CS 530 Geometric and Probabilistic Methods

Administrative Information

Instructor: Lance Williams williams@cs.unm.edu
Time: MWF 11:00 - 11:50 AM
Location: ME 210
Office Hours: Mon. 3:00-5:00, Thurs. 10:00-12:00
Office: FEC 349C
http://cs.unm.edu/~williams/cs530f10.html

TA office hours: T 9-11

Midterm 2

Below is a list summarizing what you should be familiar with for the 2nd midterm.
The Stanford EE lectures from Brad Osgood are great explanations for much of the material below.

Convolution (functional analysis)

<math>(f * g )(t)\ \ \,</math>	<math>\stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty} f(\tau)\, g(t - \tau)\, d\tau</math>
	<math>= \int_{-\infty}^{\infty} f(t-\tau)\, g(\tau)\, d\tau.</math> (commutativity)

Circulant Matrix

An <math>n\times n</math> matrix <math>\ C</math> circulant matrix takes the form

<math>

C= \begin{bmatrix} c_0 & c_{n-1} & \dots & c_{2} & c_{1} \\ c_{1} & c_0 & c_{n-1} & & c_{2} \\ \vdots & c_{1}& c_0 & \ddots & \vdots \\ c_{n-2} & & \ddots & \ddots & c_{n-1} \\ c_{n-1} & c_{n-2} & \dots & c_{1} & c_0 \\ \end{bmatrix}. </math>

Dirac Delta Function

• Integral Property
• Sifting Property

<math>\int\limits_{-\infty}^\infty f(t) \delta(t-\tau)\,dt = f(\tau)</math>

or

<math>\int\limits_{-\infty}^\infty f(\tau) \delta(t-\tau)\,d\tau = f(t)</math>

Impulse Response Function

• Completely characterizes a Linear Time-Invariant system
• The output of a system is just the convolution of the input and the Impulse Response Function (in the time domain point of view).

<math>y(t) = x(t) * h(t)\,</math>	<math>{}\quad \stackrel{\mathrm{def}}{=} \ \int_{-\infty}^{\infty} x(t-\tau)\cdot h(\tau) \, \operatorname{d}\tau</math>
	<math>{}\quad = \int_{-\infty}^{\infty} x(\tau)\cdot h(t-\tau) \,\operatorname{d}\tau,</math>

Harmonic Signal

<math>e^{2\pi i s t}</math>

• Complex function of a real variable, t.
  • Real part: <math> cos(2\pi s t) </math>
  • Imaginary part: <math> sin(2\pi s t) </math>

Fourier Series (periodic phenomena)

• Euler's Formula
  • <math>e^{ix} = \cos x + i\sin x \!</math>
  • note, <math>e^{-ix} = \cos x - i\sin x \!</math>
• Infinite Series
• Convergence
• Finding Fourier Coefficients

Fourier Transforms (non-periodic phenomena)

• Analysis/Synthesis
• The Gaussian
  • Gaussian is its own transform.

<math> e^{\pi t^2} \rightarrow e^{\pi s^2} </math>

• The Dirac Delta Function

<math> \delta (t - t_0) \rightarrow e^{-2\pi i s t_0} </math>

• The Harmonic Signal

<math> e^{-2\pi i s_0 t} \rightarrow \delta (s-s_0) </math>

• Sine

<math> sin(2\pi s_0 t) \leftrightarrow (i/2)[\delta(s + s_0) - \delta(s - s_0)] </math>

• Cosine

<math> cos(2\pi s_0 t) \leftrightarrow (1/2)[\delta(s + s_0) - \delta(s - s_0)] </math>

• Pulse or Rectangular Function

<math>\mathrm{rect}(t) = \sqcap(t) = \begin{cases}

0 & \mbox{if } |t| > \frac{1}{2} \\ \frac{1}{2} & \mbox{if } |t| = \frac{1}{2} \\ 1 & \mbox{if } |t| < \frac{1}{2}. \\ \end{cases}</math>
<math> \mathrm{rect}(t) \rightarrow \frac{sin(\pi s)}{\pi s} </math> known as the sinc function.

• the Shah Function

Fourier Transform Theorems

• Addition
• Shift
• Convolution
• Similarity, e.g. scaling in the time domain.
• Rayleigh's
• Differentiation
• Sampling

Misc. things to remember

• Gaussian

<math>f(x) = a e^{- { \frac{(x-b)^2 }{ 2 c^2} } }</math>

<math>\int_{-\infty}^\infty a e^{- { (x-b)^2 \over 2 c^2 } }\,dx=ac\cdot\sqrt{2\pi}.</math> This integral is 1 if and only if a = 1/(c√(2π)).