# Test 2 Review

## General Information

Time and date. Test 2 will be given on Wednesday, 3/23/16, at 10:20, in our usual classroom.

Bluebooks. Please bring an 8½×11 bluebook.

Office hours. I will have office hours on Monday (3/21/16), 1:40-4, and on Tuesday (3/22/15), 1-3:30 pm.

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 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), 2.3, 2.4, 3.1.1-3.1.3. The problems will be similar to ones done for homework, and as examples in class and in the text. A short table of Fourier transform properties and integrals will be provided. Here are links to practice tests: 2002 and 2009. 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 compute Fourier transforms and inverse Fourier transforms. Be able to use the properties in Theorem 2.6 to do this. §2.1-2.2.1
2. Be able to establish the simple properties listed in Theorem 2.6 of the text, and know how to use them. (You will be given a table listing these properties plus a few others, so you do not need to memorize them.) §2.2.1
• Convolutions. Be able to find the convolution of two functions and to use it to find Fourier transforms. §2.2.2
• Plancheral's Theorem. Be able to state and use Plancherel's Theorem. (See assignment 5, problem 3.)
• Filters.
1. LTI filter. Be able to define the term linear, time-invariant filter. 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.
2. Examples. Butterworth filter, "running average filter" (see problem 2.12).
3. Causal filter. Be able to define the term causal filter, and be able to determine whether an LTI filter is causal. §2.3.
• The Sampling Theorem. Be able to state and prove this theorem, and to define these terms: band-limited function, Nyquist frequency, and Nyquist rate §2.4.

### Discrete Fourier Transforms

• Definition & properties.
1. DFT. Be able state the formulas for both the DFT and inverse DFT. Be able to prove Theorem 3.3, which gives the inversion formula for the DFT.
2. Approximation fo FS coefficients. Know the connection between coefficients in a Fourier series and the DFT.
3. $\mathcal S_n$. This is the space of $n$ periodic sequences. Be able to show that the DFT and inverse DFT of an $n$ periodic sequence is an $n$ periodic sequence — i.e, that the DFT and inverse DFT transform $\mathcal S_n$ into itself.
4. FFT. Know what the fast Fourier transform (FFT) algorithm for computing the DFT is. Be able to compare the number of multiplications needed to compute the DFT vs. the number for the FFT.

Updated 3/19/2016.