Finished DSP lab 4
This commit is contained in:
Aidan Sharpe
parent
df7d52f4b1
commit
98168e00bd
@ -7,7 +7,8 @@
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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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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}
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\begin{figure}[h]
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\center
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\includegraphics[width=0.5\textwidth]{Q9_point_DFT.png}
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\caption{The 9-point DFT of $x[n]$, where $N=8$}
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\label{fig:9_point_DFT}
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\end{figure}
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\subsection{The Z-Transform}
|
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Given a discrete signal, $x[n]$, its z-transform is given by
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\begin{equation}
|
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X(z) = \sum_n x[n] z^{-n}
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||||
\end{equation}
|
||||
where $z$ is a complex variable.
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||||
where $z$ is a complex variable. The Z-transform of any discrete signal is a continuous function. Therefore, any attempt to calculate the Z-transform with a digital computer requires sampling or symbolic math. Symbolic math is quite a powerful tool, and others have worked hard to implement the Z-transform as a symbolic function. In MATLAB, this function is \verb|ztrans|.
|
||||
\\
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\\
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The Z-transform of the following signals can be found very quickly using this method:
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$$x_1[n] = a^n u[n]$$
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$$x_2[n] = (n+1)a^n u[n]$$
|
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$$x_3[n] = a^n \cos(bn) u[n]$$
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$$\mathcal{Z}\{x_1[n]\} = -{z \over a - z}$$
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$$\mathcal{Z}\{x_2[n]\} = {az \over (a - z)^2} - {z \over a - 2}$$
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$$\mathcal{Z}\{x_3[n]\} = -{z(\cos(b) - z/a) \over a{z^2\over a^2} - {2z\cos(b) \over a} + 1}$$
|
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\subsection{The Inverse Z-Transform}
|
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Similarly, the inverse Z-transform also benefits from using a symbolic calculator. For example, the inverse Z-transform of the following can be found rapidly:
|
||||
$$\mathcal{Z}\{x_4[n]\} = {z \over z + 0.5}$$
|
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$$\mathcal{Z}\{x_5[n]\} = {z^2 \over (z-0.8)^2}$$
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$$\mathcal{Z}\{x_6[n]\} = {z \over (z+0.3)(z+0.6)^2}$$
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$$x_4[n] = (-1/2)^n$$
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$$x_5[n] = 2(4/5)^n (n-1)(4/5)^n$$
|
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$$x_6[n] = {50n(-3/5)^n - 100(-3/5)^n + 100(-3/10)^n \over 9}$$
|
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||||
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\section{Conclusions}
|
||||
Overall understanding tricks for calculating geometric sums is very helpful for calculating DFTs. It is also important to use odd-number-point DFTs only, as taking an even number of samples forces almost all values to go to zero. As for Z-transforms and inverse Z-transforms, knowing how to use symbolic computation makes finding them quite easy. Additionally, symbolic computation allows for the easy manipulation of how simplified or expanded the result is.
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\end{document}
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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)
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f_c = a^n*cos(b*n)*heaviside(n)
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f_a = a^n
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f_b = (n+1)*a^n
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f_c = a^n*cos(b*n)
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% Take the z-transforms
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F_a = ztrans(f_a)
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Reference in New Issue
Block a user