2D Z-transform
The 2D Z-transform, similar to the Z-transform, is used in multidimensional signal processing to relate a two-dimensional discrete-time signal to the complex frequency domain in which the 2D surface in 4D space that the Fourier transform lies on is known as the unit surface or unit bicircle. The 2D Z-transform is defined by X z ( z 1 , z 2 ) = ∑ n 1 = 0 ∞ ∑ n 2 = 0 ∞ x ( n 1 , n 2 ) z 1 − n 1 z 2 − n 2 {\displaystyle X_{z}(z_{1},z_{2})=\sum _{n_{1}=0}^{\infty }\sum _{n_{2}=0}^{\infty }x(n_{1},n_{2})z_{1}^{-n_{1}}z_{2}^{-n_{2}}} where n 1 , n 2 {\displaystyle n_{1},n_{2}} are integers and z 1 , z 2 {\displaystyle z_{1},z_{2}} are represented by the complex numbers: z 1 = A e j ϕ 1 = A ( cos ϕ 1 + j sin ϕ 1 ) {\displaystyle z_{1}=Ae^{j\phi _{1}}=A(\cos {\phi _{1}}+j\sin {\phi _{1}})\,} z 2 = B e j ϕ 2 = B ( cos ϕ 2 + j sin ϕ 2 ) {\displaystyle z_{2}=Be^{j\phi _{2}}=B(\cos {\phi _{2}}+j\sin {\phi _{2}})\,} The 2D Z-transform is a generalized version of the 2D Fourier transform.