62 lines
1.5 KiB
Markdown
62 lines
1.5 KiB
Markdown
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- Poles
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- **X**
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- Let $X(z) = inf$
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- Let $1/X(z) = 0$
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- Roots of denominator
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- Zeros
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- **O**
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- Let $X(z) = 0$
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- Roots of numerator
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- In complex (Z for speech) domain
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[Magnitude Response From Pole/Zeros](https://www.youtube.com/watch?v=8jNjVkoZQCU)
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[MIT Pole Zero](https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf)
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Representation of rational transfer function, identifies
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- Stability
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- Causal/Anti-causal system
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- ROC
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- Minimum phase/Non minimum phase
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# BIBO Stable
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- All poles of H must lie within the unit circle of the plot
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- If we give an input less than a constant
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- Will get an output less than some constant
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# Region of Convergence
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- Depends on whether causal or anti-causal
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- Cannot contain poles
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- Goes to infinity
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## Continuous
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1. If includes imaginary axis
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- BIBO stable
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- All poles must be left of i axis
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2. Rightwards from pole with largest real-part (not infinity)
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- Causal
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3. Leftward from pole with smallest real-part (not -infinity)
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- Anti-causal
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## Discrete
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1. If includes unit circle
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- BIBO stable
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2. Outward from pole with largest (not infinite) magnitude
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- Right-sided impulse response
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- Causal (if no pole at infinity)
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3. Inward from pole with smallest (nonzero) magnitude
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- Anti-causal
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Sinusoidal when complex pair
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- $e^{-j\omega}$
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- Euler's for oscillating
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Exponential when on the axis
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- Decays, no $i$ in the exponent
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