FFT-induced spectral aliasing, fence effects, spectral leaks, spectral interference

FFT in the analysis of spectrum analysis, there will be the following four aspects of the error:

(1) Spectral aliasing: The Nyquist theorem is well known, so almost everyone knows that in order to keep the spectrum from aliasing, the spectral sampling spectrum is theoretically greater than or equal to the maximum frequency of the signal. What is the relationship between that and the time domain?

The inverse of the sampling period is the spectral resolution, and the inverse of the highest frequency is the sampling period.

Set the sampling number of N, sampling frequency FS, the highest frequency FH, so the spectral resolution f=fs/n, and FS>=2FH, so you can see the maximum frequency and spectral resolution is contradictory, improve the spectral resolution F, in the case of N determination will inevitably lead to the maximum frequency FH reduction; Increasing the maximum frequency FH will cause the increase of f, that is, the resolution becomes larger.

(2) Fence effect: Since DFT is only taking k=0,1,2,....... N-1, can only take the discrete value, if the spectrum between the large gap may be some of the intermediate information is lost, and FFT calculation DfT is inevitable, the solution is to increase the number of sampling n. Thus, the spectral interval becomes smaller and the probability of loss of information decreases.

In addition, a 0 increase allows more detailed observation of the signal on the frequency domain, but does not increase the spectral resolution.

(3) Spectrum leakage: is caused by the window function, is also the problem of computational capacity (using FFT with DFT must add Window function), time domain multiplication, frequency domain convolution, causing the spectrum distortion of the signal, only in rare cases, the spectrum leakage will not occur, most of the situation will cause leakage. such as x (n) =cos (2π/n), (n=0,1,2,3.....n-1,) n-point FFT will not leak, but 2N, or n+1, n+2, etc. will cause distortion, and cause distortion can be seen from the expression x (K) = convolution after the spectrum in 2π/n*k sampling value, so if it is a 2N DFT, for 2π/2n*k, the equivalent of the N-point DFT results in the middle of the value of a second sampling, and 2π/(n+2) *k, and the N-point FFT is completely different. solution, you can enlarge the width of the window function (time domain width, the frequency domain is narrow, (time domain has relativity), that is, the leakage of energy is small), or do not add a rectangular window function, you can add slow window function, also can let the leakage of energy smaller. because the leakage will be the expansion of the spectrum, so it may also cause the phenomenon of spectral aliasing, and the result of leakage is to reduce the spectral resolution.

Spectrum leakage will make the main line next to a lot of sidelobe, which will cause interference between the lines, more serious is the side lobe of the energy is strong to the side lobe or signal itself, this is the so-called interference between the spectrum.

FFT-induced spectral aliasing, fence effects, spectral leaks, spectral interference