Difference between revisions of "CourseNotes2 (BoutinSpring2009)" - Rhea

Latest revision as of 05:43, 16 September 2013

Course Notes, January 16, 2009

ECE438, Spring 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 dt (discrete 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) $