Sampling Theorem Formula 1-D:

One of the most important applications of digital filtering is the processing of sequences of samples derived from continuous or analog signals. This is made possible due to the results and implications of the Sampling Theorem Formula.

This theorem can be stated as follows:

“A continuous analogue function x(t) which has a limited Fourier spectrum, that is a spectrum x(jω) such that x(jω) = 0 for ω > ω_m, is uniquely described from a knowledge of its values at uniformly spaced time instants T units apart, where T = 2π/ω_s, and ω_s ≥ 2ω_m“.

Sampling Theorem 2-D:

This theorem can be stated as follows.

A function of two variables, x(x₁, x₂) whose 2-D Fourier transform is equal to zero for ω₁ > ω_1m and ω₂ > ω_2m, is uniquely determined by the values taken at uniformly spaced points in the x₁ and x₂ plane, if the spacing X₁ and X₂ satisfy the conditions x₁ ≤ π/ω_1m and x₂ ≤ π/ω_2m.

1-D Z-Transform:

Given a sequence { x(n)} with -∞ ≤ n ≤ ∞, its Z-transform is defined as

where Z is a complex variable.

The Z-transform can be inverted and x(n) can be obtained as

where C is the counter clockwise closed contour in the region of convergence of X(Z) and encircling the origin of the Z-plane.