Question

Periodicity (k₁ and k₂ are integers)

F(u, v) = F(u + k₁M, v) = F(u,v + k₂N)
= F(u + k₁M, v + k₂N)
f(x, y) = f(x + k₁M, y) = f(x, y + k₂N)
= f(x + k₁M, y + k₂N)
Where F(u,v) and f(x,y) are given as :
Discrete Fourier transform (DFT) of f(x,y)
F(u,v) = ∑ᴹ⁻¹ₓ₌₀ ∑ᴺ⁻¹ᵧ₌₀ f(x,y)e⁻ʲ²π⁽ᵘˣ/ᴹ⁺ᵛʸ/ᴺ⁾

Answer 1

The given equations demonstrate the periodicity property of the Discrete Fourier Transform (DFT), showcasing that shifting the input signal in the spatial domain by integer multiples of periods M and N produces the same transformed output. The equations illustrate that altering the variables u and v by multiples of k₁M and k₂N respectively does not change the DFT's value, affirming the periodic nature of the DFT.

Step-by-step explanation:

The equations presented highlight the periodicity property inherent in the Discrete Fourier Transform (DFT). The DFT represents a signal in the frequency domain obtained by transforming a discrete signal from the spatial domain. The property showcased in these equations emphasizes that shifting the spatial domain input signal by integer multiples of the period M in the x-direction and N in the y-direction, denoted by k₁M and k₂N respectively, results in an unchanged DFT.

The periodicity property is vital in signal processing as it underlines the repetitive nature of signals. The property showcased in the given equations indicates that the DFT remains unaltered despite shifting the input signal spatially by integer multiples of the periods M and N. This property aids in understanding the behavior of signals under translation or spatial shifts, enabling efficient signal analysis and processing.

Mathematically, the equations F(u,v) = F(u + k₁M, v) = F(u,v + k₂N) = F(u + k₁M, v + k₂N) demonstrate that the value of the DFT at different spatial positions related by integer multiples of the periods M and N remains the same. This consistency in the transformed output despite spatial shifts helps in understanding and manipulating signals in various applications like image processing, telecommunications, and data analysis.

Answer 2

Suppose f is a differentiable function of x and y, and g(u, v) = f(eu + sin(v), eu + cos(v)). Use the table of values to calculate gu(0, 0) and gv(0, 0).

f g fx fy

(0, 0) 9 -3 1 8

(1, 2) 9 9 4 3

gu(0, 0) =

gv(0, 0) =