DSP lab 4 part 1 done
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\usepackage[margin=1in]{geometry}
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\title{}
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\title{Discrete Fourier Transforms and Z-Transforms}
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\author{Aidan Sharpe \& Elise Heim}
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@ -26,16 +26,43 @@
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\subsection{The Discrete Fourier Transform (DFT)}
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Given a signal, $x[n]$, it's $N$-point DFT is given by
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\begin{equation}
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X_k = \sum_{n=0}^{N-1} x[n] W_N^{kn}
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X_k = \sum_{n=0}^{N-1} x[n] W_N^{kn},
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\label{eqn:DFT_def}
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\end{equation}
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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.
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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
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\\
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\\
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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} as a truncated geometric series, the $N$-point DFT of $x[n]$ can be found. All truncated geometric series are evaluated as
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\begin{equation}
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X_k = {1 + e^{-j2\pi k} \over 1 + e^{-j2\pi k / N}}.
|
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\label{eqn:DFT_ex}
|
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\sum_{k=0}^{n-1} a r^k =
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\begin{cases}
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an & r = 1 \\
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a\left({1 - r^n \over 1 - r}\right) & r \ne 1
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\end{cases},
|
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\end{equation}
|
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where $r$ is the common ratio between adjacent terms. For the $N$-point DFT of $x[n]$, the common ratio is $-W_N^k$, which takes a value of 1 for $k = {N\over2}$. Therefore, the $N$-point DFT of $x[n]$ is
|
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\begin{equation}
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X[k] = \begin{cases}
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N & k = {N \over 2} \\
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\left({1 - (-W_N^k)^N \over 1 - (-W_N^k)}\right) & k \ne {N \over 2}
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\end{cases}.
|
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\label{eqn:DFT_N_point}
|
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\end{equation}
|
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The $N$-point DFT of $x[n]$, where $N=8$ is seen in figure \ref{fig:N_point_DFT}. It only has a non-zero value for $k={N\over2}=4$. This is the case for all even-number-point DFTs. Therefore only odd-number-point DFTs should be used.
|
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\begin{figure}[h]
|
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\center
|
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\includegraphics[width=0.5\textwidth]{N8_point_DFT.png}
|
||||
\caption{The $N$-point DFT of $x[n]$, where $N=8$}
|
||||
\label{fig:N_point_DFT}
|
||||
\end{figure}
|
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For example, the 9-point DFT of $x[n]$, where $N=8$ is seen in figure \ref{fig:9_point_DFT}. While equation \ref{eqn:DFT_N_point} cannot be used because there are a different number of samples for the DFT and the input signal, the overall DFT is more useful than the 8-point DFT.
|
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\begin{figure}[h]
|
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\center
|
||||
\includegraphics[width=0.5\textwidth]{Q9_point_DFT.png}
|
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\label{fig:9_point_DFT}
|
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\end{figure}
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|
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|
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\subsection{The Z-Transform}
|
||||
Given a discrete signal, $x[n]$, its z-transform is given by
|
||||
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@ -4,21 +4,36 @@ import matplotlib.pyplot as plt
|
||||
N = 8
|
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|
||||
# Analytic DFT
|
||||
'''
|
||||
k = np.arange(N)
|
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W_N = np.exp(-2j*np.pi*k/N)
|
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plt.stem(k, -W_N)
|
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plt.xlabel("$k$")
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plt.ylabel("$W_N$")
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plt.show()
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'''
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# N=8
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n = np.arange(N)
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x = (-1.0)**n
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# 8-point DFT, N=8
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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.savefig("N8_point_dft.png")
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plt.close()
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# 9-point DFT, N=8
|
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DFT_9_point = np.fft.fft(x, 9)
|
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plt.stem(abs(DFT_9_point))
|
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plt.savefig("Q9_point_dft.png")
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plt.close()
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# 9-point DFT, N=9
|
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N = 9
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n = np.arange(N)
|
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x = (-1.0)**n
|
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DFT_9_point = np.fft.fft(x, 9)
|
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plt.stem(abs(DFT_9_point))
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plt.savefig("N9_point_dft.png")
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plt.show()
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Reference in New Issue
Block a user