DFT: What is the physical meaning of the symmetry about the nyquist frequency?

In summary, when taking the FFT of a real signal, there will be a reflection around the Nyquist frequency due to the periodicity of the FFT and the requirement for complex conjugates. This reflection is simply a result of the way the FFT is calculated and does not have a physical meaning.
  • #1
jsparger
1
0
When I take the fft of a set of data and plot it, there is a reflection around the nyquist. Everybody knows this, but I would like to know what the physical meaning of the second half (the reflected half) is.

The real component is the same as the first half, and the imaginary component has the opposite sign.

I can use the first half of the data to reconstruct my signal, and I understand how it relates to frequency and phase angle, but I am not clear on the second half. Can I reconstruct my signal from this half of the data as well? Why do the frequencies have the opposite phase angle? These frequencies correspond with frequencies that could be aliasing if you extend the frequencies with the same spacing (for example, I am not clear whether you should take the info to be arranged as:

0 up to near the Nyquist, back down to near zero (so that the second half is some other expression of the first half of the data, i.e. it corresponds to the same frequencies;

or

0 up to near the Nyquist, rest is junk; (second half means nothing)

or

0 up to near the Nyquist (first half), then up to near the samplingFrequency (second half), as in the second set corresponds to higher frequencies that may be aliasing.

where k is the frequency spacing = samplingFrequency/numberOfSamples

Maybe this is just an artifact of the DFT. If not, could somebody explain to me what it means? Feel free to ignore the blather above, since I really have no clue what I am talking about. Just would like to know what the reflection corresponds to physically, and why the imaginary part is opposite.

Thanks
--John
 
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  • #3
jsparger said:
When I take the fft of a set of data and plot it, there is a reflection around the nyquist. Everybody knows this, but I would like to know what the physical meaning of the second half (the reflected half) is.

The real component is the same as the first half, and the imaginary component has the opposite sign.

It's an "artifact" of starting with a real signal. In order for a complex FT to represent a real signal, the coefficient of -f must be the complex conjugate of the coefficient of +f. And there's a periodicity in a FT, repeating the spectrum every ##f_{Max}##, i.e. every ##n## points.

Putting these things together, for an ##n## point digital FT, the value at ##(n/2)+m## is the same as the value at ##(n/2) + m - n = -(n/2) + m = -[(n/2) - m]## from the periodicity, and this is the complex conjugate of the value at ##(n/2) - m##. Thus the values at ##m## above the midpoint and ##m## below the midpoint are conjugates.

It's common practice to swap the upper and lower halves for plotting, so that the corresponding frequency values go from -(n/2) to (n/2) instead of 0 to n. MATLAB has a built-in function FFTSHIFT that does this.
 

1. What is DFT?

DFT stands for Discrete Fourier Transform. It is a mathematical algorithm used to convert a signal from its original domain (often time or space) to a representation in the frequency domain.

2. What is the physical meaning of symmetry in DFT?

In DFT, symmetry refers to the fact that the magnitude and phase of the complex numbers in the frequency domain have a specific pattern. Specifically, there is symmetry about the Nyquist frequency, which is half of the sampling frequency. This means that the frequencies above the Nyquist frequency are the mirror image of the frequencies below it.

3. What is the Nyquist frequency?

The Nyquist frequency is the highest frequency that can be accurately represented in a digital signal. It is equal to half of the sampling frequency, which is the rate at which the signal is measured or sampled.

4. Why is symmetry important in DFT?

Symmetry in DFT is important because it allows us to reduce the amount of computation needed. By taking advantage of the symmetry, we can eliminate redundant calculations and make the process more efficient.

5. How is symmetry about the Nyquist frequency used in DFT?

The symmetry about the Nyquist frequency is used in DFT to reduce the number of calculations needed to compute the transform. By only calculating the frequencies up to the Nyquist frequency and taking advantage of the symmetry, we can accurately represent the signal in the frequency domain with less computational effort.

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