# Math 414-501 — Test 2 Review

## General Information

Test 2 will be given on Wednesday, 4/9/2014. Please bring an 8½×11 bluebook. Office hours: Tuesday, 10:30 - 3:00.

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 2.3, 2.4, 3.1.1-3.1.4, (also, 3.2.1; see below), 4.1-4.4, 5.1 in the text, and any material discussed in class, starting 2/26 to 4/4. The problems will be similar to ones done for homework, and as examples in class and in the text. In addition, you may be asked to define a term or state a theorem from those listed below. A short table of integrals and Fourier transform properties will be provided. Here are links to practice tests: 2002 and 2009

## Topics Covered

### Fourier Transforms

• Filters. Know what a linear, time-invariant filter is, what its connection to the convolution is, and what it's system response function and it's system function are. Given the system response function, be able to find the system function. Know what a causal filter is. Be able to filter a simple signal. §2.3.
• The Sampling Theorem. Be able to state and prove this theorem and to define these terms: band-limited function, Nyquist frequency, Nyquist rate. §2.4.

### Discrete Fourier Analysis

• Discrete Fourier transform
• Definition & properties. Be able to define the DFT, the inverse DFT. Know the connection between coefficients in a Fourier series and the DFT approximation to them, as well as the DFT approximation to the Fourier transform of a function. Be able to define the convolution of two n-periodic sequences and to show that the result is also n-periodic. Be able to show that the DFT and inverse DFT take n-periodic sequences to n-periodic sequences. Be able to prove that any of the properties in Theorem 3.4, p. 137 hold. (Chapter 3, exercise 2.) Be able to describe the FFT algorithm and to explain why itâ€™s fast. §§3.1.1-3.1.3.
• FFT and Fourier transform. Know what the connection between the DFT and FFT is. §§3.1.3.

### Multiresolution Analysis

• Haar MRA
• Haar scaling function and approximation spaces. Know what the Haar scaling function is. Be able to define its corresponding approximation spaces Vj and know the nesting and scaling properties for the Haar these spaces, and that Theorem 4.6 gives bases for them. §§4.2.1-4.2.2
• Haar wavelet and wavelet spaces. (§4.2.4) Know the definition of the Haar wavelet and Haar wavelet spaces Wj, along with their properties. §4.2.3
• Decomposition and reconstruction.
• Discrete-time signals. Know what a discrete-time signal is and how discrete-time invariant filters are defined via the (discrete-time) convolution and impulse response. This was discussed in class on 3/31. It is also covered in §3.2.1.
• Algorithms. Be able to derive the decomposition algorithm and to state the reconstruction algorithm. Be able to carry these out in simple cases similar to the ones given for homework. §§4.3.1-4.3.2.
• Filter diagrams. Know the low pass and high pass impulse response filters used in decomposition and reconstruction. Know what up sampling and down sampling are. Be able to derive the decomposition diagram, including the low pass and high pass filters involved, as we did in class on Monday, 3/30. §4.3.3
• Implementation Know the steps in the implementing the wavelet algorithms: initialization (sampling), decomposition, processing, reconstruction. §4.4
• General MRA
• Mutliresolution Analysis. Be able to define the term multiresolution analysis.
• Shannon MRA Be able to discuss the details of the Shannon MRA, including the approximation spaces (the Vj's) and scaling function, φ. Be able to show that {φ(x-k), k ∈ Z} is an o.n. set.

Updated 4/6/2014.