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
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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 $, we say signal is bounded
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- Scaling ($ y(t) = x(\frac{t}{a}) $)
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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 integerScaling and Shifting $ y(t) = x(\frac{t}{a}-t_0) $
note: $ y(0) = x(-t_0) $
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