Chapter Summary

Chapter Summary

Key Points

• 1.

  MIMO radar transmits orthogonal waveforms from $N_t$ antennas, creating a virtual array of $N_v = N_tN_r$ elements from only $N_t + N_r$ physical antennas. The virtual array steering vector is the Kronecker product $\mathbf{a}_V(\theta) = \mathbf{a}(\theta) \otimes \hat{\mathbf{a}}(\theta)$, providing a quadratic aperture gain.

• 2.

  Co-located MIMO improves angular resolution via the virtual aperture; distributed MIMO provides spatial diversity with diversity order $N_tN_r$. The phased-MIMO continuum parametrised by $L$ independent waveforms trades coherent gain against waveform diversity --- the key design knob for ISAC systems.

• 3.

  Multi-view, multi-frequency geometry creates k-space samples $\boldsymbol{\kappa}_{ij}(f_k)$ that tile the spatial frequency plane. Different frequencies provide radial diversity (range resolution $\Delta R = c / 2W$); different view angles provide angular diversity (cross-range resolution $\Delta x_\perp = \lambda / 2\sin(\Theta/2)$). The density and extent of k-space coverage determine image quality.

• 4.

  Bistatic range equals the sum of Tx-target and Rx-target distances; isorange contours are ellipses with foci at Tx and Rx. Bistatic range resolution degrades as $1/\cos(\beta/2)$ with bistatic angle $\beta$.

• 5.

  The MIMO sensing matrix $\mathbf{A}$ has rows indexed by (Tx, Rx, frequency) triples. Under far-field, narrowband conditions it factors as a Khatri-Rao product $\mathbf{A} = \mathbf{A}_f \odot (\mathbf{A}_{\mathrm{tx}} \otimes \mathbf{A}_{\mathrm{rx}})$, enabling $O(M\log Q)$ algorithms. For distributed bistatic geometries, the Kronecker structure breaks down.

• 6.

  The condition number of a Kronecker sensing matrix is the product of the factors' condition numbers: $\kappa(\mathbf{A}) = \kappa(\mathbf{A}_f) \cdot \kappa(\mathbf{A}_{\mathrm{tx}}) \cdot \kappa(\mathbf{A}_{\mathrm{rx}})$. This motivates joint optimisation of frequency grids, Tx arrays, and Rx arrays.