# Test 2 Review

## General Information

Time and date. Test 2 will be given on Wednesday, 3/21/18, at 10:20, in our usual classroom.

Bluebooks. Please bring an 8½×11 bluebook.

Office hours. I will have extra office hours -- in addition to my usual ones -- on Monday (3/19/18), 2:30-4, and on Tuesday (3/20/18), 2-3:30..

Calculators. You may use calculators to do arithmetic, although you will not need them. You may not use any calculator that has the capability of doing linear algebra, or calculus, or of storing programs or other material.

Other devices. You may not use cell phones, computers, or any other device capable of storing, sending, or receiving information.

Structure and coverage. There will be 4 to 6 questions, some with multiple parts. The test will cover sections 2.1, 2.2 (except 2.2.3), and 2.3. The problems will be similar to ones done for homework assignments 4-6 homework and as examples in class and in the text. You will be required to derive properties of the FT and/or prove various results. (See below for a list.) A small table of some Fourier transform properties and integrals will be provided. Here are links to practice tests: 2009 and 2010. Be aware that these tests cover some material that will not be on the test for this class.

## Topics Covered

### Fourier Transforms

• Computing Fourier transforms & properties
1. Be able to find Fourier transforms and inverse Fourier transforms, either by directly doing the integrals involved and/or by using properties from Theorem 2.6. §2.1-2.2.1
2. Be able to derive any of the properties listed in Theorem 2.6 of the text, and know how to use them. You will be given a small table listing some of these properties. §2.2.1
3. Know and be able to use the "quartet table" below to find Fourier transforms and inverse Fourier transforms.

 Function Fourier Transform $f(t)$ $\hat f(\lambda)$ $\hat f(t)$ $f(-\lambda)$

• Convolutions
1. Directly finding convolutions. Be able to find the convolution of two functions directly from Definition 2.9. §2.2.2
2. Convolution Theorem. Be able to state and prove this theorem. Be able to use it to find Fourier transforms of convolutions and inverse Fourier transforms of products of functions.
3. Plancheral's (or Parseval's) Theorem. Be able to state Plancherel's Theorem and to use it to find integrals, as in assignment 6, problem 2.

• Filters
1. LTI filter. Be able to define the term linear, time-invariant filter. (Remember that a definition is not a lemma, proposition or theorem!) Know what its connection to the convolution is, and what impulse response functions and frequency response (system) functions are. Given one of them, be able to find the other. Below are examples of filters that we have used.
1. Butterworth filter.
2. Running average filter. (See assignment 4, problem 2).
3. The filter in problem 6, assignment 6.
4. The "ideal" filter given in equation (2.21).
2. Finding outputs. Given an impulse response function, be able use the convolution form $L[f]=h*f$ to find the output of the filter for an input function. In addition, be able to use the convolution theorem to find the FT of the output of a filter.
3. Causal filter. Be able to define the term causal filter, and be able to prove this: If the impulse response function $h(t)=0$ for $t< 0$, the the filter is causal. Know that the converse is true: if a filter is causal, then $h(t)=0$ for $t<0$. Be able to use these to determine whether an LTI filter is causal. §2.3.

Updated 3/12/2018.