Finding Embedded Harmonics w/ Fourier Series & Integrals

In summary, the embedded harmonics in a periodic function can be found using Fourier series and integrals. This can be demonstrated through the requirement that the function must be absolutely integrable.
  • #1
Oxymoron
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1) Why can't f(x) = 1, -&inf; < x < &inf; be represented as a Fourier integral?
Is it because it must be defined on a finite interval?

2) Could someone tell me how you find the embedded harmonics in a given periodic function using Fourier series and integrals? Either a quick demonstration or outline, or a link.

Many thanks.
 
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  • #2
1) Perhaps it is because the integral &int;_infinity^infinity |f(x)|dx does not exist.
 
  • #3
1) For the Fourier integral to exist, we require that [itex]\int_{-\infty}^\infty |f(x)| dx < \infty. [/itex] i.e. the function needs to be absolutely integrable, which no non-zero constant function satisfies.
 
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1. What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as a combination of sine and cosine waves. It is commonly used in signal processing and harmonic analysis to decompose a complex signal into simpler components.

2. How does a Fourier series help in finding embedded harmonics?

A Fourier series allows us to analyze a signal and identify its constituent harmonics by representing the signal as a sum of sine and cosine waves with different frequencies and amplitudes. This makes it easier to identify and isolate embedded harmonics within a complex signal.

3. What is the role of integrals in finding embedded harmonics with Fourier series?

Integrals are used to calculate the coefficients of the sine and cosine waves in a Fourier series. These coefficients represent the amplitude and phase of each harmonic in the signal. By integrating over one period of the signal, we can determine the values of these coefficients and therefore, identify the embedded harmonics.

4. Can Fourier series be used to find embedded harmonics in non-periodic signals?

No, Fourier series can only be used to analyze signals that are periodic. Non-periodic signals require other methods, such as the Fourier transform, to identify their embedded harmonics.

5. What are some applications of finding embedded harmonics with Fourier series and integrals?

This technique is commonly used in audio and image processing, as well as in the analysis of electric circuits and other periodic systems. It can also be used to study and predict the behavior of physical systems that exhibit periodic behavior, such as ocean tides and planetary orbits.

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