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 where formula_2 are integers and formula_3 are represented by the complex numbers: The 2D Z-transform is a generalized version of the 2D Fourier transform. It converges for a much wider class of sequences, and is a helpful tool  in allowing one to draw conclusions on system characteristics such as BIBO stability. It is also used to determine the connection between the input and output of a linear Shift-invariant system, such as manipulating a difference equation to determine the system's Transfer function. Region of Convergence (ROC). The Region of Convergence is the set of points in complex space where: In the 1D case this is represented by an annulus, and the 2D representation of an annulus is known as the Reinhardt domain. From this one can conclude that only the magnitude and not the phase of a  point at formula_7 will determine whether or not it lies within the ROC. In order for a 2D Z-transform to fully define the system in which it means to describe, the associated ROC must also be know. Conclusions can be drawn on the Region of Convergence based on Region of Support (mathematics) of the original sequence formula_8. Finite Support Sequences. A sequence with a region of support that is bonded by an area formula_9 within the formula_8 plane can be represented in the z-domain as: Because the bounds on the summation are finite, as long as z1 and  z2 are finite, the 2D Z-transform will converge for all values of z1 and z2, except in some cases where z1 = 0 or z2 = 0 depending on formula_12. First Quadrant and Wedge Sequences. Sequences with a region of support in the first quadrant of the formula_8 plane have the following 2D Z-transform: From the transform if a point formula_15 lies within the ROC then any point with a magnitude also lie within the ROC. Due to these condition, the boundary of the ROC must have a negative slope or a slope of 0. This can be  assumed because if the slope was positive there would be points that meet the previous condition, but also lie outside the ROC. For example, the sequence: It is obvious that this only converges for So the boundary of the ROC is simply a line with a slope of -1 in the formula_21 plane. In the case of a wedge sequence where the region of support is less than that of a half plane. Suppose such a sequence has a region of support over the first quadrant and the region in the second quadrant where formula_22. If formula_23 is defined as  formula_24 the new 2D Z-Transform becomes: This converges if: These conditions can then be used to determine constraints on the slope of the boundary of the ROC in a similar manner to that of a first quadrant sequence. By doing this one gets: formula_27 and formula_28 Sequences with Region of Support in all Quadrants. A sequence with an unbounded Region of Support can have an ROC in any shape, and must be determined based on the sequence formula_8. A few examples are listed below: will converge for all formula_3. While: will not converge for any value of formula_3. However , These are the extreme cases, and usually, the Z-transform will converge over a finite area. A sequence with support over the entire formula_2 can be written as a sum of each quadrant sequence: Now Suppose: formula_36 and formula_37 also have similar definitions over their respective quadrants. Then the Region of convergence is simply the intersection between the four 2D Z-transforms in each quadrant. Using the 2D Z-transform to solve difference equations. A 2D difference equation relates the input to the output of a Linear Shift-Invariant (LSI) System in the following manner: formula_38 Due to the finite  limits of computation, it can be assumed that both a and b are sequences of finite extent. After using the z transform, the equation becomes: formula_39 This gives: formula_40 Thus we have defined the relation between the input and output of the LSI system. Using the 2D Z-Transform to Determine Stability. Shanks' Theorem I. For a first quadrant recursive filter in which formula_41. The filter is stable iff: formula_42 for all points formula_7 such that formula_44 or formula_45. Shanks Theorem II. For a first quadrant recursive filter in which formula_41. The filter is stable iff: formula_47 formula_48 Huang's  Theorem. For a first quadrant recursive filter in which formula_41. The filter is stable iff: formula_47 formula_51 for any formula_52 such that formula_53 Decarlo and Strintzis' Theorem. For a first quadrant recursive filter in which formula_41. The filter is stable iff: formula_55 formula_51 for any formula_52 such that formula_58 formula_59 for any formula_60 such that formula_61 Solving 2D Z-Transforms. Approach 1: Finite Sequences. For finite sequences, the 2D Z-transform is simply the sum of magnitude of each point multiplied by formula_3 raised to the inverse power of the location of the corresponding point. For example, the sequence: formula_63  has the Z-transform: formula_64 As this is a finite sequence the ROC is for all formula_3. Approach 2: Sequences with values along only formula_66 or formula_67. For a sequence with a region of support on only formula_68 or formula_69, the sequence can be treated as a 1D signal and the 1D Z-transform can be used to solve for the 2D Z-transform. For example, the sequence: formula_70 Is clearly given by formula_71. Therefore, its Z-transform is given by: formula_72 formula_73 As this is a finite sequence the ROC is for all formula_3. Approach 3: Separable Sequences. A  separable sequence is defined as formula_75 For a separable sequence finding the 2D Z-transform is as simple as separating the sequence, taking the product of the 1D Z-transform of each signal formula_76 and formula_77. For example, the sequence: formula_78 Therefore, its Z-transform is given by formula_79 The ROC is given by: formula_80 ; formula_81