25 lines
1.3 KiB
Markdown
25 lines
1.3 KiB
Markdown
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$$Y(s)=H(s)\cdot X(s)$$
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- $H(s)=\frac{Y(s)}{X(s)}=\frac{\mathcal L\{y(t)\}}{\mathcal L\{x(t)\}}$
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$$Y(z)=H(z)\cdot X(z)$$
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- $H(z)=\frac{Y(z)}{X(z)}=\frac{\mathcal Z\{y[n]\}}{\mathcal Z\{x[n]\}}$
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$$G(\omega)=\frac{|Y|}{|X|}=|H(j\omega)|$$
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- $H(j\omega)$, Frequency response
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$$\phi(\omega)=arg(Y)-arg(X)=arg\left(H\left(j\omega\right)\right)$$
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- $\phi(\omega)$, Phase shift
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$$\tau_\phi(\omega)=-\frac{\phi(\omega)}{\omega}$$
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- $\tau_\phi$, Phase delay
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- Frequency-dependent amount of delay introduced to the sinusoid by $H$
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$$\tau_g(\omega)=-\frac{d\phi(\omega)}{d\omega}$$
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- $\tau_g$, Group delay
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- Frequency-dependent amount of delay introduced to the envelope of the sinusoid by $H$
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[Partial Fractions](https://lpsa.swarthmore.edu/BackGround/PartialFraction/PartialFraction.html#Order_of_numerator_polynomial_is_not_less_than_that_of_the_denominator)
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[Partial Fractions for Laplace](https://lpsa.swarthmore.edu/LaplaceXform/InvLaplace/InvLaplaceXformPFE.html)
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[Inverse Z Transform](https://lpsa.swarthmore.edu/ZXform/InvZXform/InvZXform.html)
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[Discrete Time Systems:Impulse responses and convolution; An introduction to the Z-transform](https://homes.esat.kuleuven.be/~maapc/static/files/SYSTHEORY/Slides/Lecture5/Lecture5-Impulse%20responses%20and%20convolution%20layout.pdf)
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