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.

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) $