ECE 438 Spring 2009 mboutin Course Notes - Rhea

ECE438 Course Notes January 14, 2009

1)Definitions

ECE438 is about digital signals and systems

2) Digital Signal = a signal that can be represented by a sequence of 0's and 1's.

so the signal must be DT X(t) = t, i.e. need x(n), n belongs to Z

Signal values must be discrete

- $x(n) \in {0,1}$ <-- binary valued signal
$x(n) \in {0,1,2,...,255}$ <-- gray scale valued signal

Another example of digital signal

-the pixels in a bitmap image (grayscale) can have a value of 0,1,2,...,255 for each individual pixel. --If you concatenate all the rows of the image you can convert it to a 1 dimensional signal. i.e. $$ x = (row1,row2,row3) $$

2D Digital signal = signal that can be represented by an array of 0's and 1's

example: 128x128 gray scale image
$p_{ij} \in {0,...,255}$

matrix $A_{ij} = p_{ij}$ of size 128x128

Digital signals play an important roll in forensics applications such as: watermarking, image identification, and forgery detection among many others. Go to PSAPF and VIP's Sensor Forensics to find out more information about these applications.

Digital Systems = system that can process a ditital signal.
E.g.

Software (MATLAB,C, ...)
Firmware
Digital Hardware

Advantages of Digital Systems

precise,reproducable
easier to store data
easier to build:
- just need to represent 2 states instead of a continuous range of values

Software based digital systems

easier to build
cheap to build
adaptable
easy to fix/upgrade

Hardware-based digital systems

fast.

Continuous time world

most natural signals live here
things are easy to write, understand, conceptualize

Digital World

digital media signals live here along with computers, MATLAB, digital circuits

These world are brought together using sampling & quantization, as well as reconstruction

Signal Characteristics

Deterministic vs. random
- x(t) well defined , s.a. $x(t) = e^{j\pi t}$
- x(n) well defined , s.a. $x(n) = j^{n}$
  ex: Lena's image
Random
- x(t) drawn according to some distribution
- example: x(t) white noise
  x = rand(10) (almost) random

Periodic vs. non-periodic
- if $\exists$ positive T such that x(t+T) = x(t), $\forall t$ then we say that x(t) is periodic with period T

ECE438 Course Notes January 16, 2009

Todays Goals

Signal Characteristics
Signal Transformations
Special Signals
Singularity Functions

right sided signal:
$\exists t_{min} (n_{min})$ such that $$ x(t) = 0 $$ when $t < t_{min}$

left sided signal:
$\exists t_{max} (n_{max})$ such that $$ x(t) = 0 $$ when $t > t_{max}$
if $t_{max} \leq 0$ we say the signal is anticausal

two sided (mixed causal):
neither left sided nor right sided

Finite Duration Signal:
both right and left sided, $\exists t_{min},t_{max}$ such that $$ x(t) = 0 $$ for $t > t_{max}$ and $t < t_{min}$

Signal Metrics

Signal Energy
- $E_x = \int_{-\infty}^{\infty} |x(t)|^2\,dt$ for ct (continuous time)
  
  $E_x = \sum_{n=-\infty}^{\infty} |x(n)|^2$ for dt (discrete time)
Signal Power
- $P_x = \lim_{T \Rightarrow \infty}\frac{1}{2T}\int_{-T}^{T} |x(t)|^2\,dt$ for ct (continuous time)
  
  $P_x = \lim_{N \Rightarrow \infty}\sum_{n=-N}^{N} |x(n)|^2$ for ct (continuous time)
  
  note: for periodic signals
  $P_x = \frac{1}{N}\sum_{n=0}^{N-1}|x(n)|^2$
Signal RMS (root-mean-square)
- $X_{rms} = \sqrt{P_x}$
Signal Magnitude
- $$ m_x = max|x(t)| $$ , for CT
  
  $$ m_x = max|x(n)| $$ , for DT
  
  if $m_x < \infty$ , we say signal is bounded
Scaling ( $y(t) = x(\frac{t}{a})$ )
- note: y(0) = x(0), fixed point at t=0
  if a > 1, graph will narrow
  if a < 1, graph will expand
  
  if a>1 will not work for digital signals
  
  Down Sampler:
  $$ y(n) = x(Dn) $$ , D = integer > 1
  $x(n) \Rightarrow D\Downarrow \Rightarrow y(n)$
  
  Up Sampler: $x(n) \Rightarrow D\Uparrow \Rightarrow y(n)$
  $y(n) = x(\frac{n}{D})$ , if n/D is an integer
  
  Scaling and Shifting $y(t) = x(\frac{t}{a}-t_0)$
  note: $$ y(0) = x(-t_0) $$

ECE438 Course Notes January 21, 2009

Delta Functions

Continuous-time: (a.k.a. Dirac delta function)
$\delta(t) = \lim_{\triangle \Rightarrow 0} \frac{1}{\triangle}rect(\frac{t}{\triangle})$
Properties

$\int_{-\infty}^{\infty} \delta(t)\,dt = 1$ (unit area)
$\int_{-\infty}^{\infty} x(t) \delta(t-t_0)\,dt = x(t_0)$ (sifting property)

Discrete-time: (a.k.a. Kronecher delta fn.)
$\delta[n] = 1 | n=0, 0 | 0 > n < 0$

Sifting Property: $\sum_{n=-\infty}^{\infty} x[n] \delta[n-n_0] = x[n_0]$

Comb & Rep operators (for CT signals)

Comb operator multiplies a signal by an "impulse train".

$\sum_{k=-\infty}^{\infty} \delta(t-kT)$
$Comb_T{{x(t)}} = x(t)\sum_{k=-\infty}^{\infty} \delta(t-kT) = \sum_{k=-\infty}^{\infty} x(kT)\delta(t-kT)$

Rep operator simply replicates a signal every T units:
$rep_T{{x(t)}} = \sum_{k=-\infty}^{\infty} x(t-kT)$

Systems

A system maps an input signal x(t) to a unique output signal, y(t). $x(t) \Rightarrow \mbox {System} \Rightarrow y(t)$
$$ y(t) = S[x(t)] $$

Examples:
$y(n) = \frac{1}{3}[x(n) + x(n-1) + x(n-2)]$ (moving averaging function, seen in Lab2)

System Properties:

Linearity
- Definition: A system S is linear if for any two input signals $$ x_1(t) $$ and $$ x_2(t) $$ , and any (complex) constant, a, it satisfies the following two properties:</li?
- Superposition: $$ S[x_1(t) + x_2(t)] = S[x_1(t)] + S[x_2(t)] $$
- Homogeneity: $$ S[ax_1(t)] = aS[x_1(t)] $$
Time-Invariance
- Definition: A system S is time-invariant(TI) if delaying the input signal results only in an identical delayin the output signal.
- If $$ y_1(t) = S[x_1(t)] $$ and $$ y_2(t) = S[x_1(t-t_0)] $$ than $$ y_2(t) = y_1(t-t_0) $$

ECE438 Course Notes January 23, 2009

Side Note: Issues with Rhea

Issue #1: Scaling horizontally of entire page causes top navigation to shrink too far and become ridiculous...
Solution: CSS Styling min-width: 800px; or something around that on the div containing the black navigation header will resolve this issue;

Issue #2: Drop Down menus do not work on Internet Explorer 6
Solution: Javascript plugins can be used to create drop down menus on an If IE6 check

Issue #3: Creating Completely new pages isnt clear on how to do
Solution: Be more clear and have more informative navigation
The features of this site are completely within the bounds of IE6 with a little work...Far more difficult has been accomplished in ie6

Professor Mimi's Definitions of System Properties
Use of diagrams makes this difficult to add in, please see previous days lecture notes for the properties of Linearity and Time-Invariance.

Causality
- if for any $t_0\,,x_1(t) = x_2(t), \forall t \leq t_0$
  $\Rightarrow y_1(t)=y_2(t), \forall t \leq t_0$
  output at t0 depends on input at $t \leq t_0$ only.
BIBO Stable (Bounded-Input-Bouned-Output Stability)
- if any bounded input yields a bounded output.
  i.e. if x(t) / x(n) st $M_x < \infty$
  $\Rightarrow y(t)/ y(n)$ s.t. $M_y < \infty$
- To show stability mathematically use triangle inequality. We assume x(n) is bounded and try to show y(n) is then also bounded