Two explanations of frequency resolution

Frequency resolution can be understood as the minimum frequency interval on the frequency axis when DFT is used, where n is the number of sampling points, sampling frequency and sampling interval. So it is the time length t of the analog signal before sampling, so the longer the signal length, the better the frequency resolution. Is the frequency resolution higher with more sampling points?

The number of sampling points is related to the required frequency resolution. For example, if the machine speed is 3000 rpm =50Hz, if the estimated fault frequency to be analyzed is less than 8 times and the frequency resolution on the spectrum is required to be Δ f =1Hz, the sampling frequency and the number of sampling points are set as follows: the highest analysis frequency FM = 8.50hz = 400hz, sampling frequency FS = 2.56fm = 2.56400hz =/kloc-. Sampling number n = 2.56 (FM/δ f) = 2.56 (400 Hz/1Hz) =1024 spectral line number M=N/2.56= 1024/2.56=400.

According to FFT transform, we actually get the spectral line of 1024, but we know that there is a negative frequency in mathematical calculation, which is symmetrical. So in fact, we are concerned about the spectral lines corresponding to the positive frequency, which means that the positive frequency has 5 12 lines. Why do we usually say 400 lines in this case? Because in general, due to the influence of frequency aliasing and time domain truncation, it is usually considered as 400 lines.

In addition, the number of sampling points is not arbitrarily set, that is, the larger the better, and vice versa. For rotating machinery, it is necessary to meet the requirements of full-cycle sampling to eliminate frequency anomalies. Simply improving the resolution can't eliminate the frequency anomaly. Some people think that the longer the data, the better the signal length in time domain. In fact, it is unclear in some concepts, such as full-cycle sampling.

The lowest sampling frequency Fs without frequency aliasing is required to be twice as high as the highest analysis frequency Fm, and the reason why it is adopted is mainly related to the binary representation of the computer. Its main purpose is to avoid signal confusion and ensure that high-frequency signals are not distorted into low-frequency signals. The selection of sampling length t should first ensure that it can reflect the whole picture of the signal, and the transient signal should include the whole transient process; For periodic signals, in theory, it is enough to collect one periodic signal. Secondly, frequency resolution should be considered. When the maximum analysis frequency Fm is determined, the sampling length t is inversely proportional to the frequency resolution △f, that is, the longer the t is, the smaller the frequency resolution is. General analysis software sets the number of spectral lines m, the number of sampling points n = 2.56 m, and the sampling points commonly used in signal analysis are 5 12, 1024, 2048, 4096, etc. It is equivalent to the number of spectral lines such as 200, 400, 800, 1600, and the number of sampling points in spectral analysis is generally selected to the integer power of 2. △f=Fm/M, the more visible spectral lines m, the smaller the frequency resolution △f and the higher the frequency resolution. In the fault diagnosis of motor, in order to find that the sideband interval is the peak of the extremely pass frequency (generally below 1Hz), high resolution (below 1Hz) is often needed. Generally, 2 10 HZFM and 6400 spectral lines are selected. As for the whole period sampling, it is difficult to realize, and it will inevitably lead to leakage because of signal truncation. In order to avoid these mistakes, we should adopt the method of adding windows. Frequency resolution can also be understood as the ability of an algorithm (such as power spectrum estimation) to keep two very close spectral peaks in the original signal separate. This is an index used to compare and test the performance of different algorithms. In the signal system, the frequency domain pattern of a rectangular pulse with a width of n is a sinc function, and the width between two first-order zeros is 4 π/n. Because the truncation of the time domain signal is equivalent to multiplying the time domain signal by a window function, the frequency domain of the signal is equivalent to the convolution of a sinc function, that is, the frequency domain is modulated by the sinc function. According to the convolution property, the difference W0 between the circumferential frequencies of the two signals must be greater than 4 π/n ... Therefore, if the number n of data points increases, that is, the data length increases, the frequency resolution can also be improved, which is the same as the first explanation. At the same time, the influence of data truncation by window function should be considered, and of course the characteristics of window function should also be considered. If the spectrum of the window function is an impact function, it is equivalent to no truncation, but this situation does not exist. The main functions of the window are as follows:

1. The main lobe width b is the smallest (equivalent to 4π/N in the rectangular window, the width between two zero-crossing points in the frequency domain).

2. The maximum sidelobe peak value A is the smallest (so the sidelobe leakage is small and some high-frequency components are less lost)

3. The asymptotic decay speed d of the sidelobe peak is the largest (which also reduces the sidelobe leakage).

At present, there are four common time-frequency analysis methods, which are based on short-time Fourier transform, wavelet transform, Choi-Williams distribution and Hilbert-Huang transform. The experimental results show that Hilbert-Huang method has the highest frequency resolution.