# Discrete Fourier transform mirrored?

• lordchaos
In summary, the discrete Fourier transform produces two peaks for a single sine wave because of aliasing due to sampling. The spectrum ends halfway through the transform and then reappears as a mirror image, with a frequency component at +\omega and -\omega. The use of this mirror image is for extracting the magnitude and phase of an oscillation. If the input is real, the second half can be discarded as it is a mirror image of the first half with even symmetry for the real part and odd symmetry for the imaginary part.
lordchaos
Why does a discrete Fourier transform seems to produce two peaks for a single sine wave? It seems to be the case that the spectrum ends halfway through the transform and then reappears as a mirror image; why is that? And what is the use of this mirror image? If I want to recover the frequency, phase and magnitude of an oscillation, do I need to use any data from this mirror image?

because

$$\cos(\omega t + \phi) = \frac{1}{2} \left( e^{+i \omega t} + e^{-i \omega t} \right)$$

so there is a frequency component at $+\omega$ and at $-\omega$.

because of aliasing due to sampling, negative frequencies are displayed in the upper half of the output of the DFT.

Thanks for that. Does this affect how I should extract the magnitude & phase from the transform? Or is it OK to throw the second half away for that purpose?

if your input to the DFT is real (i.e. they are complex numbers, but the imaginary part is zero), then yes, the second half is a mirror image of the first half. the real part (or the magnitude) of the DFT output has even symmetry and the imaginary part (or the phase) has odd symmetry.

## 1. What is a discrete Fourier transform (DFT) mirrored?

A discrete Fourier transform mirrored is a mathematical operation that converts a signal from its original time domain to a frequency domain. This means that it breaks down a signal, which is a function of time, into its component frequencies.

## 2. What is the purpose of using a DFT mirrored?

The purpose of using a DFT mirrored is to analyze and understand the frequency components of a signal. It can help identify the dominant frequencies in a signal, which can be useful in various applications such as signal processing, data compression, and filtering.

## 3. How is a DFT mirrored different from a regular DFT?

A DFT mirrored differs from a regular DFT in the way it displays the frequency components. In a regular DFT, the frequency components are displayed in a linear order, while in a DFT mirrored, they are displayed in a mirrored order, with the highest frequency at the center and decreasing frequencies on either side.

## 4. What are some common applications of DFT mirrored?

DFT mirrored has various applications in different fields, including audio and image processing, radar and sonar signal analysis, and data compression. It is also used in physics and engineering for analyzing signals in experiments and designing filters for noise reduction.

## 5. Are there any limitations to using DFT mirrored?

One limitation of using DFT mirrored is that it assumes the signal is periodic, which may not always be the case in real-world applications. It also requires a large amount of computation, which can be time-consuming. Additionally, DFT mirrored is not suitable for analyzing non-stationary signals, meaning signals that change over time.

• General Math
Replies
12
Views
1K
• General Math
Replies
5
Views
1K
• Calculus
Replies
4
Views
2K
• General Math
Replies
2
Views
3K
• General Math
Replies
2
Views
961
• General Math
Replies
2
Views
1K
• General Math
Replies
3
Views
2K
• Electrical Engineering
Replies
4
Views
934
• General Math
Replies
1
Views
2K