# Fourier transform of a even/odd function

• Jalo
In summary, a Fourier transform of an even function is a mathematical operation that breaks down the function into sine and cosine waves with different frequencies and amplitudes. It differs from a Fourier transform of an odd function in that it is symmetric and only contains real-valued cosine waves. This is because the sine waves, which are imaginary, cancel out due to the symmetry of the function. The Fourier transform of an even function is commonly used in signal processing for filtering and noise reduction. An odd function can also have a non-zero Fourier transform, but it will be antisymmetric and only contain imaginary values at non-zero frequencies.
Jalo

## Homework Statement

Is the Fourier transform of a even/odd function also even/odd ?

## The Attempt at a Solution

So far this result seems to be true. I can't find a confirmation however...

Daniel.

As you do the transform on any odd function do you see a place where the even component is always zero? similarly for even functions?

## 1. What is a Fourier transform of an even function?

A Fourier transform of an even function is a mathematical operation that decomposes an even function into a series of sine and cosine waves with different frequencies and amplitudes.

## 2. How is a Fourier transform of an even function different from that of an odd function?

A Fourier transform of an even function is symmetric, meaning that it has the same values at positive and negative frequencies. On the other hand, a Fourier transform of an odd function is antisymmetric and has opposite values at positive and negative frequencies.

## 3. Why is the Fourier transform of an even function real-valued?

Since an even function is symmetric, its Fourier transform only contains cosine waves, which are real-valued. This is because the sine waves, which are imaginary, cancel out due to the symmetry of the function.

## 4. How can the Fourier transform of an even function be used in signal processing?

The Fourier transform of an even function is commonly used in signal processing for filtering and noise reduction. By decomposing the signal into its frequency components, we can manipulate the amplitudes and phases of these components to achieve the desired effect on the original signal.

## 5. Can an odd function have a non-zero Fourier transform?

Yes, an odd function can have a non-zero Fourier transform. However, since the transform will be antisymmetric, it will have a zero value at zero frequency and will only contain imaginary values at non-zero frequencies.

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