Alternate expressions of Fourier series formula

In summary, the Fourier series formula F(t)=\frac{1}{2}a_{0}+\sum^{\infty}_{n=1}(a_{n}cos(nwt)+b_{n}sin(nwt)) can be expressed as F(t)=\frac{1}{2}a_{0}+\sum^{\infty}_{n=1}c_{n}cos(nwt-\phi_{n}). Relate the coefficients c_{n} to a_{n} and b_{n}.
  • #1
Esran
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Homework Statement



Show that the Fourier series formula [tex]F(t)=\frac{1}{2}a_{0}+\sum^{\infty}_{n=1}(a_{n}cos(nwt)+b_{n}sin(nwt))[/tex] can be expressed as [tex]F(t)=\frac{1}{2}a_{0}+\sum^{\infty}_{n=1}c_{n}cos(nwt-\phi_{n})[/tex]. Relate the coefficients [tex]c_{n}[/tex] to [tex]a_{n}[/tex] and [tex]b_{n}[/tex].

Homework Equations



We have the usual equations for the coefficients of a Fourier series.

The Attempt at a Solution



I'm really just checking the integrity of my solution here. I want to be sure I did not misunderstand the nature of [tex]\phi_{n}[/tex] or anything else.

Let [tex]n\in Z^{+}[/tex]. Pick [tex]\phi_{n}[/tex] such that [tex]\frac{a_{n}}{cos(\phi_{n})}=\frac{b_{n}}{sin(\phi_{n})}[/tex]. We know we can do this since we could just choose [tex]\phi_{n}=nw_{0}t[/tex], where [tex]w_{0}=2 \pi f[/tex], the fundamental frequency of the Fourier series. Let [tex]c_{n}=\frac{a_{n}}{cos(\phi_{n})}=\frac{b_{n}}{sin(\phi_{n})}[/tex].

[tex]F(t)=\frac{1}{2}a_{0}+\sum^{\infty}_{n=1}(a_{n}cos(nwt)+b_{n}sin(nwt))[/tex]
[tex]F(t)=\frac{1}{2}a_{0}+\sum^{\infty}_{n=1}(c_{n}cos(\phi_{n})cos(nwt)+c_{n}sin(\phi_{n})sin(nwt))[/tex]
[tex]F(t)=\frac{1}{2}a_{0}+\sum^{\infty}_{n=1}(c_{n}cos(nwt-\phi_{n}))[/tex]

Which completes the problem.
 
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  • #2
Your proof seems fine, but the one thing I would point out is that φn is supposed to be a constant. So you shouldn't pick φn = nω0t, because that would make the phase constant a linear function of time. Instead, just solve the equation
[tex]\frac{a_n}{\cos\phi_n} = \frac{b_n}{\sin\phi_n}[/tex]
for φn, getting
[tex]\phi_n = \tan^{-1}\frac{b_n}{a_n}[/tex]
(I'll leave the investigation of the case an=0 to you :wink:) Using that value of φn, you can define cn the same way you did, and the rest of the proof should be unchanged.
 
  • #3
Thanks!
 

Related to Alternate expressions of Fourier series formula

What is an alternate expression of Fourier series formula?

An alternate expression of Fourier series formula is a mathematical representation of a periodic function in terms of a sum of sine and cosine functions with different amplitudes and frequencies. It is used to decompose a complex function into simpler components.

How is the alternate expression of Fourier series formula different from the standard formula?

The alternate expression of Fourier series formula involves using different coefficients and a different form of the sine and cosine functions. It is typically used to solve specific types of problems, such as those involving even or odd functions.

What is the purpose of using an alternate expression of Fourier series formula?

The purpose of using an alternate expression of Fourier series formula is to simplify the representation of a function and make it easier to analyze. It can also be used to find the Fourier coefficients of a function more efficiently.

What are some common applications of the alternate expression of Fourier series formula?

The alternate expression of Fourier series formula is commonly used in signal processing, image processing, and data compression. It is also used in various fields of science and engineering to analyze and model periodic phenomena.

How can I calculate the coefficients in the alternate expression of Fourier series formula?

The coefficients in the alternate expression of Fourier series formula can be calculated using various methods, such as the Euler formula, the Fourier sine and cosine series, or the complex exponential form. The choice of method depends on the properties of the function being analyzed and the desired accuracy of the coefficients.

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