1.Be able to distinguish linear and non-linear systems. 2.Be able to distinguish space-invariant from space-varying systems. 3.Describe and evaluate convolution as a graphical operation. 4.Describe and sketch the output of a convolution operation when input functions are simple. 5.Describe a linear system with a modulation transfer function (MTF) (H(f), H(w), or H(u,v)). What information does the MTF provide? Linear System Learning Objectives

1.Be able to distinguish linear and non-linear systems. 2.Be able to distinguish space-invariant from space-varying systems. 3.Describe and evaluate convolution as a graphical operation. 4.Describe and sketch the output of a convolution operation when input functions are simple. 5.Describe a linear system with a modulation transfer function (MTF) (H(f), H(w), or H(u,v)). What information does the MTF provide? Linear System Learning Objectives

Continuous Fourier Theory Forward and inverse transform equation What are the basis functions of a Fourier transform? Basic Fourier transform pairs –rect, gaussian, delta function, cosine, sine, triangle, comb function Fourier Theorems –Linearity, scaling, shift, convolution, integration, derivative, zero moment Duality Relationship between even/odd functions and real/imaginary channels Hankel Transform ( equation, scaling)

2D Continuous Fourier Theory Forward and inverse transform equation What are the basis functions of 2D Fourier transform? –What do they look like Basic Fourier transform pairs –2D rect, gaussian, delta function, cosine, sine, triangle, comb function, bed of nails, circle Fourier Theorems –Linearity, scaling, shift, convolution, integration, derivative, zero moment, separable functions Duality Relationship between even/odd functions and real/imaginary channels View 2D transform as 2 1D transforms

Sampling How do we mathematically represent sampling? Sampling and replication duality Sketch replication in the opposite domain for one and two dimensions Restoration using a sinc interpolation filter Units of 2D sampling – cycles/ mm

Discrete Fourier Transform Relationship between number of points sampled, sampling rate, and length of sampling (N= 2BL) in terms of the frequency domain. Understand what the DFT calculates Be able to execute and interpret 1D and 2D FFTs using Matlab –Match and verify your conceptual understanding with Matlab