Syncing to work on interview with entrepreneur essay
This commit is contained in:
Aidan Sharpe
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|
||||
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Load Diff
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@ -734,4 +734,39 @@ $$h[n] = -7(0.9)^n u[-n-1] - 6(0.8)^n u[n]$$
|
||||
{
|
||||
For FIR systems, if the Z-transform does not converge at $|z|= 0$ or $|z| = \infty$, they are not considered poles, because only IIR systems can have poles.
|
||||
}
|
||||
|
||||
\chapter{The Discrete Fourier Transform}
|
||||
The discrete Fourier transform is the sampled DTFT. Since the DTFT is a continuous function, it cannot be analyzed in the same way as a discrete signal. The N-point DFT is defined as:
|
||||
$$X[k] = \sum_{n=0}^{N-1} x[n] W_N^{kn}$$
|
||||
where $W_N^k = e^{-j2\pi k / N}$.
|
||||
|
||||
\ex{}
|
||||
{
|
||||
Find the N-point DFT of $x[n] = u[n] - u[n-N]$.
|
||||
$$\sum_{n=0}^{N-1} s^n = \begin{cases}
|
||||
{1 - s^N \over 1 - s} & s \ne 1 \\
|
||||
N & s = 1
|
||||
\end{cases}$$
|
||||
|
||||
When $k = 0$:
|
||||
$$X[0] = \sum_{n=0}^{N-1} (1)^n = N$$
|
||||
When $k \ne 0$:
|
||||
$$X[k] = {1 - W_N^{kN} \over 1 - W_N^k} = 0$$
|
||||
|
||||
These points are zero because they correspond to the zero-crossings of the $\text{sinc}$ function, which is the Fourier transform of a unit pulse.
|
||||
}
|
||||
|
||||
\section{The Inverse DFT}
|
||||
\ex{}
|
||||
{
|
||||
The 10-point DFT of $x[n]$ is given as:
|
||||
$$X[k] = 2\delta[k] + 1, k \in [0,9] \cap \mathbb{Z}$$
|
||||
Find the 10-point signal $x[n]$
|
||||
}
|
||||
|
||||
\chapter{FIR Filter Design}
|
||||
Consider $h[n]$ with finite support $0 \le n \le M$. The filter is said to have an order $M$ with $M+1$ taps or coefficients. FIR systems are always BIBO stable. Focus will be on Type-1 FIR filter, where $M$ is even, so the number of taps is odd, and $h[n]$ is symmetric over the center coefficient $h\lt[M\over2\rt]$.
|
||||
|
||||
|
||||
|
||||
\end{document}
|
||||
|
||||
@ -33,4 +33,7 @@
|
||||
\contentsline {chapter}{\numberline {4}The Z-Transform}{19}{chapter.4}%
|
||||
\contentsline {section}{\numberline {4.1}Z Transform Transfer Functions}{20}{section.4.1}%
|
||||
\contentsline {section}{\numberline {4.2}The Inverse Z-Transform}{22}{section.4.2}%
|
||||
\contentsline {chapter}{\numberline {5}The Discrete Fourier Transform}{24}{chapter.5}%
|
||||
\contentsline {section}{\numberline {5.1}The Inverse DFT}{24}{section.5.1}%
|
||||
\contentsline {chapter}{\numberline {6}FIR Filter Design}{25}{chapter.6}%
|
||||
\contentsfinish
|
||||
|
||||
Reference in New Issue
Block a user