Problems on Fourier Series

1. For problem IX.3 #8, how many terms of the cosine series do you need in order to get the error under 0.01? How many terms of the sine series do you need to get the error under the same amount? Graph the error in each case.

2. Suppose that a spacecraft near Neptune has measured a quantity A and sent it to earth in the form of a periodic signal A cos$\pi$t of amplitude A. On its way to earth, the signal picks up periodic noise, containing only second and higher ``degree'' Fourier terms. Suppose that one period of the signal h(t) actually received on earth is graphed below. Determine the signal that the spacecraft originally sent and hence the value A of the measurement.¹

   

3. Below is a graph of a function f of period 2$\pi$, and a table containing some y values, measured from the graph at five equal time intervals. (The grid on the graph is drawn with lines every $\pi$/2 on the x-axis.)
   

   t	- $\pi$	- $\pi$/2	0	$\pi$/2	$\pi$
   f (t)	-2.2	2.6	2.2	3.0	-2.2

   (a)

   Use the trapezoid rule to find a₀ through a₂, b₁, and b₂.

   (b)

   Graph your Fouier polynomial and compare it to the original graph.

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Problems on Fourier Series

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Footnotes

... measurement.¹

answer is $\approx$ 27