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Find the complex Fourier series representations of the following signals without explicitly calculating Fourier integrals. What is the signal's period in each case?
Find the Fourier series representation for the following periodic signals . For the third signal, find the complex Fourier series for the triangle wave without performing the usual Fourier integrals. Hint: How is this signal related toone for which you already have the series?
We can learn about phase distortion by returning to circuits and investigate the following circuit .
fourier2.m
might be
useful.Often, we want to approximate a reference signal by a
somewhat simpler signal. To assess the quality of anapproximation, the most frequently used error measure is
the mean-squared error. For a periodic signal
$s(t)$ ,
$$\epsilon ^{2}=\frac{1}{T}\int_{0}^{T} (s(t)-\tilde{s}(t))^{2}\,d t$$ where
$s(t)$ is the reference signal and
$\tilde{s}(t)$ its approximation. One convenient way of findingapproximations for periodic signals is to truncate their
Fourier series.
$$\tilde{s}(t)=\sum_{k=-K}^{K} {c}_{k}e^{i\frac{2\pi k}{T}t}$$ The point of this problem is to analyze whether thisapproach is the best (
The daily temperature is a consequence of several effects, one of them being the sun's heating. If thiswere the dominant effect, then daily temperatures would be proportional to the number of daylight hours. The plot shows that the average daily high temperature does not behave that way.
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