Lower bounds for parametric estimation with constraints

Model-based MR parameter mapping with sparsity constraint

“…In this subsection, we derive the sparsity-constrained Cramer-Rao lower bound (CRLB) [15,16] on θ , which provides useful insights into the benefits of incorporating the sparsity constraint into the parameter estimation problem. Furthermore, it can also be used as a benchmark to evaluate our solution algorithm.…”

“…Excluding the sparsity constraint, we can calculate the unconstrained CRLB as follows: $$\text{Cov}false(\mathrm{bold\theta ̂}\text{)}\ge {\mathbf{W}}^H\mathrm{boldZ}\mathrm{boldW},$$ where $\mathrm{boldZ}=false({\mathbf{J}}_{\mathrm{boldc},\mathrm{boldc}}^{†}+{\mathbf{J}}_{\mathrm{boldc},\mathrm{boldc}}^{†}{\mathbf{J}}_{\mathrm{boldc},\mathrm{bold\rho }}{\mathbf{Q}}^{†}{\mathbf{J}}_{\mathrm{bold\rho },\mathrm{boldc}}{\mathbf{J}}_{\mathrm{boldc},\mathrm{boldc}}^{†}false)$ and $\mathrm{boldQ}={\mathbf{J}}_{\mathrm{bold\rho },\mathrm{bold\rho }}-{\mathbf{J}}_{\mathrm{bold\rho },\mathrm{boldc}}{\mathbf{J}}_{\mathrm{boldc},\mathrm{boldc}}^{†}{\mathbf{J}}_{\mathrm{boldc},\mathrm{bold\rho }}.$ It has been shown in [15, 16] that W H ZW ≥ W H Z s W . Therefore, incorporating sparsity constraint can be theoretically demonstrated to be beneficial.…”

Joint optimization of MIMO radar waveform and biased estimator with prior information in the presence of clutter

“…∂x is assumed to have full row rank with g(x) being the equality constraint set on x and U is the tangent hyperplane of g(x) [20].…”

“…It is noted that the CRB provides a lower bound on the variance when any unbiased estimator is used without employing any prior information. In fact, some prior information may be available in many array signal processing fields (see, e.g., [20][21][22]), which can be regarded as a constraint on the estimated parameter space. A variant of the CRB for this kind of the constrained estimation problem was developed in [20,22], which is called the constrained CRB.…”

“…In fact, some prior information may be available in many array signal processing fields (see, e.g., [20][21][22]), which can be regarded as a constraint on the estimated parameter space. A variant of the CRB for this kind of the constrained estimation problem was developed in [20,22], which is called the constrained CRB. Moreover, a biased estimator can lower the resulting variance obtained by any unbiased estimator generally [23][24][25][26][27][28].…”