Linear System Learning Objectives

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

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

3. 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)

4. 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

5. 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

6. 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