Two-Dimensional Frequency Domain

Lecture and notes by Prof. Brian L. Evans (UT Austin)

Scribe: Er-Hsien Fu (UT Austin)

Based on notes by Prof. Russell M. Mersereau (Georgia Tech)

Stability:

A LSI system is stable if and only if its impulse response h(n1, n2) satisfies

 

Support:

A sequence x(n1,n2) has support on R if

A system has support on R if its impulse response has support on R

 

Frequency response of 2-D systems

A 2-D System h with input x and output y. The input is a 2-D complex exponential exp(j n1 w1 + j n2 w2).

Periodic in both w 1 and w 2 with period 2p

 

Example:

 Consider the following impulse response whose center value occurs at the 2-D origin: 

    n2
     ^
     |
     |

1/8 1/4 1/8

1/4 1/2 1/4  --> n1 

1/8 1/4 1/8

Symmetric (four-fold)

Separable: you can write the impulse response as h(n1,n2) = f(n1) g(n2) where

 

Inverting the 2-D Frequency Response

The frequency response of a system can be calculated from the system's impulse response.

 

- The opposite is also true

- Follows from 2-D Fourier series expansion

2-D Fourier Transform

Example:

Example:

The impulse response is plotted below.

Properties of the Fourier Transform

Substitute w(n1,n2) = x(n1,n2) for a perhaps more familiar form of the relation.

 

The result is a little surprising since the product of two functions in the discrete-time domain is a convolution in the frequency domain.

 

Why not a convolution?