Math 414-501 — Test 2 Review

General Information

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 2.1-2.4, 3.1.1-3.1.4, 3.2.1, 3.2.2 (pgs. 149-150), 4.1-4.3.1 in the text, and any material discussed in class, starting 2/23 to 4/13. 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

• Finding Fourier transforms Be able to find Fourier transforms; inverse Fourier transforms; convolutions; and integrals via Plancherel's Theorem. You may use any property of the Fourier transform to do the calculation. A brief table of integrals will be supplied. The problems will be similar to those done in class or for homework.
• Filters. Know what a linear, time-invariant filter is, what its connection to the convolution is, and what it's impulse response function and it's frequency response (system) function are. Given the impulse response function, be able to find the frequency response 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 it’s fast. §§3.1.1-3.1.3.
  • FFT and Fourier transform. Know what the connection between the DFT and FT is. §§3.1.3.
• Discrete-time signals Know what a discrete-time signal is. Be able to calculate Z-transforms in simple cases. Be able to state the discrete-time convolution theorem. §§3.2.1-3.2.2

Haar Wavelet Analysis

• Haar scaling function and approximation spaces. Know what the Haar scaling function, $\phi$, is and be able to derive its two-scale relation. Be able to define its corresponding approximation spaces $V_j$. Know the nesting and scaling properties for these spaces. Be able to use the $\{\phi(2^jx-k\}_{k=-\infty}^\infty$ basis for $V_j$. §§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 W_j, along with their properties. §4.2.3
• Decomposition and reconstruction. Be able to do simple decomposition and reconstruction problems similar to the ones done for homework.

Updated 4/12/2014.