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) $.
The above symbols are brought to you with thanks to Brian Thomas
Signal Metrics
- Signal Energy
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$ 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)
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- Signal Power
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$ 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 $
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- Signal RMS (root-mean-square)
- $ X_{rms} = \sqrt{P_x} $
- Signal Magnitude
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$ m_x = max|x(t)| $, for CT
$ m_x = max|x(n)| $, for DT
if $ m_x < \infty $, we say signal is bounded
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