# Property of real-valued Fourier transformation

• MHB
• mathmari
In summary, when a signal's Fourier transformation is real-valued, the signal is not necessarily real-valued. The signal can be imaginary, but the imaginary part must be odd and the real part must be even. These properties are known and can be derived. However, the even part may have an imaginary component, as seen in the example provided.
mathmari
Gold Member
MHB
Hey!

When it is given that a signal $x(t)$ has a real-valued Fourier transformation $X(f)$ then is the signal necessarily real-valued?

I have thought the following:

$X_R(ω)=\frac{1}{2}[X(ω)+X^{\star}(ω)]⟺\frac{1}{2}[x(t)+x^{\star}(−t)]=x_e(t) \\ X_I(ω)=\frac{1}{2i} [X(ω)−X^{\star}(ω)]⟺ \frac{1}{2i}[x(t)−x^{\star}(−t)]=−i⋅x_o(t)$

where $X_R(ω)$ and $X_I(ω)$ are the real and imaginary parts of $X(ω)$, and $x_e(t)$ and $x_o(t)$ are the even and odd parts of $x(t)$, respectively.So the odd part of $x$ is $0$ and the even one is real-valued, and so the signal $x(t)$ is real-valued.Is everything correct? Are the above properties known or do we have to derive them? (Wondering)

Hey mathmari!

Wiki lists such a property.
If $X(\omega)$ is real, then $x(t)$ is Hermitian. That is, $x(-t)=x^*(t)$.
It still means that $x(t)$ can be imaginary, but the imaginary part must be odd. Additionally the real part must be even. (Nerd)

Consider for instance $X(\omega)=2\pi\delta(\omega-1)$. It's real isn't it?
Its inverse Fourier transform is $x(t)=\cos t+i\sin t$.
As you can see the real part is even and the imaginary part is odd.
Furthermore, the odd part $x_o(t)$ is indeed $0$, but the even part $x_e(t)$ has an imaginary component. (Worried)

## 1. What is the real-valued Fourier transformation?

The real-valued Fourier transformation is a mathematical operation that decomposes a function into its constituent frequencies. It is commonly used in signal processing and data analysis to analyze the frequency components of a signal or dataset.

## 2. How is the real-valued Fourier transformation different from the complex-valued Fourier transformation?

The main difference between the two is that the complex-valued Fourier transformation allows for complex-valued input and output, while the real-valued Fourier transformation only deals with real numbers. This makes the real-valued Fourier transformation more suitable for analyzing real-world data that is often represented by real numbers.

## 3. What are the applications of the real-valued Fourier transformation?

The real-valued Fourier transformation has a wide range of applications in various fields, including signal processing, image processing, data compression, and data analysis. It is also used in solving differential equations and in quantum mechanics.

## 4. How is the real-valued Fourier transformation calculated?

The real-valued Fourier transformation is calculated by taking the integral of the function over the entire real line. This integral is then evaluated at different frequencies to determine the amplitude and phase of each frequency component in the original function.

## 5. What is the inverse real-valued Fourier transformation?

The inverse real-valued Fourier transformation is the mathematical operation that reconstructs a function from its frequency components. It is the reverse process of the real-valued Fourier transformation and is used to recover the original function from its frequency representation.

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