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− | So many symbols, so little time... Here's a quick lookup table for our commonly-used symbols! | + | =List of Symbols= |
+ | So many symbols, so little time... Here's a quick lookup table for our commonly-used symbols in [[ECE]], especially [[ECE301]] and [[ECE438]]! | ||
*<math>\omega_s</math>: Sampling frequency; equal to <math>\frac{2\pi}{T}</math> | *<math>\omega_s</math>: Sampling frequency; equal to <math>\frac{2\pi}{T}</math> | ||
*<math>\omega_m</math>: Maximum frequency in a band-limited signal (<math> = max(\{|w|\ :\ w \neq 0\})</math> | *<math>\omega_m</math>: Maximum frequency in a band-limited signal (<math> = max(\{|w|\ :\ w \neq 0\})</math> | ||
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*<math>X(s)</math>: The Laplace Transform of <math>x(t)</math>. | *<math>X(s)</math>: The Laplace Transform of <math>x(t)</math>. | ||
The above symbols are brought to you with thanks to Brian Thomas | The above symbols are brought to you with thanks to Brian Thomas | ||
+ | |||
+ | <h3>Signal Metrics</h3> | ||
+ | <br/> | ||
+ | <ul style="list-style:none;"> | ||
+ | <li><strong>Signal Energy</strong> | ||
+ | <ul style="list-style:none;"> | ||
+ | <li> | ||
+ | <p><math>E_x = \int_{-\infty}^{\infty} |x(t)|^2\,dt</math> for ct (continuous time)</p> | ||
+ | <p><math>E_x = \sum_{n=-\infty}^{\infty} |x(n)|^2</math> for dt (discrete time)</p> | ||
+ | </li> | ||
+ | </ul> | ||
+ | </li> | ||
+ | <li><strong>Signal Power</strong> | ||
+ | <ul style="list-style:none;"> | ||
+ | <li> | ||
+ | <p><math>P_x = \lim_{T \Rightarrow \infty}\frac{1}{2T}\int_{-T}^{T} |x(t)|^2\,dt</math> for ct (continuous time)</p> | ||
+ | <p><math>P_x = \lim_{N \Rightarrow \infty}\sum_{n=-N}^{N} |x(n)|^2</math> for dt (discrete time)</p> | ||
+ | <p>note: for periodic signals <br/> | ||
+ | <math>P_x = \frac{1}{N}\sum_{n=0}^{N-1}|x(n)|^2</math> | ||
+ | </p> | ||
+ | </li> | ||
+ | </ul> | ||
+ | </li> | ||
+ | <li><strong>Signal RMS (root-mean-square)</strong> | ||
+ | <ul style="list-style:none;"> | ||
+ | <li> | ||
+ | <math>X_{rms} = \sqrt{P_x}</math> | ||
+ | </li> | ||
+ | </ul> | ||
+ | </li> | ||
+ | <li><strong>Signal Magnitude</strong> | ||
+ | <ul style="list-style:none;"> | ||
+ | <li> | ||
+ | <p><math>m_x = max|x(t)|</math>, for CT</p> | ||
+ | <p><math>m_x = max|x(n)|</math>, for DT</p> | ||
+ | <p> if <math>m_x < \infty</math>, we say signal is bounded</p> | ||
+ | </li> | ||
+ | </ul> | ||
+ | </li> | ||
+ | ---- | ||
+ | [[ECE|Back to ECE]] | ||
+ | |||
+ | [[ECE301|Back to ECE 301]] | ||
+ | |||
+ | [[ECE438|Back to ECE 438]] |
Latest revision as of 11:08, 12 November 2010
List of Symbols
So many symbols, so little time... Here's a quick lookup table for our commonly-used symbols in ECE, especially ECE301 and ECE438!
- $ \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
-
Back to ECE 438