Question

I need the Cramer Rao Bound for frequency estimation of two-dimensional two-component complex sinusoidal signal whose components are closely spaced at each dimension.

The signal is given as:

A 2-D multiple sinusoid observed in the presence of additive white Gaussian noise (AWGN) can be represented by
Sₗ(n₁,n₂)=L
ΣAₗeʲ²π⁽ ᶠ¹,ˡⁿ¹ᶠˢ¹⁺ᶠ²,ˡⁿ²/ᶠˢ²⁾+ω(n₁,n₂)
l=1
where n₁ 0≤n₁≤N −1 and 0≤n₂ ≤N₂ −1. Here, Nₖ represents the signal length for dimension k, where k can be 1 or 2. fₖ,ₗ represents the frequencies of the lᵗʰ individual sinusoidal component in dimension k, where l=1,2,⋯,L, and Fₛₖ is the sampling frequency for dimension
k.
Aₗ is the constant complex amplitude of the lᵗʰ individual sinusoidal component, and the phases of complex amplitudes are uniformly distributed over (0,2π]. The additive noise ω(n₁,n₂)is assumed to be circularly symmetric AWGN with zero mean and a variance of σ².Given this signal model, you can proceed to derive the Cramer-Rao Bound for the estimation of the frequencies f₁.ₗ and f₂.ₗ in both dimensions. This bound will provide insight into the minimum achievable variance of frequency estimation for the given signal model and noise characteristics.

Answer 1

The Cramer-Rao Bound is a mathematical inequality that represents the minimum achievable variance of frequency estimation for a given signal model and noise characteristics. In the case of a 2D two-component complex sinusoidal signal with closely spaced components, we can derive the Cramer-Rao Bound for frequency estimation.

Step-by-step explanation:

The Cramer-Rao Bound is a mathematical inequality that represents the minimum achievable variance of frequency estimation for a given signal model and noise characteristics. In the case of a 2D two-component complex sinusoidal signal with closely spaced components, we can derive the Cramer-Rao Bound for frequency estimation.

The Cramer-Rao Bound provides insight into the accuracy of frequency estimation and helps determine the theoretical lower limit of the variance in the estimated frequencies. It can be derived using the Fisher Information Matrix, which is a measure of the amount of information the data contains about the unknown parameters.

To derive the Cramer-Rao Bound for the estimation of the frequencies f₁ and f₂ in both dimensions, we need to calculate the Fisher Information Matrix and then compute its inverse to obtain the bound.