stem/Signal Proc/Pole-Zero.md
andy 5a94c5ff1a vault backup: 2023-06-06 17:01:49
Affected files:
STEM/AI/Kalman Filter.md
STEM/Signal Proc/Convolution.md
STEM/Signal Proc/Image/Tracking.md
STEM/Signal Proc/Pole-Zero.md
STEM/Signal Proc/Transfer Function.md
STEM/Speech/Linguistics/Consonants.md
STEM/Speech/Linguistics/Linguistics.md
STEM/Speech/Linguistics/README.md
STEM/Speech/Linguistics/Terms.md
STEM/Speech/Linguistics/Vowels.md
STEM/Speech/NLP/Jargon.md
STEM/Speech/NLP/NLP.md
STEM/Speech/NLP/README.md
STEM/Speech/NLP/Recognition.md
STEM/Speech/Perception/Perception.md
STEM/Speech/Perception/README.md
STEM/Speech/Speech Processing/Applications.md
STEM/Speech/Speech Processing/README.md
STEM/Speech/Speech Processing/Source-Filter.md
STEM/Speech/Speech Processing/Vocal Tract.md
STEM/img/english-phoneme-table.png
STEM/img/formant.png
STEM/img/pole-zero-attenuation.png
STEM/img/pole-zero-feedback.png
STEM/img/pole-zero-stable.png
STEM/img/roc-right-left.png
STEM/img/roc-two-sided.png
STEM/img/spectrum-vocal-tract.png
STEM/img/transfer-stable-unstable.png
STEM/img/vowel-chart.png
STEM/img/vowel-spaces.png
2023-06-06 17:01:49 +01:00

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