Syncing to work on interview with entrepreneur essay
This commit is contained in:
Aidan Sharpe
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6th-Semester-Spring-2024/DSP/Labs/Lab-04/lab-4.aux
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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{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}
|
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\end{document}
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import numpy as np
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||||
import matplotlib.pyplot as plt
|
||||
|
||||
N = 8
|
||||
n = np.arange(N)
|
||||
x = (-1.0)**n
|
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DFT_8_point = np.fft.fft(x, 8)
|
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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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syms n a b
|
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||||
% Signals to take the z-transform of
|
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f_a = a^n;
|
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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)
|
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F_b = ztrans(f_b)
|
||||
F_c = ztrans(f_c)
|
||||
11
6th-Semester-Spring-2024/DSP/Labs/Lab-04/lab4_3.m
Normal file
11
6th-Semester-Spring-2024/DSP/Labs/Lab-04/lab4_3.m
Normal file
@ -0,0 +1,11 @@
|
||||
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)
|
||||
f_c = simplify(iztrans(F_c))
|
||||
Reference in New Issue
Block a user