page_title: 2D Z-transform
section_title: First Quadrant and Wedge Sequences
hierarchical_section_title: 2D Z-transform / Region of Convergence (ROC) / First Quadrant and Wedge Sequences
caption_attribution_description: English: Sequence with Region of support over a wedge and its corresponding ROC
context_page_description: 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 are integers and 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.
context_section_description: Sequences with a region of support in the first quadrant of the  plane have the following 2D Z-transform:
  From the transform if a point z₀₁,z₀₂ 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:
   has the z transform
  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 ln(z₀₁),ln(z₀₂) 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 n₀₁ = -Ln₀₂. If l is defined as l = ₀₁+Ln₀₂ 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:
   and
caption: Sequence with Region of support over a wedge and its corresponding ROC
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