Difference between revisions of "ECE 438 Spring 2009 mboutin Symbols" - Rhea

Revision as of 13:50, 27 January 2009

So many symbols, so little time... Here's a quick lookup table for our commonly-used symbols!

• $ \omega_s $: Sampling frequency; equal to $ \frac{2\pi}{T} $
• $ \omega_m $: Maximum frequency in a band-limited signal ($ = max(\{|w|\ :\ w \neq 0\}) $
• $ \omega_c $: Cutoff frequency of a filter ($ \omega_c > 0 $). (For instance, lowpass filters are nonzero in the range $ \omega \in [-\omega_c, \omega_c] $.)
• $ T $: Sampling period; equal to $ \frac{2\pi}{\omega_s} $
• NR, or "Nyquest Rate": $ =2\omega_m $. If $ \omega_s > NR = 2\omega_m $, then the band-limited signal can be uniquely reconstructed from the sampled signal.
• $ p(t) $: "Impulse train" -- equivalent to $ \sum_{n=-\infty}^{\infty} \delta(t-nT) $
• $ s $: A complex number -- often expressed as a sum of it's parts, $ a+j\omega $, where $ a, \omega \in \mathbb{R} $
• $ X(s) $: The Laplace Transform of $ x(t) $.

The above symbols are brought to you with thanks to Brian Thomas

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