Power Electronics notes from March 19th
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|
||||
\section{Introduction}
|
||||
|
||||
\section{Results \& Discussion}
|
||||
|
||||
\subsection{Uniform Convergence}
|
||||
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.
|
||||
\\
|
||||
\\
|
||||
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 $x[0] = a$.
|
||||
\\
|
||||
\\
|
||||
Additionally, $x[n]$ takes the form of a geometric series, so its sum is given by
|
||||
\begin{equation}
|
||||
\sum_{n=m}^\infty s^n = {s^m \over 1-s}, |s| < 1.
|
||||
\end{equation}
|
||||
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}.
|
||||
|
||||
\begin{figure}[h]
|
||||
\center
|
||||
\includegraphics[width=\textwidth]{abs_sum_anun.png}
|
||||
\caption{The signal value of $x[n]=a^n u[n]$ and its absolute sum}
|
||||
\label{fig:abs_sum_anun}
|
||||
\end{figure}
|
||||
|
||||
|
||||
|
||||
\section{Conclusions}
|
||||
|
||||
\end{document}
|
||||
27
6th-Semester-Spring-2024/DSP/Labs/Lab-03/lab-03-1.py
Normal file
27
6th-Semester-Spring-2024/DSP/Labs/Lab-03/lab-03-1.py
Normal file
@ -0,0 +1,27 @@
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
|
||||
n = np.arange(0,200)
|
||||
|
||||
a = 0.9
|
||||
|
||||
# x[n] = a^n u[n]
|
||||
x = a**n * np.heaviside(n, 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()
|
||||
Reference in New Issue
Block a user