-time Fourier transform is

$$G[u,k_f] =\sum_{\ell=0}^{M_w-1} r_{\theta_0}[uM_H+\ell] w_\mathrm{md}[\ell] e^{-j2\pi k_f\ell/M_\mathrm{md}}.$$

$|G[u,k_f]|^2$ shows fine motion over time. Beamforming before the STFT can suppress other directions that occupy the same range cell.