Signal Processing.

The Nyquist–Shannon sampling theorem sets the minimum rate at which a continuous signal must be sampled to be perfectly reconstructed — sample too slowly and distinct frequencies become indistinguishable (aliasing).

A critical-density Gabor Riesz basis cannot have a window well localized in both time and frequency.

Engineering far-field half-power beamwidth approximation for a uniformly illuminated continuous linear aperture.

Time and angular-frequency standard deviations of a finite-energy signal obey the Fourier uncertainty principle.

Engineering high-resolution SQNR approximation for a full-scale sinusoid quantized by an ideal N-bit uniform quantizer.

Gaussian compressed sensing with basis pursuit has a sharp asymptotic sparsity–undersampling transition.

A subgaussian measurement matrix satisfies k-order restricted isometry with high probability at logarithmic oversampling.

Every k-sparse solution of Ax=y is unique exactly when the measurement matrix spark exceeds 2k.

Covariance of an unbiased estimator is bounded below by inverse Fisher information.

Van Trees lower bound combines expected data Fisher information and prior information.

Estimation MSE is bounded through binary hypothesis-testing error probabilities.

N Euclidean points admit a low-dimensional embedding preserving all pairwise distances within 1±epsilon.

A bank of generalized sampling channels must collectively provide Nyquist density to reconstruct an arbitrary B-bandlimited signal.

M unit vectors in an N-dimensional complex space cannot all have smaller pairwise coherence.

Stable sampling density cannot be below occupied spectral measure when frequency is measured in hertz.

Approximately 2BT eigenmodes are well concentrated in both a time interval of duration T and two-sided baseband bandwidth B.

Maximum in-interval energy fraction of a bandlimited signal is the largest prolate-spheroidal concentration eigenvalue.

Using cyclic frequency, Gaussian windows uniquely attain the minimum uncertainty product up to symmetries.

Maximum zero-inter-symbol-interference symbol rate of an ideal real baseband channel of bandwidth B.

Conventional Rayleigh angular/separation criterion for two incoherent point sources through an ideal circular aperture.

Conventional coherent periodic-structure resolution criterion requiring capture of diffraction orders.