Spectrum mixing caused by FFT, fence effect, spectrum leakage, and inter-spectrum interference

During spectrum analysis, FFT has the following four errors:

(1) spectrum mixing: The nequest theorem has been well known, so almost everyone knows that in order to avoid spectral mixing, theoretically the sampled spectrum is greater
The highest frequency of the signal. What is the relationship with the time domain?

The reciprocal of the sampling period is the spectrum resolution, and the highest frequency is the sampling period.

Set the number of sampling points to N, sampling frequency fs, the highest frequency FH, so the spectrum resolution F = FS/N, and FS> = 2fh, so we can see the most
High frequency conflicts with the resolution of the spectrum. while increasing the resolution of F, when n is determined, it will inevitably lead to the highest frequency.
The decrease of rate FH. Similarly, increasing the maximum frequency of FH will also lead to the increase of F, that is, the resolution will increase.

(2) fence effect: Because DFT only takes K = 0, 1, 2 ,....... n-1, can only get discrete value, if the spectrum is large, it may
The information is lost, while FFT is inevitable. The solution is to increase the number of sampling points n. In this way, the spectrum Interval
This reduces the probability of information loss.

In addition, adding 0 allows you to observe the signal in the frequency domain in more detail, but does not increase the spectrum resolution.

The effect of zeros on resolution is as follows:

(3) spectrum leakage: it is caused by the addition of a window function. It is also a problem of computing workload. (A window function must be added when FFT is used .)
The convolution results in spectrum distortion. Only in rare cases, spectrum leakage does not occur. In most cases
Leakage. For example, x (n) = cos (2 π/N), (n = 0, 1, 2, 3 ..... n-1,) N points of FFT will not leak, but 2n, or n + 1,
N + 2 and so on will cause distortion, which can be seen from the expression.X (K) = after convolution, the spectrum is in the 2 π/N * k
Sample value,Therefore, if it is 2n DFT, It is 2 π/2n * K, which is equivalent to a value of N points in the middle of each value of the DFT result.
2 π/(n + 2) * k is totally different from the N-point FFT. The solution is to expand the width of the window function (the width in the time domain, the frequency domain ).
The upper limit is narrow (relative to the time domain), that is, the leaked energy is smaller), or do not add a rectangular window function.
A slow window function can also change the leaked energy.

Because the leakage may expand the spectrum, it may also lead to spectrum mixing. The consequence of the leakage is to reduce the spectrum.
Resolution.

Spectrum leakage will cause many sides next to the main spectral line, which will cause interference between the spectral lines. The more serious is that the power of the two sides is not high.
It is also called inter-spectral interference.