.. _a212_heatmap_capon: Capon Heatmap Processor ======================= Introduction ^^^^^^^^^^^^ The Capon heatmap processor — also known as the Minimum Variance Distortionless Response (MVDR) beamformer — replaces the Bartlett quadratic form with the reciprocal of :math:`\mathbf{a}^H R^{-1} \mathbf{a}`. The result is a much narrower main lobe than the conventional beamformer for the same array aperture, at the cost of inverting the channel covariance matrix at every range bin and being sensitive to mismatch between the assumed :term:`steering vector ` and the true channel response. This mismatch is small enough, in practice, for Capon to perform reliably as a DOA estimator. The small residual mainly affects amplitude estimates; see :doc:`Robust Capon `. Typical use case ^^^^^^^^^^^^^^^^ * High-resolution :term:`direction-of-arrival ` applications where two or more closely spaced targets must be resolved. * Applications with many targets whose side lobes could mask each other when using conventional beamforming. * When the application only demands :term:`DOA` estimates (no power estimate) and :term:`SNR` is good. Because A212 uses offline :term:`steering-vector ` calibration, the accuracy is not high enough to use Capon as an amplitude estimator. The accuracy of the calibration is, however, high enough to use it as a high-performance :term:`DOA` estimator. (See :doc:`robust Capon `.) * As the input to a :term:`CFAR` detector — Capon's narrow main lobes and low side lobes give clean peaks for adaptive thresholding. Pros and cons ^^^^^^^^^^^^^ **Pros** * Markedly higher angular resolution than conventional beamforming. * Lower side lobes and noise level; better at finding and separating targets. * Lower spectrum variance. * Adaptive: nulls on strong interference. **Cons** * Requires inverting :math:`R` at every range bin every frame — more expensive than conventional. * Sensitive to :term:`steering-vector ` mismatch: not suited for amplitude estimation. (See :doc:`Robust Capon ` for amplitude workaround.) * The number of samples used for one heatmap needs to exceed the number of channels used. This is needed to ensure that the covariance matrix is invertible. * Sensitive to low :term:`SNR`. Math and theory ^^^^^^^^^^^^^^^ Let :math:`R \in \mathbb{C}^{N_{ch} \times N_{ch}}` be the smoothed estimate of the channel covariance matrix at a given range bin and :math:`\mathbf{a}(\theta, \phi) \in \mathbb{C}^{N_{ch}}` the :term:`steering vector ` for direction :math:`(\theta, \phi)`. The Capon (MVDR) output is .. math:: P_{\text{capon}}(\theta, \phi) = \frac{1}{\mathbf{a}(\theta, \phi)^H R^{-1} \mathbf{a}(\theta, \phi)}. This corresponds to the optimization problem of minimizing the output power subject to a unit-gain constraint in the steering direction, which both sharpens peaks on real targets and suppresses energy arriving from other directions (interference). See P. Stoica and R. Moses, *Spectral Analysis of Signals* (Prentice Hall, 2005) for the full derivation. Example spectrum plots ^^^^^^^^^^^^^^^^^^^^^^ Compared to Conventional, Capon gives well-separated peaks for the three targets. The amplitude of the peaks does, however, correlate poorly with the actual amplitude of the signals. .. figure:: /images/processing/a212_heatmap_processing/figures/heatmap_capon.png :align: center Capon D–Az heatmap on a recording of three corner reflectors. The three colored markers and dashed lines mark the true reflector locations. The 1D plots show the distance and azimuth spectra cut at each of the three true peak locations. Recorded using :term:`Profile` 3. .. figure:: /images/processing/a212_heatmap_processing/figures/el_spectrum_capon.png :align: center Elevation spectrum of the Capon heatmap at each of the three true peak locations. Recorded using :term:`Profile` 3. Configuration ^^^^^^^^^^^^^ Shared parameters are documented in :doc:`common`. Processor-specific fields: :attr:`~acconeer.exptool.a2.algo.heatmap.capon.HeatmapConfig.covariance_matrix_time_constant_s` Time constant of the exponential smoothing applied to the per-range covariance matrices across frames. Capon is more sensitive to a noisy :math:`R` than the conventional processor, so this parameter typically deserves more care. See the :ref:`shared notes ` for more tuning guidance. :attr:`~acconeer.exptool.a2.algo.heatmap.capon.HeatmapConfig.high_pass_filter_time_constant_s` Time constant of the per-channel high-pass filter that runs on the IQ data to suppress static signals before the covariance is built. The Capon processor always runs this filter since it cannot handle perfectly correlated signals (static signals will always be perfectly correlated). See the :ref:`shared notes ` for more tuning guidance. Typical processing cost ^^^^^^^^^^^^^^^^^^^^^^^ Per frame, the Capon processor does everything the conventional processor does, plus a per-range matrix inverse (:math:`\mathcal{O}(N_d \cdot N_{ch}^3)`). Beyond that, the beamforming quadratic form is structurally the same as in conventional. On the configuration the RSS computation time test measures, 12 virtual channels and 25 azimuth angles, this corresponds to **~1.5 ×** the per-frame cost of Conventional. API reference ^^^^^^^^^^^^^ .. autoclass:: acconeer.exptool.a2.algo.heatmap.capon.HeatmapProcessor :members: :undoc-members: .. autoclass:: acconeer.exptool.a2.algo.heatmap.capon.HeatmapConfig :members: :undoc-members: