6th Semester Files
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6th-Semester-Spring-2024/DSP/Labs/Lab-02/dsp-lab-02.aux
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6th-Semester-Spring-2024/DSP/Labs/Lab-02/dsp-lab-02.aux
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\documentclass{IEEEtran}[journal]
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\usepackage{graphicx}
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\usepackage{listings}
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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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\maketitle
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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{Analysis of Amplitude Modulation}
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A message, $m(t)$, with a bandwidth, $B=2[\text{kHz}]$ modulates a cosine carrier with a frequency of 10[kHz]. The combined signal is $s(t) = m(t)\cos(20000\pi t)$. Using a Fourier transform on $s(t)$ reveals a maximum frequency at 12[kHz]. In fact, as seen in figure \ref{fig:AM_Fourier_Whitenoise}, by filling the band that $m(t)$ occupies with white noise, the Fourier transform of $s(t)$ contains white noise centered on the carrier frequency with twice the bandwidth of the original signal. The spike that occurs at 10[kHz] is the result of the original signal having a DC term and the carrier frequency having a value of 10[kHz].
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\begin{figure}[h]
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\includegraphics[width=0.48\textwidth]{lab2_1a.png}
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\caption{The Fourier transform of a white noise signal carried at 10[kHz]}
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\label{fig:AM_Fourier_Whitenoise}
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\end{figure}
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If instead, $m(t)$ had a triangular spectrum of amplitude 1, the spectrum of $s(t)$ will be two triangles touching at the base at 10[kHz] as seen in figure \ref{fig:carried_triangle}.
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\begin{figure}[h]
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\includegraphics[width=0.48\textwidth]{lab2_1b.png}
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\caption{The Fourier transform of a signal with a triangular spectrum carried at 10[kHz]}
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\label{fig:carried_triangle}
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\end{figure}
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\subsection{Periodicity and Sampling Frequency}
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Consider the signal $x(t) = \cos(2\pi t/7)$. Given the standard forms, $\cos(2\pi f t)$ and $\cos(\omega t)$, where f is linear frequency and $\omega$ is angular frequency, $f = {1\over7}$ and $\omega = {2\pi \over 7}$. Given a sampling frequency of 1[Hz], the sampling theorem is satisfied. To determine if a sampled signal is periodic, the condition $\omega N = 2 \pi r$, where $r$ is the smallest integer such that $N$ is an integer, must be satisfied.
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\section{Conclusions}
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\end{document}
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import matplotlib.pyplot as plt
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PI = np.pi
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fs = 20000
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t = np.arange(0, 1, 1/fs)
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fB = np.arange(0,2001,1)
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fC = 10000
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m = np.zeros_like(t)
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for f in fB:
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m += np.cos(2*PI*f*t)*(-abs(f-1000)/1000 + 1)
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c = np.cos(2*PI*fC*t)
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s = m*c
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plt.plot(np.arange(0,fs), abs(np.fft.fft(s)))
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plt.show()
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fs = 20000
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t = np.arange(0, 1, 1/fs)
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x = np.cos(2*PI*t/7)
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plt.plot(np.arange(0,fs), abs(np.fft.fft(s)))
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plt.show()
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import matplotlib.pyplot as plt
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def x(n):
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return (1/n) * np.heaviside(n, 0.5)
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n = np.arange(1,100)
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y = x(n)
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for i in range(len(y)):
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E_x_total += abs(y[i])**2
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E_x[i] = E_x_total
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fig, ax = plt.subplots()
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ax.scatter(n,y, label="x[n]")
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ax.scatter(n, E_x, c='red', label="E_x")
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ax.axhline(np.pi**2 / 6, linestyle='-')
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ax.legend()
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plt.show()
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Reference in New Issue
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