Syncing to work on interview with entrepreneur essay
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Aidan Sharpe
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\usepackage[margin=1in]{geometry}
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\title{}
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\title{Uniform Convergence, Convolutions, and Correlation of Signals}
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\author{Aidan Sharpe \& Elise Heim}
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\maketitle
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\end{titlepage}
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\begin{abstract}
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\end{abstract}
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\section{Introduction}
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\section{Results \& Discussion}
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\subsection{Uniform Convergence}
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Consider the discrete time signal, $x[n] = a^n u[n]$, where $u[n]$ is the unit-step and $a=0.9$. Let $X(e^{j\omega})$ be the discrete time Fourier transform (DTFT) of $x[n]$. For the DTFT of a signal to exist, it must be absolutely summable over all $n \in \mathbb{Z}$. If the signal is absolutely summable, then it must also be bounded, meaning $|x[n]| < \infty, \forall n \in \mathbb{Z}$. In other words, $x[n]$ must have a finite maximum value.
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\subsection{Simple Example of Uniform Convergence}
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Consider the discrete time signal, $x[n] = a^n u[n]$, where $u[n]$ is the unit-step and $a=0.9$. Let $X(e^{j\omega})$ be the discrete time Fourier transform (DTFT) of $x[n]$. For the DTFT of a signal to exist, it must be absolutely summable over all $n \in \mathbb{Z}$. If the signal is absolutely summable, then it must also be bounded, meaning there exists some non-negative real number $B$, such that $|x[n]| \le B, \forall n \in \mathbb{Z}$. At a bare minimum, $x[n]$ must have a finite maximum value.
|
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\\
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\\
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In this case, $x[n]$ is bounded because $a^n$ grows with smaller values of $n$, and $u[n]$ is zero when $n<0$. Therefore, the maximum value of $x[n] = a^n u[n]$ is $x[0] = a$.
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@ -41,15 +36,211 @@ Additionally, $x[n]$ takes the form of a geometric series, so its sum is given b
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\end{equation}
|
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In this case, $m=0$, because the signal has no value for $n<0$. Since $a=0.9$, the sum evaluates to is ${1 \over 1-0.9} = 10$. Considering that $x[n]$ is always a positive real number, each term is its own absolute value, so the sum and the absolute sum are equivalent. By taking the absolute sum of the first 200 terms of $x[n]$, it becomes clear that it approaches 10 in the limit, seen in figure \ref{fig:abs_sum_anun}.
|
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\begin{figure}[h]
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\begin{figure}[H]
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\center
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\includegraphics[width=\textwidth]{abs_sum_anun.png}
|
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\caption{The signal value of $x[n]=a^n u[n]$ and its absolute sum}
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\label{fig:abs_sum_anun}
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||||
\end{figure}
|
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\noindent
|
||||
Since the signal is absolutely summable, the DTFT, $X(e^{j\omega})$ exists, and can be found by evaluating the sum:
|
||||
\begin{equation}
|
||||
X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n}.
|
||||
\end{equation}
|
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Evaluating this sum reveals that
|
||||
$$X(e^{j\omega}) = {1\over 1 - a e^{-j\omega}},$$
|
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where $a = 0.9$. The plot for $X(e^{j\omega})$ is seen in figure \ref{fig:dtft_anun}.
|
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\begin{figure}[H]
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\includegraphics[width=\textwidth]{dtft_anun.png}
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\caption{The DTFT of $x[n] = a^n u[n]$}
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\label{fig:dtft_anun}
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\end{figure}
|
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\noindent
|
||||
In this case, the DTFT can be easily evalueated because the sum is a geometric series. However, for more complicated signals, the sum can only be approximated. These non-infinite sums are called truncated DTFTs and take the form:
|
||||
\begin{equation}
|
||||
X(e^{j\omega}) = \sum_{-K}^K x[n] e^{-j\omega n}.
|
||||
\end{equation}
|
||||
Seen in figure \ref{fig:truncated_DTFTs_anun}, the approximation seems to be quite poor for small values of $K$, but the accuracy seems to quickly increases for slightly larger values of $K$. The actual performance, quantified by the maximum error for a given value of $K$, lines up with these observations, as seen in figure \ref{fig:max_error_anun}. It also becomes apparent through figure \ref{fig:max_error_anun} that large values of $K$ have diminishing returns.
|
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\begin{figure}[H]
|
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\includegraphics[width=\textwidth]{truncated_DTFTs_anun.png}
|
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\caption{The truncated DTFT of $x[n] = a^n u[n]$ for various values of $K$}
|
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\label{fig:truncated_DTFTs_anun}
|
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\end{figure}
|
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\begin{figure}[H]
|
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\includegraphics[width=\textwidth]{max_error_anun.png}
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\caption{The maximum error of the truncated DTFT of $x[n] = a^n u[n]$ for various values of $K$}
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\label{fig:max_error_anun}
|
||||
\end{figure}
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||||
\subsection{More Complex Example of Uniform Convergence}
|
||||
Consider the signal $x[n] = n a^n u[n-1]$. Again, to determine if the DTFT of the signal exists, it must be absolutely summable. For $a<1$, its sum is given by:
|
||||
\begin{equation}
|
||||
\sum_{n=1}^\infty n a^n = {a \over (1-a)^2}.
|
||||
\label{eqn:n_geometric_series}
|
||||
\end{equation}
|
||||
For $a=0.9$, the infinite series evaluates to 90, and this result is verified by the sum of the first 200 samples of $x[n]$ seen in figure \ref{fig:abs_sum_nanun-1}.
|
||||
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{abs_sum_nanun-1.png}
|
||||
\caption{The signal value of $x[n] = n a^n u[n-1]$ and its absolute sum}
|
||||
\label{fig:abs_sum_nanun-1}
|
||||
\end{figure}
|
||||
|
||||
\noindent
|
||||
The signal, $x[n]$ has a maximum amplitude of 3.4868 at $n=9$ and $n=10$. Therefore, the signal is bounded by 3.4868. It is also clear that the signal must be bounded because it is absolutely summable. If the signal was unbounded, then its absolute sum would also be unbounded.
|
||||
\\
|
||||
\\
|
||||
The DTFT of $x[n]$ can be found using equation \ref{eqn:n_geometric_series}. In this case, the value of $a$ in the equation is $a e^{-j\omega}$. Therefore, the DTFT of $x[n]$ is
|
||||
\begin{equation}
|
||||
X(e^{j\omega}) = \sum_{-\infty}^\infty n a^n u[n-1] e^{-j\omega n} = \sum_1^\infty n\left(a e^{-j\omega} \right)^n = {a e^{-j\omega} \over \left(1 - a e^{-j\omega} \right)^2}.
|
||||
\end{equation}
|
||||
\\
|
||||
\\
|
||||
The truncated DTFT is in this case is a variation on the geometric series of the form $k r^k$. The sum of the first $n$ terms is given by:
|
||||
\begin{equation}
|
||||
\sum_{k=1}^n k r^k = {r - r^{n+2} \over (1-r)^2} - {(n+1) r^{n+1} \over 1 - r}.
|
||||
\end{equation}
|
||||
Applying this formula allows for varying degrees of approximation of the DTFT, as seen in figure \ref{fig:truncated_dtfts_nanun-1}.
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{truncated_DTFTs_nanun.png}
|
||||
\caption{Varying levels of approximation of the DTFT of $x[n] = n a^n u[n-1]$}
|
||||
\label{fig:truncated_dtfts_nanun-1}
|
||||
\end{figure}
|
||||
It can clearly be seen that as $K$ gets larger, the approximation of the DTFT becomes increasingly better. By plotting the maximum error for increasing values of $K$, as seen in figure \ref{fig:max_error_nanun-1}, it becomes clear that, once again, there are diminishing returns for larger values.
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{max_error_nanun-1.png}
|
||||
\caption{The maximum error of the truncated DTFT of $x[n] = n a^n u[n-1]$ for increasing values of $K$}
|
||||
\label{fig:max_error_nanun-1}
|
||||
\end{figure}
|
||||
|
||||
\subsection{The Inverse DTFT}
|
||||
An ideal lowpass filter, whose frequency response is seen in figure \ref{fig:ideal_lowpass_fresponse}, has unity gain for all frequencies less than the cutoff frequency, $\omega_c$, and zero gain for all frequencies greater than the cutoff frequency. Therefore, the DTFT of a lowpass signal would be a pulse of amplitude one from $-\omega_c$ to $\omega_c$. To recover the corresponding signal, the inverse DTFT should be applied. This operation is defined as:
|
||||
\begin{equation}
|
||||
x[n] = {1\over 2\pi} \int\limits_{-\pi}^{\pi} X(e^{j\omega}) e^{j\omega n} d\omega
|
||||
\end{equation}
|
||||
where $X(e^{j\omega})$ is the DTFT of the signal $x[n]$.
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{ideal_lowpass_fresponse.png}
|
||||
\caption{The frequency response of an ideal lowpass filter}
|
||||
\label{fig:ideal_lowpass_fresponse}
|
||||
\end{figure}
|
||||
Considering that $X(e^{j\omega})$ has a value of one between $-\omega_c$ and $\omega_c$ and 0 elsewhere, the integral can be rewritten as:
|
||||
\begin{equation}
|
||||
x[n] = {1\over 2\pi} \int\limits_{-\omega_c}^{\omega_c} e^{j\omega n} d\omega.
|
||||
\end{equation}
|
||||
Evaluating the integral gives:
|
||||
\begin{equation}
|
||||
x[n] = {\omega_c \over \pi} \sinc(\omega_c n),
|
||||
\end{equation}
|
||||
where $\sinc(x)$ is the continuous normalized $\sinc$ function defined as:
|
||||
\begin{equation}
|
||||
\sinc(x) = \begin{cases}
|
||||
1 & x = 0 \\
|
||||
{\sin(\pi x) \over \pi x} & x \ne 0
|
||||
\end{cases}
|
||||
\end{equation}
|
||||
\noindent
|
||||
The plot of $x[n]$, where $\omega_c = 0.4\pi$ is seen in figure \ref{fig:recovered_lowpass}.
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{recovered_lowpass_signal.png}
|
||||
\caption{The inverse DTFT of an ideal lowpass filter}
|
||||
\label{fig:recovered_lowpass}
|
||||
\end{figure}
|
||||
\noindent
|
||||
Given this signal, finite sum approximations of the DTFT yield oscillating frequency responses that resemble the ideal case, as seen in figure \ref{fig:truncated_DTFTs_lowpass}. Similar to other approximations of DTFTs, the more terms in the summation, the better the accuracy of the approximation.
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{truncated_DTFTs_lowpass.png}
|
||||
\caption{Finite sum approximations of an ideal lowpass filter with varying bounds of summation}
|
||||
\label{fig:truncated_DTFTs_lowpass}
|
||||
\end{figure}
|
||||
|
||||
\subsection{Convolutions: Pulse Response}
|
||||
Consider a linear, time-invariant signal with the impulse response $h[n] = [1, 2, 1]$. If the input of the system is given as $x[n] = u[n] - u[n-2]$ where $u[n]$ is the unit step, the output of the system can be calculated analytically.
|
||||
\begin{equation}
|
||||
y[n] = x[0]h[n] + x[1]h[n-1] = h[n] + h[n-1]
|
||||
\end{equation}
|
||||
$$y[0] = 1 + 0 = 1$$
|
||||
$$y[1] = 2 + 1 = 3$$
|
||||
$$y[2] = 1 + 2 = 3$$
|
||||
$$y[3] = 0 + 1 = 1$$
|
||||
|
||||
\noindent
|
||||
With this in mind, one can use MATLAB to check and confirm the answer, as shown below in figure \ref{fig:number4}.
|
||||
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{num4.png}
|
||||
\caption{The response of $x[n] = u[n] - u[n-2]$ to an LTI system}
|
||||
\label{fig:number4}
|
||||
\end{figure}
|
||||
|
||||
\subsection{Convolutions: Exponential Decay Response}
|
||||
Consider a linear, time-invariant system with the impulse response $h[n] = u[n]$, where $u[n]$ is the unit step. If the input to the system is an expression for $y[n]$ can be calculated, as shown below.
|
||||
$$y[n] = \sum_{k=0}^{99} x[k] h[n-k]$$
|
||||
The plot for $y[n]$ is seen in figure \ref{fig:number5a}.
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{num5a.png}
|
||||
\caption{Convolution of $x[n] = 0.95^n u[n]$ with $h[n]$}
|
||||
\label{fig:number5a}
|
||||
\end{figure}
|
||||
|
||||
The unit step response occurs when $x[n] = u[n]$, and it is seen in figure \ref{fig:number5b}.
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{num5b.png}
|
||||
\caption{Step response of an LTI system}
|
||||
\label{fig:number5b}
|
||||
\end{figure}
|
||||
This system is BIBO stable because its impulse response is a bounded signal.
|
||||
|
||||
\subsection{Correlation Video Summaries}
|
||||
The first video discusses correlation. As mentioned by the video, “correlation is a measure of how similar signals are.” The video provides examples of three different signals: x, y, and z. It also displays a formula for calculating a correlation measurement. A greater correlation measurement between signals with similar energy means that they are correlated, however the measurement tells no information on how similar or different two signals are when the signals have significantly different energy. As demonstrated by altering the signals, the correlation measurement on its own does not provide an accurate representation of correlation. This is why it is necessary to use normalized correlation.
|
||||
\\
|
||||
\\
|
||||
In the second video, a new formula is presented to determine the normalized correlation. This takes our original equation and divides it by the square of the sum of the energies of the two signals being compared. The denominator is a scaling factor comprised of the energy of the signals being compared. Instead of taking the form of an integer, like a regular correlation, the resulting normalized correlation now takes the form of a value between -1 and 1. The more similar the signals, the greater the value. With this in mind, it might seem as though calculating correlation without normalizing would be trivial. However, through a demonstration with a MATLAB script, the value of non-normalized correlation is displayed. By calculating the correlation of multiple signals, it is easy to determine that some signals are twice as strongly present as other signals. However, the same information cannot be gleaned from the normalized correlation.
|
||||
\\
|
||||
\\
|
||||
The third and final video explains crosscorrelation, which is a measure of similarity between signals at different time lag positions. It introduces two signals, which are displayed such that there is zero lag. This means that the first, second, third, etc. samples are aligned vertically with each other. In order to calculate the correlation, it is necessary to multiply the vertically aligned samples and sum them all. At this zero lag position, a correlation value is calculated. If sample number zero of the first signal is aligned vertically with sample number one of the second signal, a lag of one is introduced. In this way, the vertically aligned samples can be multiplied then summed, and another correlation value can be determined. For each lag position, including negative lags, a new correlation value can be calculated to create a correlation sequence. Such a sequence can be generated and plotted in MATLAB.
|
||||
|
||||
\subsection{Crosscorrelation}
|
||||
Crosscorrelation between two real signals $x[n]$ and $h[n]$ at different lags $n$ is defined as:
|
||||
\begin{equation}
|
||||
c_{xh}[n] = \sum_k x[k] h[k-n]
|
||||
\end{equation}
|
||||
\\
|
||||
\\
|
||||
If $x[n] = [1 3 -2 4]$ and $h[n] = [2, 3, -1, 3]$, the crosscorrelation can be computed for $c_{xh}[0]$, $c_{xh}[1]$, and $c_{xh}[-1]$, as shown below.
|
||||
$$c_{xh}[0] = 25$$
|
||||
$$c_{xh}[1] = -4$$
|
||||
$$c_{xh}[-1] = -6$$
|
||||
The relationship of crosscorrelation versus lag can be plotted, as demonstrated by figure \ref{fig:number7}.
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{num7.png}
|
||||
\caption{Cross correlation of $x[n]$ and $h[n]$}
|
||||
\label{fig:number7}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\subsection{Autocorrelation}
|
||||
The autocorrelation of a real signal $x[n]$ is defined as:
|
||||
\begin{equation}
|
||||
c_{xx}[n] = \sum_k x[k] x[k-n]
|
||||
\end{equation}
|
||||
which is the crosscorrelation of a signal with itself. For example, the autocorrelation of $x[n] = [1, 3, -2, 4]$ is seen in figure \ref{fig:number8}.
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{num8real.png}
|
||||
\caption{Autocorrelation of $x[n] = [1,3,-2,4]$}
|
||||
\label{fig:number8}
|
||||
\end{figure}
|
||||
\subsection{Crosscorrelation and Autocorrelation}
|
||||
Given the signal $x[n] = \begin{bmatrix}1&1&1&1&1&1&1&1\end{bmatrix}$, its autocorrelation at zero lag is 1.5.
|
||||
\\
|
||||
\\
|
||||
Consider the signals $v[n] = 2x[n]$, $w[n] = -x[n]$, and $y[n] = \begin{bmatrix}1&1&-1&-1&1&1&-1&-1\end{bmatrix}$.
|
||||
$$c_{xv}[0] = 3$$
|
||||
$$c_{xw}[0] = -1.5$$
|
||||
$$c_{xy}[0] = 1$$
|
||||
Since $w[n] = -x[n]$, it makes sense that the crosscorrelation of $x[n]$ and $w[n]$ is the negation of the autocorrelation of $x[n]$.
|
||||
|
||||
\section{Conclusions}
|
||||
Multiple concepts were reinforced through this lab. Discrete-Time Fourier Transformations (DTFT), convolutions, as well as correlation were explored through a variety of exercises. This lab showed how the inverse DTFT can be utilized to model a filter, as well as how the DTFT is very easy to approximate with non-infinite sums. Additionally, a variety of different signals were represented in MATLAB to gain a visual understanding of the signals. Through some helpful sources that were included within the lab, one can learn a great deal about correlation values and how they can be used to compare signals.
|
||||
|
||||
\end{document}
|
||||
|
||||
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6th-Semester-Spring-2024/DSP/Labs/Lab-03/dtft_anun.png
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@ -1,11 +1,10 @@
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
|
||||
import scipy.signal
|
||||
n = np.arange(0,200)
|
||||
|
||||
a = 0.9
|
||||
|
||||
# x[n] = a^n u[n]
|
||||
# $x[n] = a^n u[n]$
|
||||
x = a**n * np.heaviside(n, 1)
|
||||
|
||||
asums = np.zeros(len(n))
|
||||
@ -25,3 +24,42 @@ plt.plot(n, asums)
|
||||
plt.xlabel("$n$")
|
||||
plt.ylabel("$\sum_n x[n]$", rotation="horizontal")
|
||||
plt.show()
|
||||
|
||||
# Compute DTFT of $x[n]$
|
||||
X = scipy.signal.freqz(1, (1, -a))
|
||||
omega, h = X
|
||||
|
||||
# Plot DTFT of $x[n]$
|
||||
plt.plot(omega, np.abs(h))
|
||||
plt.ylabel("Amplitude")
|
||||
plt.xlabel("Frequency [rad/sample]")
|
||||
plt.show()
|
||||
|
||||
# Set up plot for DTFT of $x[n]$ for comparrison with truncated DTFTs
|
||||
plt.plot(omega, np.abs(h), label="Actual DTFT")
|
||||
plt.ylabel("Amplitude")
|
||||
plt.xlabel("Frequency [rad/sample]")
|
||||
|
||||
# Calculate the truncated DTFTs of $x[n]$ as a function of $K$, $\sum_{n=-K}^K x[n] e^{-j\omega n}$
|
||||
for K in (3, 10, 20):
|
||||
# Finite geometric series formula
|
||||
X_K = (1 - (a**(K+1) * np.exp(-1j * omega * (K+1)))) / (1 - a*np.exp(-1j*omega))
|
||||
plt.plot(omega, np.abs(X_K), label=f"Truncated DTFT ($K={K}$)")
|
||||
|
||||
plt.legend()
|
||||
plt.show()
|
||||
|
||||
# Frequency of maximum difference between actual and truncated DTFT
|
||||
K_range = np.arange(1,200+1)
|
||||
max_diffs = np.zeros(K_range.shape)
|
||||
for K in K_range:
|
||||
X_K = (1 - a**(K+1) * np.exp(-1j * omega * (K+1))) / (1 - a*np.exp(-1j*omega))
|
||||
abs_diff = np.abs(X_K - h)
|
||||
max_diffs[K-1] = np.max(abs_diff)
|
||||
|
||||
plt.plot(K_range, max_diffs)
|
||||
plt.xlabel("K")
|
||||
plt.ylabel("Maximum Error")
|
||||
plt.show()
|
||||
|
||||
|
||||
|
||||
70
6th-Semester-Spring-2024/DSP/Labs/Lab-03/lab-03-2.py
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70
6th-Semester-Spring-2024/DSP/Labs/Lab-03/lab-03-2.py
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@ -0,0 +1,70 @@
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
import scipy.signal
|
||||
|
||||
n = np.arange(0,200)
|
||||
|
||||
a = 0.9
|
||||
|
||||
# $x[n] = a^n u[n]$
|
||||
x = n*a**n * np.heaviside(n-1, 1)
|
||||
|
||||
asums = np.zeros(len(n))
|
||||
|
||||
for i in range(len(n)):
|
||||
asums[i] = np.sum(x[0:i])
|
||||
|
||||
# Plot settings for $x[n]$
|
||||
plt.subplot(121)
|
||||
plt.plot(n, x)
|
||||
plt.xlabel("$n$")
|
||||
plt.ylabel("$x[n]$", rotation="horizontal")
|
||||
|
||||
# Plot settings for sum of $x[n]$
|
||||
plt.subplot(122)
|
||||
plt.plot(n, asums)
|
||||
plt.xlabel("$n$")
|
||||
plt.ylabel("$\sum_n x[n]$", rotation="horizontal")
|
||||
plt.show()
|
||||
|
||||
# Compute DTFT of $x[n]$
|
||||
X = scipy.signal.freqz((0, a), (1, -2*a, a*a))
|
||||
omega, h = X
|
||||
|
||||
# Set up plot for DTFT of $x[n]$
|
||||
plt.plot(omega, np.abs(h))
|
||||
plt.ylabel("Amplitude")
|
||||
plt.xlabel("Frequency [rad/sample]")
|
||||
plt.show()
|
||||
|
||||
# Set up plot of actual DTFT of $x[n]$ for comparisson with truncated DTFTs
|
||||
plt.plot(omega, np.abs(h), label="Actual DTFT")
|
||||
plt.ylabel("Amplitude")
|
||||
plt.xlabel("Frequency [rad/sample]")
|
||||
|
||||
# Calculate the truncated DTFTs of $x[n]$ as a function of $K$, $\sum_{n=-K}^K x[n] e^{-j\omega n}$
|
||||
for K in (3, 10, 20, 40):
|
||||
# Finite geometric series formula
|
||||
#X_K = ((a*np.exp(-1j*omega)) - (a*np.exp(-1j*omega))**(K+3)) / (1 - a*np.exp(-1j*omega))**2 - (K+2)*(a*np.exp(-1j*omega))**(K+2) / (1 - a*np.exp(-1j*omega))
|
||||
n_K = np.arange(-K,K+1)
|
||||
X_K = np.zeros(omega.shape, np.complex128)
|
||||
for n in n_K:
|
||||
x_n = n* a**n * np.heaviside(n-1, 1)
|
||||
X_K += x_n * np.exp(-1j * n * omega)
|
||||
plt.plot(omega, np.abs(X_K), label=f"Truncated DTFT ($K={K}$)")
|
||||
|
||||
plt.legend()
|
||||
plt.show()
|
||||
|
||||
# Frequency of maximum difference between actual and truncated DTFT
|
||||
K_range = np.arange(1,200+1)
|
||||
max_diffs = np.empty(K_range.shape)
|
||||
for K in K_range:
|
||||
X_K = ((a*np.exp(-1j*omega)) - (a*np.exp(-1j*omega))**(K+3)) / (1 - a*np.exp(-1j*omega))**2 - (K+2)*(a*np.exp(-1j*omega))**(K+2) / (1 - a*np.exp(-1j*omega))
|
||||
abs_diff = np.abs(X_K - h)
|
||||
max_diffs[K-1] = np.max(abs_diff)
|
||||
|
||||
plt.plot(K_range, max_diffs)
|
||||
plt.xlabel("K")
|
||||
plt.ylabel("Maximum Error")
|
||||
plt.show()
|
||||
39
6th-Semester-Spring-2024/DSP/Labs/Lab-03/lab-03-3.py
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39
6th-Semester-Spring-2024/DSP/Labs/Lab-03/lab-03-3.py
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@ -0,0 +1,39 @@
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
|
||||
omega_c = 0.4*np.pi
|
||||
omega = np.linspace(-np.pi, np.pi, 300)
|
||||
|
||||
# Inverse DTFT of Ideal Lowpass: $x[n] = {\omega_c \over \pi} \sinc(\omega_c n)$
|
||||
|
||||
# Plot $x[n]$ for values of $n$ between -50 and 50
|
||||
n = np.arange(-50,51)
|
||||
x = np.sinc(omega_c*n/np.pi)*omega_c/np.pi
|
||||
plt.stem(n,x)
|
||||
plt.ylabel("$x[n]$")
|
||||
plt.xlabel("$n$")
|
||||
plt.show()
|
||||
|
||||
# Plot the frequency response of $X(e^{j\omega})$
|
||||
f_response = np.compress(np.where(omega >= 0, True, False), omega)
|
||||
X_f_response = np.where(np.abs(f_response) <= omega_c, 1, 0)
|
||||
plt.plot(f_response, X_f_response)
|
||||
plt.ylabel("Amplitude")
|
||||
plt.xlabel("Frequency [rad/sample]")
|
||||
plt.show()
|
||||
|
||||
# Calculate truncated DTFTs of $x[n]$
|
||||
plt.plot(f_response, X_f_response, label="Actual DTFT of $x[n]$")
|
||||
for K in (10, 20, 30):
|
||||
n_K = np.arange(-K, K+1)
|
||||
X_K = np.zeros(f_response.shape, np.complex128)
|
||||
for n in n_K:
|
||||
x_n = np.sinc(omega_c*n/np.pi)*omega_c/np.pi
|
||||
X_K += x_n * np.exp(-1j*f_response*n)
|
||||
plt.plot(f_response, X_K, label=f"Truncated DTFT (K = {K})")
|
||||
|
||||
# Plot truncated DTFTs of $x[n]$
|
||||
plt.ylabel("Amplitude")
|
||||
plt.xlabel("Frequency [rad/sample]")
|
||||
plt.legend()
|
||||
plt.show()
|
||||
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8
6th-Semester-Spring-2024/DSP/Labs/Lab-04/lab-4.aux
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8
6th-Semester-Spring-2024/DSP/Labs/Lab-04/lab-4.aux
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@ -0,0 +1,8 @@
|
||||
\relax
|
||||
\@writefile{toc}{\contentsline {section}{\numberline {1}Results \& Discussion}{1}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}The Discrete Fourier Transform (DFT)}{1}{}\protected@file@percent }
|
||||
\newlabel{eqn:DFT_def}{{1}{1}}
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {1.2}The Z-Transform}{1}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {1.3}The Inverse Z-Transform}{1}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {section}{\numberline {2}Conclusions}{1}{}\protected@file@percent }
|
||||
\gdef \@abspage@last{2}
|
||||
72
6th-Semester-Spring-2024/DSP/Labs/Lab-04/lab-4.fdb_latexmk
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72
6th-Semester-Spring-2024/DSP/Labs/Lab-04/lab-4.fdb_latexmk
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@ -0,0 +1,72 @@
|
||||
# Fdb version 4
|
||||
["pdflatex"] 1712868153.51878 "lab-4.tex" "lab-4.pdf" "lab-4" 1712868154.22707 0
|
||||
"/usr/share/texlive/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df ""
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6th-Semester-Spring-2024/DSP/Labs/Lab-04/lab-4.pdf
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\documentclass{article}
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\usepackage{amsmath}
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\usepackage{amsfonts}
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\usepackage{graphicx}
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\usepackage{listings}
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\usepackage{caption}
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\usepackage{subcaption}
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\usepackage{float}
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\usepackage[margin=1in]{geometry}
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\title{}
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\author{Aidan Sharpe \& Elise Heim}
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\DeclareMathOperator{\sinc}{sinc}
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\begin{document}
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\begin{titlepage}
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\maketitle
|
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\end{titlepage}
|
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||||
\section{Results \& Discussion}
|
||||
|
||||
\subsection{The Discrete Fourier Transform (DFT)}
|
||||
Given a signal, $x[n]$, it's $N$-point DFT is given by
|
||||
\begin{equation}
|
||||
X_k = \sum_{n=0}^{N-1} x[n] W_N^{kn}
|
||||
\label{eqn:DFT_def}
|
||||
\end{equation}
|
||||
where $W_N = e^{-j2\pi/N}$. The discrete Fourier transform is the sampled version of the discrete time Fourier transform (DTFT), which is a continuous function. More specifically, the $N$-point DFT contains $N$ samples from the continuous DTFT.
|
||||
|
||||
For example, consider the signal $x[n] = (-1)^n$ for $0 \le n \le N-1$. By evaluating the sum shown in equation \ref{eqn:DFT_def}, the $N$-point DFT of $x[n]$ is found to be
|
||||
\begin{equation}
|
||||
X_k = {1 + e^{-j2\pi k} \over 1 + e^{-j2\pi k / N}}.
|
||||
\label{eqn:DFT_ex}
|
||||
\end{equation}
|
||||
|
||||
\subsection{The Z-Transform}
|
||||
Given a discrete signal, $x[n]$, its z-transform is given by
|
||||
\begin{equation}
|
||||
X(z) = \sum_n x[n] z^{-n}
|
||||
\end{equation}
|
||||
where $z$ is a complex variable.
|
||||
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\subsection{The Inverse Z-Transform}
|
||||
|
||||
\section{Conclusions}
|
||||
\end{document}
|
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||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
|
||||
N = 8
|
||||
n = np.arange(N)
|
||||
x = (-1.0)**n
|
||||
DFT_8_point = np.fft.fft(x, 8)
|
||||
DFT_9_point = np.fft.fft(x, 9)
|
||||
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||||
plt.stem(abs(DFT_8_point))
|
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plt.show()
|
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plt.stem(abs(DFT_9_point))
|
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plt.show()
|
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6th-Semester-Spring-2024/DSP/Labs/Lab-04/lab4_2.m
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|
||||
syms n a b
|
||||
|
||||
% Signals to take the z-transform of
|
||||
f_a = a^n;
|
||||
f_b = (n+1)*a^n*heaviside(n)
|
||||
f_c = a^n*cos(b*n)*heaviside(n)
|
||||
|
||||
% Take the z-transforms
|
||||
F_a = ztrans(f_a)
|
||||
F_b = ztrans(f_b)
|
||||
F_c = ztrans(f_c)
|
||||
11
6th-Semester-Spring-2024/DSP/Labs/Lab-04/lab4_3.m
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|
||||
syms z
|
||||
|
||||
% Given z-transforms
|
||||
F_a = z/(z+0.5)
|
||||
F_b = z^2/(z-0.8)^2
|
||||
F_c = z/((z+0.3)*(z+0.6)^2)
|
||||
|
||||
% Inverse z-transforms
|
||||
f_a = iztrans(F_a)
|
||||
f_b = iztrans(F_b)
|
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f_c = simplify(iztrans(F_c))
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File diff suppressed because it is too large
Load Diff
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@ -734,4 +734,39 @@ $$h[n] = -7(0.9)^n u[-n-1] - 6(0.8)^n u[n]$$
|
||||
{
|
||||
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.
|
||||
}
|
||||
|
||||
\chapter{The Discrete Fourier Transform}
|
||||
The discrete Fourier transform is the sampled DTFT. Since the DTFT is a continuous function, it cannot be analyzed in the same way as a discrete signal. The N-point DFT is defined as:
|
||||
$$X[k] = \sum_{n=0}^{N-1} x[n] W_N^{kn}$$
|
||||
where $W_N^k = e^{-j2\pi k / N}$.
|
||||
|
||||
\ex{}
|
||||
{
|
||||
Find the N-point DFT of $x[n] = u[n] - u[n-N]$.
|
||||
$$\sum_{n=0}^{N-1} s^n = \begin{cases}
|
||||
{1 - s^N \over 1 - s} & s \ne 1 \\
|
||||
N & s = 1
|
||||
\end{cases}$$
|
||||
|
||||
When $k = 0$:
|
||||
$$X[0] = \sum_{n=0}^{N-1} (1)^n = N$$
|
||||
When $k \ne 0$:
|
||||
$$X[k] = {1 - W_N^{kN} \over 1 - W_N^k} = 0$$
|
||||
|
||||
These points are zero because they correspond to the zero-crossings of the $\text{sinc}$ function, which is the Fourier transform of a unit pulse.
|
||||
}
|
||||
|
||||
\section{The Inverse DFT}
|
||||
\ex{}
|
||||
{
|
||||
The 10-point DFT of $x[n]$ is given as:
|
||||
$$X[k] = 2\delta[k] + 1, k \in [0,9] \cap \mathbb{Z}$$
|
||||
Find the 10-point signal $x[n]$
|
||||
}
|
||||
|
||||
\chapter{FIR Filter Design}
|
||||
Consider $h[n]$ with finite support $0 \le n \le M$. The filter is said to have an order $M$ with $M+1$ taps or coefficients. FIR systems are always BIBO stable. Focus will be on Type-1 FIR filter, where $M$ is even, so the number of taps is odd, and $h[n]$ is symmetric over the center coefficient $h\lt[M\over2\rt]$.
|
||||
|
||||
|
||||
|
||||
\end{document}
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||||
|
||||
@ -33,4 +33,7 @@
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||||
\contentsline {chapter}{\numberline {4}The Z-Transform}{19}{chapter.4}%
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||||
\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}%
|
||||
\contentsline {chapter}{\numberline {5}The Discrete Fourier Transform}{24}{chapter.5}%
|
||||
\contentsline {section}{\numberline {5.1}The Inverse DFT}{24}{section.5.1}%
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||||
\contentsline {chapter}{\numberline {6}FIR Filter Design}{25}{chapter.6}%
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||||
\contentsfinish
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||||
|
||||
BIN
6th-Semester-Spring-2024/DSP/Quizzes/answer_sheet_3.docx
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6th-Semester-Spring-2024/DSP/Quizzes/answer_sheet_3.docx
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\relax
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\documentclass[12pt, english]{article}
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bindingoffset=0.2in,
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|
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|
||||
\author{Aidan Sharpe}
|
||||
\title{Scott Catron In-Class Presentation Takeaways}
|
||||
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\doublespacing
|
||||
During the first part of his in-class presentation, guest speaker, Scott Catron discussed several startup-building strategies. He started by discussing the important qualities of startups and their foundes, using Silicon Valley as an exapmle. He stressed the importance of having repeat founders, since they have already made the common mistakes and learned from them. Mr. Catron also listed several other important team members such as co-founders and engineers. Beyond the team, he also highlighted the importance of choosing a location, and having a strong and broad support network.
|
||||
|
||||
In the second part of his presentation, Mr. Catron discussed "no-code" app-building platforms that enable rapid proof-of-concept app development. These platforms provide the necessary tools for user experience designers to get the look and feel of the app in the form of a high-res, functional prototype. He also discussed how AI and machine learning may play into app development for better, more complex high-res prototypes in the near future.
|
||||
|
||||
In the final portion of his presentation, Mr. Catron discussed statup accelerators. He again stressed the importance of having repeat founders, as accelerators understand the risks associated with a team of first-time founders.
|
||||
|
||||
As someone who has built mobile apps in the past, being able to test features without having to write any code sounds really nice. Testing new features, even basic ones, took a lot more headache than it would have if I had used a no-code approach. The one major downside that I can see with a codeless design approach is that the tools will not be able to incorporate advanced data structures and algorithms. I am also skeptical about the degree of customization that these platforms offer. Regardless, they remain valuable tools for prototyping user experience features.
|
||||
|
||||
|
||||
\end{document}
|
||||
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6th-Semester-Spring-2024/EnI/Group Work/ProblemSolution.pptx
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|
||||
#include <Arduino_LSM6DS3.h>
|
||||
#include <WiFiNINA.h>
|
||||
#include "data.h"
|
||||
|
||||
#define SAMPLES_PER_MINUTE 256
|
||||
#define SENSITIVITY 5
|
||||
#define THRESHOLD 0.04
|
||||
#define ID 1
|
||||
#define DEBUG false
|
||||
|
||||
float accelX, accelY, accelZ;
|
||||
|
||||
float calX, calY, calZ;
|
||||
unsigned char calibration;
|
||||
|
||||
float oneMinSamples[SAMPLES_PER_MINUTE];
|
||||
unsigned char oneMinSampleIndex;
|
||||
float oneMinMax;
|
||||
int startTime, deltaTime;
|
||||
|
||||
void setup()
|
||||
{
|
||||
// Serial only used in debug mode
|
||||
if (DEBUG)
|
||||
{
|
||||
Serial.begin(9600);
|
||||
while (!Serial);
|
||||
}
|
||||
|
||||
if (!IMU.begin())
|
||||
{
|
||||
if (DEBUG)
|
||||
{
|
||||
Serial.println("Failed to initialize IMU");
|
||||
}
|
||||
|
||||
while(1);
|
||||
}
|
||||
|
||||
// Setup LED
|
||||
pinMode(LEDR, OUTPUT);
|
||||
pinMode(LEDG, OUTPUT);
|
||||
pinMode(LEDB, OUTPUT);
|
||||
|
||||
oneMinSampleIndex = 0;
|
||||
calibration = 0;
|
||||
calX = 0.0;
|
||||
calY = 0.0;
|
||||
calZ = 0.0;
|
||||
}
|
||||
|
||||
void loop()
|
||||
{
|
||||
startTime = micros();
|
||||
// Calculate and update the square of the magnitude of acceleration
|
||||
if (IMU.accelerationAvailable())
|
||||
{
|
||||
IMU.readAcceleration(accelX, accelY, accelZ);
|
||||
accelX *= SENSITIVITY;
|
||||
accelY *= SENSITIVITY;
|
||||
accelZ *= SENSITIVITY;
|
||||
}
|
||||
|
||||
|
||||
// Calibration Stage
|
||||
if (calibration < 0xFF)
|
||||
{
|
||||
calibration++;
|
||||
if (DEBUG)
|
||||
{
|
||||
Serial.print("Calibrating (");
|
||||
Serial.print(calibration/2.55);
|
||||
Serial.print("%)\x0d");
|
||||
}
|
||||
digitalWrite(LEDR, LOW);
|
||||
digitalWrite(LEDG, LOW);
|
||||
digitalWrite(LEDB, HIGH);
|
||||
|
||||
// Sum all accelerations over the calibration period
|
||||
calX += accelX;
|
||||
calY += accelY;
|
||||
calZ += accelZ;
|
||||
|
||||
// Set the calibration values to the mean over the calibration period
|
||||
if (calibration == 0xFF)
|
||||
{
|
||||
calX /= calibration;
|
||||
calY /= calibration;
|
||||
calZ /= calibration;
|
||||
}
|
||||
}
|
||||
// Post-Calibration Stage
|
||||
else
|
||||
{
|
||||
// Adjust the result by the calibrated offset
|
||||
accelX -= calX;
|
||||
accelY -= calY;
|
||||
accelZ -= calZ;
|
||||
|
||||
// Add the sample to the sample history
|
||||
oneMinSamples[oneMinSampleIndex] = sqrt(accelX*accelX + accelY*accelY + accelZ*accelZ);
|
||||
|
||||
// Continuously calculate the maximum square of acceleration per minute
|
||||
oneMinMax = arrMaxf(oneMinSamples, SAMPLES_PER_MINUTE);
|
||||
if (DEBUG)
|
||||
{
|
||||
Serial.print("id:");
|
||||
Serial.print(ID);
|
||||
Serial.print(", equipmentStatus:");
|
||||
Serial.print((oneMinMax >= THRESHOLD) ? "Taken":"Free");
|
||||
Serial.print(", oneMinMax:");
|
||||
Serial.print(oneMinMax);
|
||||
Serial.print(", accelerationX:");
|
||||
Serial.print(accelX);
|
||||
Serial.print(", accelerationY:");
|
||||
Serial.print(accelY);
|
||||
Serial.print(", accelerationZ:");
|
||||
Serial.print(accelZ);
|
||||
Serial.print(", accelerationMag:");
|
||||
Serial.println(oneMinSamples[oneMinSampleIndex]);
|
||||
}
|
||||
if (oneMinMax >= THRESHOLD)
|
||||
{
|
||||
digitalWrite(LEDR, HIGH);
|
||||
digitalWrite(LEDG, LOW);
|
||||
digitalWrite(LEDB, LOW);
|
||||
}
|
||||
else
|
||||
{
|
||||
digitalWrite(LEDR, LOW);
|
||||
digitalWrite(LEDG, HIGH);
|
||||
digitalWrite(LEDB, LOW);
|
||||
}
|
||||
}
|
||||
|
||||
// Calculate the time in microseconds the function took
|
||||
deltaTime = micros() - startTime;
|
||||
|
||||
// 256 samples per minute
|
||||
delayMicroseconds(234375 - deltaTime);
|
||||
oneMinSampleIndex++;
|
||||
}
|
||||
@ -0,0 +1,11 @@
|
||||
// Calculate the maximum value in an array of floats
|
||||
float arrMaxf(float* arr, unsigned int length);
|
||||
|
||||
// Calculate the minimum value in an array of floats
|
||||
float arrMinf(float* arr, unsigned int length);
|
||||
|
||||
// Calculate the index of the maximum value in an array of floats
|
||||
unsigned int arrMaxfIndex(float* arr, unsigned int length);
|
||||
|
||||
// Calculate the index of the minimum value in an array of floats
|
||||
unsigned int arrMinfIndex(float* arr, unsigned int length);
|
||||
@ -0,0 +1,70 @@
|
||||
float arrMaxf(float* arr, unsigned int length)
|
||||
{
|
||||
float max = *arr;
|
||||
float current;
|
||||
|
||||
for (unsigned int i = 1; i < length; ++i)
|
||||
{
|
||||
current = *(arr+i);
|
||||
if (current > max)
|
||||
{
|
||||
max = current;
|
||||
}
|
||||
}
|
||||
|
||||
return max;
|
||||
}
|
||||
|
||||
float arrMinf(float* arr, unsigned int length)
|
||||
{
|
||||
float min = *arr;
|
||||
float current;
|
||||
|
||||
for (unsigned int i = 1; i < length; ++i)
|
||||
{
|
||||
current = *(arr+i);
|
||||
if (current < min)
|
||||
{
|
||||
min = current;
|
||||
}
|
||||
}
|
||||
|
||||
return min;
|
||||
}
|
||||
|
||||
unsigned int arrMaxfIndex(float* arr, unsigned int length)
|
||||
{
|
||||
float max = *arr;
|
||||
float current;
|
||||
unsigned int maxIndex = 0;
|
||||
|
||||
for (unsigned int i = 1; i < length; ++i)
|
||||
{
|
||||
current = *(arr+i);
|
||||
if (current > max)
|
||||
{
|
||||
max = current;
|
||||
maxIndex = i;
|
||||
}
|
||||
}
|
||||
|
||||
return maxIndex;
|
||||
}
|
||||
|
||||
unsigned int arrMinfIndex(float* arr, unsigned int length)
|
||||
{
|
||||
float min = *arr;
|
||||
float current;
|
||||
unsigned int minIndex = 0;
|
||||
|
||||
for (unsigned int i = 1; i < length; ++i)
|
||||
{
|
||||
current = *(arr+i);
|
||||
if (current < min)
|
||||
{
|
||||
min = current;
|
||||
minIndex = i;
|
||||
}
|
||||
}
|
||||
return minIndex;
|
||||
}
|
||||
@ -30,3 +30,101 @@ Color-brand association
|
||||
- 10% accent color
|
||||
=== Consumer's Culture ===
|
||||
Different cultures have different perceptions of color
|
||||
|
||||
== Make Teamwork Work ==
|
||||
*Group work*: working in a group of people on the same task
|
||||
*Team work*: creating an environment in which team member talents are facitlitated to achieve a common goal
|
||||
=== Teamwork Activity ===
|
||||
*BOSS*: describe the vision and details of a design to the secretary
|
||||
*SECRETARY*: communicate the vision to the designer and relay questions to the boss
|
||||
*DESIGNER*: draw the design and ask questions based on verbal instructions from the secretary
|
||||
=== Key Takeaways ===
|
||||
- Everyone has different strengths
|
||||
- Different roles suit different people better
|
||||
|
||||
|
||||
== Lunch & Learn "Unnecessary Inventions" ==
|
||||
|
||||
|
||||
|
||||
== Re-Investing Funds (Monday, March 25th, 2024) ==
|
||||
=== Fund Types ===
|
||||
- Prize money
|
||||
- Don't have to give up equity
|
||||
- Grants
|
||||
- Profit
|
||||
- Self-funds
|
||||
- Betting on yourself
|
||||
- Loans
|
||||
- Usually have interest payments
|
||||
- Family & Friend borrowing
|
||||
|
||||
=== Why is Re-Investing Important ===
|
||||
- Investors like to see you have skin in the game
|
||||
- Might not need to give up equity
|
||||
- Accelerate first to market strategy
|
||||
|
||||
=== Where to Use Funds ===
|
||||
- Product Development
|
||||
- Inventory for more sales
|
||||
- Cash flow
|
||||
- Improving products
|
||||
- Professional packaging
|
||||
- Legal/Formation Fees
|
||||
- IP protection
|
||||
- Legal calls for advice
|
||||
- LLC, Corporation, 501 C3, ...
|
||||
- Contracts
|
||||
- Marketing
|
||||
- Flyers, pens, magnets
|
||||
- (Boosted) Social media ads
|
||||
- SEO
|
||||
- Conferences
|
||||
- Getting More Hands
|
||||
- Fiverr/Upwork
|
||||
- Compensation contracts based on performance
|
||||
- Full time work
|
||||
- Interns
|
||||
- Complimentary Products
|
||||
- Cross sell to consumers
|
||||
- Similar products in different industries
|
||||
|
||||
|
||||
|
||||
== Thinking Outside the Box ==
|
||||
- Brainstormed in groups of 3/4 and 2/3 to come up with problem and solution statements for people that we may not relate to.
|
||||
|
||||
=== Vlad the Vampire ===
|
||||
- *PROBLEM*: Vlad can't be in the sun without getting bad sunburn or risk dying
|
||||
- *SOLUTION*: Heavy duty sunscreen that blocks out all sunlight from reaching Vlad's skin
|
||||
|
||||
=== Frankie the Monster ===
|
||||
|
||||
|
||||
|
||||
== Pitch Deck Workshop ==
|
||||
=== Common Components ===
|
||||
- Title
|
||||
- Problem
|
||||
- Highlight problem through story
|
||||
- Solution
|
||||
- Market Size/Demand
|
||||
- Offer
|
||||
- Competitor matrix
|
||||
- Team
|
||||
- Founders
|
||||
- Advisors
|
||||
- Milestones
|
||||
- Proof of Concept
|
||||
- Revenue Model
|
||||
- B2C
|
||||
- B2B
|
||||
- Freemium
|
||||
- Projections
|
||||
- Use of Funds
|
||||
- R&D
|
||||
- Inventory/supply
|
||||
- Re-investing
|
||||
- Marketing
|
||||
- Call to action
|
||||
- QR code to download app
|
||||
|
||||
BIN
6th-Semester-Spring-2024/SysCon/Labs/Lab7/SysConLab7.mlx
Normal file
BIN
6th-Semester-Spring-2024/SysCon/Labs/Lab7/SysConLab7.mlx
Normal file
Binary file not shown.
Binary file not shown.
25
6th-Semester-Spring-2024/SysCon/routh_hurwitz.m/license.txt
Normal file
25
6th-Semester-Spring-2024/SysCon/routh_hurwitz.m/license.txt
Normal file
@ -0,0 +1,25 @@
|
||||
Copyright (c) 2019, Varun Bhatia
|
||||
All rights reserved.
|
||||
|
||||
Redistribution and use in source and binary forms, with or without
|
||||
modification, are permitted provided that the following conditions are met:
|
||||
|
||||
* Redistributions of source code must retain the above copyright notice, this
|
||||
list of conditions and the following disclaimer.
|
||||
|
||||
* Redistributions in binary form must reproduce the above copyright notice,
|
||||
this list of conditions and the following disclaimer in the documentation
|
||||
and/or other materials provided with the distribution
|
||||
* Neither the name of nor the names of its
|
||||
contributors may be used to endorse or promote products derived from this
|
||||
software without specific prior written permission.
|
||||
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
|
||||
AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
|
||||
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
|
||||
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE
|
||||
FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
|
||||
DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
|
||||
SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
|
||||
CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
|
||||
OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
|
||||
OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
|
||||
341
6th-Semester-Spring-2024/SysCon/routh_hurwitz.m/routh_hurwitz.m
Normal file
341
6th-Semester-Spring-2024/SysCon/routh_hurwitz.m/routh_hurwitz.m
Normal file
@ -0,0 +1,341 @@
|
||||
%% Routh-Hurwitz Stability Criterion Table Generator V1.1
|
||||
|
||||
% Author: Varun Bhatia [bhatiav16@berkeley.edu]
|
||||
|
||||
% Date: July 28, 2019
|
||||
|
||||
% This function goes through the process of setting up a Routh-Hurwitz
|
||||
% table to determine information regarding the in/stability of a control
|
||||
% system given a closed/open-loop transfer function. More specifically, this
|
||||
% will solve for and output how many closed/open-loop poles are in the
|
||||
% right-half plane, the left-half plane, and on the jw-axis. Additionally,
|
||||
% this function will take into account and generate the Routh_Hurwitz table
|
||||
% given two special cases:
|
||||
% 1)First element of a row is 0.
|
||||
% 2)Entire Row is 0.
|
||||
|
||||
%Recall that this function operates under the assumption that you are
|
||||
%inputting a CLOSED LOOP TRANSFER FUNCTION. Later developments will
|
||||
%incorporate inputs of open loop forward transfer functions with unity
|
||||
%and/or nonunity feedback.
|
||||
|
||||
%Any usage of MATLAB functions is accredited to MathWorks.
|
||||
|
||||
%UPDATE 1.1: This program now takes into account UNITY feedback.
|
||||
|
||||
%%
|
||||
|
||||
function []= routh_hurwitz()
|
||||
|
||||
prompt_n = 'Enter highest power in numerator: ';
|
||||
num_hp = input(prompt_n);
|
||||
num_hp_added = num_hp + 1;
|
||||
num = zeros(1, num_hp_added);
|
||||
|
||||
for i = 1:num_hp+1
|
||||
num(i) = input('Enter coefficients, starting with the highest power: ');
|
||||
end
|
||||
|
||||
prompt_d = 'Enter highest power in denominator: ';
|
||||
den_hp = input(prompt_d);
|
||||
den_hp_added = den_hp + 1;
|
||||
den = zeros(1, den_hp_added);
|
||||
|
||||
for i = 1:den_hp+1
|
||||
den(i) = input('Enter coefficients, starting with the highest power: ');
|
||||
end
|
||||
|
||||
prompt_TF = 'Are you entering an open loop or closed loop TF? If closed loop type "C", if open loop type "O" with quotes around letter: ';
|
||||
TF_type = input(prompt_TF);
|
||||
|
||||
if TF_type == "O"
|
||||
|
||||
H = tf(num,den);
|
||||
K = 1;
|
||||
G = feedback(H,K)
|
||||
[~,d] = tfdata(G, 'v');
|
||||
den = d; %to acquire the new denominator coefficients from post-feedback
|
||||
|
||||
elseif TF_type == "C"
|
||||
|
||||
G = tf(num,den)
|
||||
|
||||
end
|
||||
|
||||
|
||||
|
||||
if den_hp == 0
|
||||
cols = 1;
|
||||
elseif den_hp == 1 | den_hp == 2
|
||||
cols = 2;
|
||||
elseif den_hp == 3 | den_hp == 4
|
||||
cols = 3;
|
||||
elseif den_hp == 5 | den_hp == 6
|
||||
cols = 4;
|
||||
elseif den_hp == 7 | den_hp == 8
|
||||
cols = 5;
|
||||
elseif den_hp == 9 | den_hp == 10
|
||||
cols = 6;
|
||||
elseif den_hp == 11 | den_hp == 12
|
||||
cols = 7;
|
||||
elseif den_hp == 13 | den_hp == 14
|
||||
cols = 8;
|
||||
end
|
||||
|
||||
%the matrix with proper number of rows and columns
|
||||
RH_Table = zeros(den_hp+1, cols);
|
||||
|
||||
|
||||
%set up the first element in row 1 & row 2 since MATLAB doesn't do 0 indexing
|
||||
if cols == 1
|
||||
RH_Table(1,1) = den(1);
|
||||
|
||||
elseif cols > 1
|
||||
RH_Table(1,1) = den(1);
|
||||
RH_Table(2,1) = den(2);
|
||||
|
||||
end
|
||||
|
||||
if cols == 1
|
||||
fprintf('\n *Only one element in the RH Table, thus, you have a stable system.* \n');
|
||||
|
||||
elseif cols > 1
|
||||
first_count = 3; %since added first element (element 1) above so now skip one and start at 3rd element for first row
|
||||
second_count = 4; %since added second element (element 2) above so now skip one and start at 4th element for second row
|
||||
|
||||
%filling in the first row of the RH table
|
||||
for i = 2:cols %start at 2 because filled in first element above
|
||||
if first_count <= den_hp_added
|
||||
RH_Table(1,i) = den(first_count);
|
||||
first_count = first_count + 2;
|
||||
else
|
||||
RH_Table(1,i) = 0;
|
||||
end
|
||||
end
|
||||
|
||||
%filling in the second row of the RH table
|
||||
for i = 2:cols %start at 2 because filled in first element above
|
||||
if second_count <= den_hp_added
|
||||
RH_Table(2,i) = den(second_count);
|
||||
second_count = second_count + 2;
|
||||
else
|
||||
RH_Table(2,i) = 0;
|
||||
end
|
||||
end
|
||||
|
||||
%now fill in remaining elements of RH Table with determinants
|
||||
X = zeros(2,2);
|
||||
|
||||
for i = 1:(den_hp-1)
|
||||
for j = 1:(cols-1)
|
||||
X(1,1) = RH_Table(i,1);
|
||||
X(2,1) = RH_Table(i+1,1);
|
||||
X(1,2) = RH_Table(i,j+1);
|
||||
X(2,2) = RH_Table(i+1,j+1);
|
||||
|
||||
RH_Table(i+2,j) = -det(X)/RH_Table(i+1,1);
|
||||
end
|
||||
end
|
||||
|
||||
RH_Table % to show full RH table
|
||||
|
||||
%now analyze the first column of the table for sign changes to
|
||||
%determine how many poles in the right half-plane.
|
||||
sign_changes = 0;
|
||||
for i = 1:(den_hp)
|
||||
if (RH_Table(i,1) * RH_Table(i+1,1) < 0)
|
||||
sign_changes = sign_changes + 1;
|
||||
end
|
||||
end
|
||||
|
||||
if sign_changes > 0
|
||||
fprintf(['\n *You have %d right-half poles and %d left-half poles.',...
|
||||
' Thus, you have an unstable system.* \n'],sign_changes,den_hp-sign_changes)
|
||||
else
|
||||
fprintf(['\n *You have %d right-half poles and %d left-half poles.',...
|
||||
' Thus, you have a stable system.* \n'],sign_changes,den_hp-sign_changes)
|
||||
end
|
||||
|
||||
end
|
||||
|
||||
|
||||
%Special Case #1: Check if there exists a full row of zeros. If so, the
|
||||
%following code will be utilized.
|
||||
|
||||
%first check if any array contains a row of zeros
|
||||
row_of_zeros = 0;
|
||||
tol = 1.e-6;
|
||||
for i = 1:(den_hp + 1)
|
||||
if abs(RH_Table(i,:) - 0) < tol
|
||||
row_of_zeros = 1;
|
||||
row = i;
|
||||
break; %break because you have identified a row of zeros
|
||||
end
|
||||
end
|
||||
|
||||
|
||||
%If the check shows that you do indeed have a row of zeros, the following
|
||||
%code will run
|
||||
|
||||
if (row_of_zeros == 1)
|
||||
row_above = RH_Table(row-1,:);
|
||||
aux_poly = poly2sym(row_above);
|
||||
diff_poly = diff(aux_poly);
|
||||
diff_poly_coeffs = flip(coeffs(diff_poly));
|
||||
|
||||
for j = 1:cols
|
||||
if numel(diff_poly_coeffs) < cols
|
||||
diff_poly_coeffs = [diff_poly_coeffs, 0]; %add zero to the end of coefficients to make sure dimensions match when adding back to table
|
||||
else
|
||||
break;
|
||||
end
|
||||
end
|
||||
|
||||
RH_Table(row,:) = diff_poly_coeffs; %replacing row in RH Table with the coefficients of row above, differentiated
|
||||
|
||||
X = zeros(2,2);
|
||||
|
||||
for i = row-1:(den_hp-1)
|
||||
for j = 1:(cols-1)
|
||||
X(1,1) = RH_Table(i,1);
|
||||
X(2,1) = RH_Table(i+1,1);
|
||||
X(1,2) = RH_Table(i,j+1);
|
||||
X(2,2) = RH_Table(i+1,j+1);
|
||||
|
||||
RH_Table(i+2,j) = -det(X)/RH_Table(i+1,1);
|
||||
end
|
||||
end
|
||||
|
||||
%Split up the sign changes process into two:
|
||||
%1) Check sign changes until two rows above row of zeros normally and
|
||||
%store these sign changes.
|
||||
%2) Check sign changes from one row above row of zeros down to bottom.
|
||||
%By symmetry, the right half poles must equal the left half poles, and
|
||||
%anything left over are jw-axis poles. Additionally, the row above the
|
||||
%row of zeros is required to be an even polynomial.
|
||||
|
||||
%1)
|
||||
first_sign_changes = 0;
|
||||
second_sign_changes = 0;
|
||||
for k = 1:(row-2)
|
||||
if (RH_Table(k,1) * RH_Table(k+1,1) < 0)
|
||||
first_sign_changes = first_sign_changes + 1;
|
||||
end
|
||||
end
|
||||
|
||||
%2)
|
||||
for k = row-1:den_hp
|
||||
if (RH_Table(k,1) * RH_Table(k+1,1) < 0)
|
||||
second_sign_changes = second_sign_changes + 1;
|
||||
end
|
||||
end
|
||||
|
||||
total_sign_changes = first_sign_changes + second_sign_changes;
|
||||
left_half_poles =((row-2)-first_sign_changes + second_sign_changes);
|
||||
jw_axis_poles = den_hp - (row-2) - (2*second_sign_changes);
|
||||
|
||||
RH_Table
|
||||
|
||||
if (total_sign_changes) > 0
|
||||
fprintf(['\n *Upon changes made to the RH Table, you now have %d right-half',...
|
||||
' poles and %d left-half poles. Additionally, you have %d jw-axis poles. Thus, you have an unstable',...
|
||||
' system.* \n'],total_sign_changes,left_half_poles,jw_axis_poles);
|
||||
elseif (total_sign_changes == 0)
|
||||
fprintf(['\n *Upon changes made to the RH Table, you now have %d',...
|
||||
' right-half poles and %d left-half poles. Additionally, you have %d jw-axis poles.',...
|
||||
' Thus, you have a marginally stable system.* \n'],total_sign_changes,left_half_poles, jw_axis_poles);
|
||||
end
|
||||
end
|
||||
|
||||
|
||||
%Special Case #2: Check if zero exists in first column and if so, form reciprocal
|
||||
%polynomial and redo process above to get determinants. Have a separate
|
||||
%method for this portion, however, for the purposes of File Exchange, I've
|
||||
%attached that portion below.
|
||||
|
||||
for i = 1:(den_hp+1)
|
||||
if RH_Table(i,1) == 0
|
||||
den = flip(den);
|
||||
if cols == 1
|
||||
RH_Table(1,1) = den(1);
|
||||
elseif cols > 1
|
||||
RH_Table(1,1) = den(1);
|
||||
RH_Table(2,1) = den(2);
|
||||
end
|
||||
|
||||
if cols == 1
|
||||
fprintf('\n *Only one element in the RH Table, thus, you have a stable system.* \n');
|
||||
elseif cols > 1
|
||||
first_count = 3;
|
||||
second_count = 4;
|
||||
for k = 2:cols
|
||||
if first_count <= den_hp_added
|
||||
RH_Table(1,k) = den(first_count);
|
||||
first_count = first_count + 2;
|
||||
else
|
||||
RH_Table(1,k) = 0;
|
||||
end
|
||||
end
|
||||
|
||||
for l = 2:cols
|
||||
if second_count <= den_hp_added
|
||||
RH_Table(2,l) = den(second_count);
|
||||
second_count = second_count + 2;
|
||||
else
|
||||
RH_Table(2,l) = 0;
|
||||
end
|
||||
end
|
||||
|
||||
X = zeros(2,2);
|
||||
for m = 1:(den_hp-1)
|
||||
for j = 1:(cols-1)
|
||||
X(1,1) = RH_Table(m,1);
|
||||
X(2,1) = RH_Table(m+1,1);
|
||||
X(1,2) = RH_Table(m,j+1);
|
||||
X(2,2) = RH_Table(m+1,j+1);
|
||||
|
||||
RH_Table(m+2,j) = -det(X)/RH_Table(m+1,1);
|
||||
end
|
||||
end
|
||||
|
||||
sign_changes = 0;
|
||||
for k = 1:(den_hp)
|
||||
if (RH_Table(k,1) * RH_Table(k+1,1) < 0) %if the element * next element is negative, must be a sign change
|
||||
sign_changes = sign_changes + 1;
|
||||
end
|
||||
end
|
||||
|
||||
RH_Table
|
||||
if sign_changes > 0
|
||||
fprintf(['\n *Upon changes made to the RH Table, you now have %d right-half',...
|
||||
' poles and %d left-half poles. Thus, you have an unstable',...
|
||||
' system.* \n'],sign_changes,den_hp-sign_changes)
|
||||
else
|
||||
fprintf(['\n *Upon changes made to the RH Table, you now have %d',...
|
||||
' right-half poles and %d left-half poles.',...
|
||||
' Thus, you have a stable system.* \n'],sign_changes,den_hp-sign_changes)
|
||||
end
|
||||
end
|
||||
end
|
||||
end %end special case #2%
|
||||
|
||||
|
||||
end
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
Loading…
Reference in New Issue
Block a user