Digital Signal Processing (DSP) with Python Programming

5Hints and Solutions

5.1. Useful Maths

5.1.1.– (Module and phase joint law of a 2D Gaussian r.v.) (see p. 20) As this substitution is bijective and its Jacobian is equal to r, which is positive, the joint distribution of the pair (R, Θ) has a density of:

Following property [1.3] we derive from p_RΘ(r, θ) = g(r)h(θ), where g(r) = and h(θ) = (2π)⁻¹ (θ ∈ (0, 2π)), that the random variables R and Θ are independent. Therefore, Θ is uniform on (0, 2π) and R has a Rayleigh distribution.

5.1.2.– (δ-method) (see p. 21) We shall use the hypothesis cov (X₀, X₁) = σ²I₂. The Jacobian, here noted ∂g, of g : (X₀, X₀) → (R, θ) is deduced from the Jacobian ∂h of h : (R, θ) → (X₀, X₁) following:

We have:

and, from [1.49]:

We deduce:

[5.1]

This result is therefore correct on the condition that ...