Question 23.18: Find the Fourier series representation of the sawtooth wavef......
Find the Fourier series representation of the sawtooth waveform described at the beginning of this section (see Figure 23.12).
Step-by-Step
This function is defined by f(t)=t,-\pi<t<\pi, \text { and has period } T=2 \pi \text {. } It is an odd function and hence a_n = 0 for all n. To find the b_n we must evaluate
\begin{aligned} b_n & =\frac{1}{\pi} \int_{-\pi}^\pi t \sin n t \mathrm{~d} t \\ & =\frac{1}{\pi}\left\{\left[\frac{-t \cos n t}{n}\right]_{-\pi}^\pi+\int_{-\pi}^\pi \frac{\cos n t}{n} \mathrm{~d} t\right\} \\ & =\frac{1}{n \pi}\{-\pi \cos n \pi-\pi \cos n \pi\} \end{aligned}since the last integral vanishes. Therefore,
b_n=-\frac{2}{n} \cos n \pi=-\frac{2}{n}(-1)^nWe conclude that b_1=2, b_2=-1, b_3=\frac{2}{3}, \ldots . Therefore f (t) has Fourier series
f(t)=2\left\{\sin t-\frac{1}{2} \sin 2 t+\frac{1}{3} \sin 3 t-\frac{1}{4} \sin 4 t+\frac{1}{5} \sin 5 t-\cdots\right\}Related Answered Questions
Question: 23.22
Verified Answer:
For the capacitor,
v_{\mathrm{o}}=\frac{i}{...
Question: 23.21
Verified Answer:
We find
\begin{aligned} c_n & =\frac{1}...
Question: 23.20
Verified Answer:
We note that v{t) is periodic with period T = 2π: ...
Question: 23.19
Verified Answer:
The function illustrated in Figure 23.24 is given ...
Question: 23.17
Verified Answer:
As usual we sketch the function first (Figure 23.1...
Question: 23.16
Verified Answer:
Assume that f (t) can be expressed in the form
[la...
Question: 23.15
Verified Answer:
As usual we sketch f (t), as shown in Figure 23.16...
Question: 23.14
Verified Answer:
As usual we sketch f (t) first as this often provi...
Question: 23.13
Verified Answer:
The function f (t) is shown in Figure 23.14. Note ...
Question: 23.12
Verified Answer:
We must evaluate
\int_{-\pi / \omega}^{\pi ...