From 996a2322d8dc4ec95fefe08d92344e0cbdc266a2 Mon Sep 17 00:00:00 2001 From: Adog64 Date: Wed, 6 Mar 2024 08:34:10 -0500 Subject: [PATCH] Lots of updates --- .../DSP/Notes/dsp-notes.aux | 6 +- .../DSP/Notes/dsp-notes.fdb_latexmk | 8 +- .../DSP/Notes/dsp-notes.log | 40 ++++-- .../DSP/Notes/dsp-notes.pdf | Bin 256873 -> 271477 bytes .../DSP/Notes/dsp-notes.synctex.gz | Bin 119086 -> 141816 bytes .../DSP/Notes/dsp-notes.tex | 91 ++++++++++++ .../DSP/Notes/dsp-notes.toc | 2 + .../Learning-Assignment-Summary-3.pptx | Bin 0 -> 1990731 bytes .../EnI/Group Work/GymUsageMockup001.xcf | Bin 0 -> 11778278 bytes .../EnI/Group Work/low-res-prototyping.wiki | 19 +++ ... Mindset (2020) - libgen.li.pdf.crdownload | Bin 0 -> 1043486 bytes .../EnI/Notes/EnI-notes.aux | 16 ++- .../EnI/Notes/EnI-notes.fdb_latexmk | 8 +- .../EnI/Notes/EnI-notes.log | 30 ++-- .../EnI/Notes/EnI-notes.pdf | Bin 98013 -> 103716 bytes .../EnI/Notes/EnI-notes.synctex.gz | Bin 13591 -> 18877 bytes .../EnI/Notes/EnI-notes.tex | 21 +++ .../EnI/Notes/EnI-notes.toc | 2 + .../Power-Electronics/Homework/Homework-3/-o | 0 .../Homework/Homework-3/ChargingCircuit.png | Bin 0 -> 1845 bytes .../Homework/Homework-3/Homework-3.pdf | Bin 0 -> 310464 bytes .../Power_Electronics_Homework_3.pdf | Bin 0 -> 237316 bytes .../Homework/Homework-3/homework3.aux | 6 + .../Homework/Homework-3/homework3.fdb_latexmk | 29 ++++ .../Homework/Homework-3/homework3.fls | 131 ++++++++++++++++++ .../Homework/Homework-3/homework3.log | 119 ++++++++++++++++ .../Homework/Homework-3/homework3.pdf | Bin 0 -> 64242 bytes .../Homework/Homework-3/homework3.synctex.gz | Bin 0 -> 1728 bytes .../Homework/Homework-3/homework3.tex | 23 +++ .../Homework/Homework-3/iL100v.png | Bin 0 -> 15075 bytes .../Homework/Homework-3/iL150v.png | Bin 0 -> 15000 bytes .../Homework/Homework-3/iL50v.png | Bin 0 -> 15205 bytes .../Notes/power-electronics-notes.aux | 24 +++- .../Notes/power-electronics-notes.fdb_latexmk | 13 +- .../Notes/power-electronics-notes.fls | 62 ++++++++- .../Notes/power-electronics-notes.log | 79 ++++++++--- .../Notes/power-electronics-notes.pdf | Bin 149523 -> 172622 bytes .../Notes/power-electronics-notes.synctex.gz | Bin 25216 -> 34888 bytes .../Notes/power-electronics-notes.tex | 37 +++++ .../Notes/power-electronics-notes.toc | 6 + .../SysCon/Labs/Lab4/Lab4_p1.slx | Bin 0 -> 76554 bytes .../SysCon/Labs/Lab4/Lab4_p2.slx | Bin 0 -> 84867 bytes .../SysCon/Labs/Lab4/Lab4_p3.slx | Bin 0 -> 83557 bytes .../SysCon/Labs/Lab4/Lab4_p4.slx | Bin 0 -> 87464 bytes .../SysCon/Labs/Lab4/SysConLab4.mlx | Bin 0 -> 236316 bytes .../SysCon/Labs/Lab4/pole_effects_l1.png | Bin 0 -> 33664 bytes .../SysCon/Labs/Lab4/pz1a.png | Bin 0 -> 12504 bytes .../SysCon/Labs/Lab4/simulink_p1.png | Bin 0 -> 12546 bytes .../SysCon/Labs/Lab4/simulink_p2.png | Bin 0 -> 27192 bytes .../SysCon/Labs/Lab4/simulink_p3.png | Bin 0 -> 23946 bytes .../SysCon/Labs/Lab4/simulink_p4.png | Bin 0 -> 31676 bytes .../SysCon/Labs/Lab4/trans_effects_l3.png | Bin 0 -> 61450 bytes .../SysCon/Labs/Lab4/trans_effects_l4.png | Bin 0 -> 40123 bytes .../SysCon/Labs/Lab4/trans_effects_pl2.png | Bin 0 -> 33309 bytes .../SysCon/Labs/Lab4/trans_effects_pl3.png | Bin 0 -> 29009 bytes .../SysCon/Labs/Lab4/trans_effects_pl4.png | Bin 0 -> 20549 bytes .../SysCon/Labs/Lab4/zero_effects_l2.png | Bin 0 -> 34437 bytes .../SysCon/Labs/Lab5/SysConLab5.mlx | Bin 0 -> 146284 bytes .../SysCon/Labs/Lab5/l1.slx | Bin 0 -> 71479 bytes .../SysCon/Labs/Lab5/l1_cascade.png | Bin 0 -> 10432 bytes .../SysCon/Labs/Lab5/l1_combined.png | Bin 0 -> 10443 bytes .../SysCon/Labs/Lab5/l1_system.png | Bin 0 -> 6388 bytes .../SysCon/Labs/Lab5/l2.slx | Bin 0 -> 80225 bytes .../SysCon/Labs/Lab5/l2_combined.png | Bin 0 -> 8653 bytes .../SysCon/Labs/Lab5/l2_parallel.png | Bin 0 -> 8328 bytes .../SysCon/Labs/Lab5/l3.slx | Bin 0 -> 94441 bytes .../SysCon/Labs/Lab5/l3_equiv.png | Bin 0 -> 13064 bytes .../SysCon/Labs/Lab5/l3_feedback.png | Bin 0 -> 13124 bytes .../SysCon/Labs/Lab5/l3_system.png | Bin 0 -> 33822 bytes .../SysCon/Labs/Lab5/l4_equiv.png | Bin 0 -> 13190 bytes .../SysCon/Labs/Lab5/l4_system.png | Bin 0 -> 22592 bytes .../SysCon/Labs/Lab5/l5_equiv.png | Bin 0 -> 13364 bytes .../SysCon/Labs/Lab5/pl4_system.png | Bin 0 -> 4973 bytes .../SysCon/Labs/Lab5/pl5_system.png | Bin 0 -> 4948 bytes CourseMap.xlsx | Bin 309477 -> 309477 bytes CybersecurityEngineering.png | Bin 0 -> 74433 bytes 76 files changed, 713 insertions(+), 59 deletions(-) create mode 100644 6th-Semester-Spring-2024/EnI/Assignments/Learning-Assignment-Summary-3.pptx create mode 100644 6th-Semester-Spring-2024/EnI/Group Work/GymUsageMockup001.xcf create mode 100644 6th-Semester-Spring-2024/EnI/Group Work/low-res-prototyping.wiki create mode 100644 6th-Semester-Spring-2024/EnI/Heidi M. Neck_ Christopher P. Neck_ Emma L. 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- Glide +- Look for possible problems +- Prioritize features or functionality +- Get the general idea across + +== Going from low-res to high-res == +- OnShape for 3D modelling diff --git a/6th-Semester-Spring-2024/EnI/Heidi M. 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b/6th-Semester-Spring-2024/Power-Electronics/Homework/Homework-3/homework3.tex new file mode 100644 index 0000000..ae9d54f --- /dev/null +++ b/6th-Semester-Spring-2024/Power-Electronics/Homework/Homework-3/homework3.tex @@ -0,0 +1,23 @@ +\documentclass{article} +\usepackage{graphicx} + +\begin{document} +\begin{figure} + \center + \includegraphics[width=0.5\textwidth]{ChargingCircuit.png} + \caption{Charging circuit with 50v DC source as load} +\end{figure} + +\begin{figure} + \includegraphics[width=\textwidth]{iL50v.png} + \caption{Current through inductor with 50v DC source as load} +\end{figure} +\begin{figure} + \includegraphics[width=\textwidth]{iL100v.png} + \caption{Current through inductor with 100v DC source as load} +\end{figure} +\begin{figure} + \includegraphics[width=\textwidth]{iL150v.png} + \caption{Current through inductor with 150v DC source as load} +\end{figure} +\end{document} diff --git a/6th-Semester-Spring-2024/Power-Electronics/Homework/Homework-3/iL100v.png 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$r$ is the magnitude and $\theta$ is the phase angle. \noindent If a region of convergence (ROC) does not include the unit circle, then its DTFT does not exist except for special case signals. If the ROC \emph{does} contain the unit circle, then the DTFT can be found by substituting $z = e^{j\omega}$. +\section{Z Transform Transfer Functions} +Recall that the impulse response of an LTI system completely characterizes the system. It can be used to determine responses of other systems via convolution, and it is a useful tool in determining the BIBO stability of the system. +\\ +\\ +Given an impulse response to a discrete LTI system, the Z-transform of the impulse response is called the \emph{transfer function} +$$H(z) = Z[h[n]] = \sum_n h[n]z^{-n}$$ +An LTI system is causal if $h[n]$ is a causal signal. This is the same as if the ROC of $H(z)$ includes $z=\infty$. These conditions are equivalent because the ROC of a Z-transform only contains $z=\infty$ for causal signals. +\\ +\\ +Additionally, an LTI system is BIBO stable if $h[n]$ is absolutely summable. +$$\sum_n |h[n]| < \infty$$ +The equivalent condition for $H(z)$ is the ROC of $H(z)$ must include the unit circle. These conditions are equivalent because the DTFT exists when the unit circle is in the ROC, and the DTFT only exists when the signal is absolutely summable. +\nt{ + For causal systems only, BIBO stability is achieved if all the poles of $H(z)$ are inside the unit circle. Since non-causal systems do not include $z=\infty$ in the ROC, this test will not hold for non-causal systems. +} +\noindent +If the impulse response $h[n]$ has finite support, then the output, $y[n]$, given an input, $x[n]$, can be found by convolution. +$$y[n] = h[n] * x[n] = \sum_{k=0}^{N-1} h[k] x[n-k]$$ +However, if the impulse response $h[n]$ has infinite support, the output, $y[n]$, is given by a constant coefficient difference equation. +$$\sum_{j=0}^N a[j]y[n-j] = \sum_{k=0}^M b[k]x[n-k]$$ +where $a[j]$ are the constant coefficient of the output, and $b[k]$ are the constant coefficients of the input. + +\ex{} +{ + Given the infinite impulse response: + $$h[n] = a^n u[n]$$ + The transfer function of the system is: + $$H(z) = {z \over z - a}$$ + with a ROC of $|z| > |a|$. Since this includes $z=\infty$, the system is causal. If $|a| < 1$, then the system is also BIBO stable. + + $$H(z) = {z \over z-a} = {Y(z) \over X(z)}$$ + $${Y(z) \over X(z)} = {1 \over 1-az^{-1}}$$ + $$Y(z)-az^{-1}Y(z) = X(z)$$ + Taking the inverse Z-transform: + $$y[n] - a y[n-1] = x[n]$$ + $$y[n] = x[n] + ay[n-1]$$ + Since the output relies on both the input and the previous output, the recursive implementation of the input-output relationship is used. +} + +\ex{} +{ + Given the impulse response, + $$h[n] = na^n u[n]$$ + the corresponding transfer function is + $$H(z) = {az \over (z-a)^2}$$ + with ROC, $|z| > |a|$. Since the impulse response is causal, the system is causal. Equivalently, since the ROC includes $z=\infty$, the system is causal. + Given this region of convergence, the system is BIBO stable if and only if $|a| < 1$. + $$H(z) = {az\over (z-a)^2} = {Y(z) \over X(z)}$$ + $$H(z) = {az \over z^2 - 2az + a^2} = {az \over z^2(1 -2az^{-1} + a^2z^{-2})}$$ + $$az^{-1}X(z) = Y(z) - 2az^{-1}Y(z) + a^2z^{-2}Y(z)$$ + Taking the inverse Z-transform to find the input-output relationship: + $$x[n-1] = y[n] - 2ay[n-1] + a^2y[n-2]$$ + $$y[n] = x[n-1] + 2ay[n-1] - a^2y[n-2]$$ +} + +\ex{} +{ + $$h[n] = u[n] - u[n-4]$$ + $$h[n] = \begin{bmatrix} 1 & 1 & 1 & 1 \end{bmatrix}, 0 \le n \le 3$$ + $$H(z) = \sum_n h[n] z^{-n} = 1 + z^{-1} + z^{-2} + z^{-3}$$ + For finite support signals, the Z-transform is the entire Z-plane with the possible exception of $z=0$ and $z=\infty$. Checking $z=0$ reveals a pole, but $z=\infty$ works, so the system is causal. All finite support impulse responses (FIR) are BIBO stable. + $$y[n] = \sum_{k=0}^3 h[k]x[n-k]$$ + $$y[n] = h[0]x[n] + h[1]x[n-1] + h[2]x[n-2] + h[3]x[n-3]$$ +} + +\section{The Inverse Z-Transform} +Given the causal signal $x[n] = a^n u[n]$, its Z-transform is $X(z) = {z \over z - a}$ with region of convergence $|z| > |a|$. Given an anti-causal signal $x[n] -a^n u[-n-1]$, its Z-transform is $X(z) = {z \over z - a}$ with region of convergence $|z| < |a|$. +\\ +\\ +Consider the following second-order LTI system whose transfer function is given by: +$$H(z) = {z (z - 0.2) \over (z-0.8)(z-0.9)}$$ +If the system is causal, then the ROC includes $\infty$, so the ROC must be $|z|>0.9$. In this case the system would be BIBO stable. However, if the system is anti-causal, then the ROC must include 0 and therefore would be $|z|<0.8$. In this case, the system would not be BIBO stable. If the system is two-sided, then the ROC would be $0.8 < |z| < 0.9$. In this case the system would also not be BIBO stable. +\\ +\\ +Find the partial fraction expansion of $H(z) \over z$: +$${H(z) \over z} = {z - 0.2 \over (z-0.8)(z-0.9)} = {A \over z - 0.9} + {B \over z - 0.8}$$ +$$z - 0.2 = (z-0.8)A + (z-0.9)B$$ +$$B = -6, A = 7$$ +$${H(z) \over z} = {7 \over z-0.9} - {6 \over z-0.8}$$ +$$H(z) = {7z \over z-0.9} - {6z \over z-0.8}$$ +Apply the inverse Z-transform on both sides. If the system is causal: +$$h[n] = 7(0.9)^n u[n] - 6(0.8)^n u[n]$$ +If the system is anti-causal: +$$h[n] = -7(0.9)^n u[-n-1] + 6(0.8)^n u[-n-1]$$ +If the system is two-sided: +$$h[n] = -7(0.9)^n u[-n-1] - 6(0.8)^n u[n]$$ + +\nt +{ + For FIR systems, if the Z-transform does not converge at $|z|= 0$ or $|z| = \infty$, they are not considered poles, because only IIR systems can have poles. +} \end{document} diff --git a/6th-Semester-Spring-2024/DSP/Notes/dsp-notes.toc b/6th-Semester-Spring-2024/DSP/Notes/dsp-notes.toc index 3a2109e..3659e6e 100644 --- a/6th-Semester-Spring-2024/DSP/Notes/dsp-notes.toc +++ b/6th-Semester-Spring-2024/DSP/Notes/dsp-notes.toc @@ -31,4 +31,6 @@ \contentsline {subsection}{\numberline {3.1.2}Properties of the Discrete Time Fourier Transform}{17}{subsection.3.1.2}% \contentsline {subsection}{\numberline {3.1.3}Special Cases}{18}{subsection.3.1.3}% \contentsline {chapter}{\numberline {4}The Z-Transform}{19}{chapter.4}% +\contentsline {section}{\numberline {4.1}Z Transform Transfer Functions}{20}{section.4.1}% +\contentsline {section}{\numberline {4.2}The Inverse Z-Transform}{22}{section.4.2}% \contentsfinish diff --git a/6th-Semester-Spring-2024/EnI/Assignments/Learning-Assignment-Summary-3.pptx b/6th-Semester-Spring-2024/EnI/Assignments/Learning-Assignment-Summary-3.pptx new file mode 100644 index 0000000000000000000000000000000000000000..8e26d52ca564af5d54a3c13b012cc73564287e46 GIT binary patch literal 1990731 zcmeFYWo%?km!)fFhO%8|W@aukGc%@{nVIb}Gcz+YSDBd^%FN7gU2k{Ke0OxG??^NA zuT#pCsZ{Dn@$4P3_KF=xUJ4Wp4G0Pf3g{P!u{zMdykNh+>p1|d9O>!)UaR6|q(BAW zLN4FY2i7}QRFUGc&P9SYg-;Nnm{$u(flfH?nmA(79P#{Zt&Y>1ROd ze55AY3cU4$(I6CW+*Kwa`BCNJm<7I077d`U>8bhi!!JUJ6yL@5cLTSs=^rB%2u}vp zMGN6wU11RUe7eHbs9_6E_)wDCVU@ly&AzbA-gx_uMCb`&YkeCFcnH$vUTDg(AiOx* ze3L}Yy5NYlY$FQDYD*=1r?M6KiC>6YKr;D9X{vsm+r#F>m?yGvnWV5Y5+_-dT|WpZ z66O5`rT*wj*Jy(&t%PFmI1E&hWV7%T>h?x^H`Wv5A%t!c-Yc{FO7ol8!rHPXmb+5{ 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