Give an analytical approximation for the frequency response

Question

Questions -

Q1. Compute the two-dimensional continuous-space Fourier transform of the following signal: The two-dimensional zero-one function p_A(x, y) where A is an elliptical region, with semi-minor axis X and semi-major axis 2X, oriented at 45^o as shown in the figure

Q2. A square image (pw = ph) is sampled on the hexagonal lattice Λ generated by the sampling matrix. Where X =1/512 ph. Design a Gaussian FIR filter with unit sample response h[x] = cexp(||x||²/2r²) for x ∈ B having a 3dB bandwidth of 0.2/x c/ph. The region of support of the FIR filter is B = {x ∈ Λ| ||x|| ≤ 3X}. Having determined the correct values of c and r, give the coefficient of the filter. Give an analytical approximation for the frequency response of the filter. Make a contour plot and a perspective plot of the frequency response of the filter over frequency range -2/X ≤ u ≤ 2/X, -4/√(3X) ≤ v ≤ 4/√(3X).

Sketch, by hand (or other means) on the contour plot, the points of the reciprocal lattice Λ* and a Voronoi unit cell of Λ*, and comment on the periodicity of the frequency response. Recall that the Voronoi unit cell consists of all points in Λ* closer to the origin than to any other point of Λ*.

Q3. Consider an ideal discrete-space circularly symmetric low pass filter defined on a rectangular lattice with horizontal and vertical sample spacing X and Y. The pass band is C_w = {(u, v) | v² + v² ≤ W²} and the unit cell of the reciprocal lattice is P* = {(u, v) | -(1/2X) ≤ u < 1/2X, -(1/2Y) ≤ v < 1/2Y}. Assume that W < min (1/2X, 1/2Y) so that C_w ⊂ P*

Where of course H(u + k/X, v + l/Y) = H(u, v) for all integers k, l. Show that the unit sample response of this filter is given by

h[mX, nY] = (WXY/√(X²m²+Y²n²))J₁(2πW√(X²m²+Y²n²))

Where J₁(s) is the Bessel function of the first kind and first order. You may use the following identities:

J₀(s) = 1/2π ₀∫^2πexp[js cos(θ+φ)]dθ for any φ

₀∫^xsJ₀(s)ds = sJ₁(s)

Alternatively, you may use Table 2.2, but carefully justify your answer. Simplify the expression in the Case X = Y.

Q4. For each of the following pairs of lattices Λ₁ and Λ₂, state whether Λ₁ ⊂ Λ₂, Λ₂ ⊂ Λ₁ or neither. If neither, find (by inspection) the least dense lattice Λ₃ such that Λ₁ ⊂ Λ₃ and Λ₂ ⊂ Λ₃. For each lattice Λ₁, Λ₂ and Λ₃ (if required), determine and sketch the reciprocal lattice and a unit cell of the reciprocal lattice. Also, determine a system to convert a signal f(x) defined on Λ₁ to a signal g(x) defined on Λ₂. Assume that ideal low-pass filters are used where filters are required; give an expression for their frequency response and sketch their pass-band and gain in the frequency domain.