• 基本运算：FFT

FFT是频谱分析的基础，以下的四种分析功能均以FFT变换的结果为基础，再进行二次处理得到。

FFT变换的结果为复数，含有原始信号的相位信息。

• 自功率谱：auto power spectrum，Gxx 1.9.5以后版本的APS带声学记权功能 To compute the spectra listed above, follow these steps:

Step 1

A window is applied:

x(k) = w(k) x(k)’

where x(k)’ is the original data and x(k) is the data used for a Fourier transform.

Step 2

The FFT is applied to x(t) to compute Sx

Now, periodogram method is used to compute the spectra with area correction

Step 3.1

Power Spectrum Sxx = Sx Sx* / (LineCorr)2

Step 3.2

Power Spectral Density = Sx Sx* T / AreaCorr

Step 3.3

Energy Spectral Density = Sx Sx* T2 / AreaCorr

Step 3.4

RMS Spectrum =

where T is the time duration of the capture. The symbol * is for complex conjugation. AreaCorr is a factor for area correction, which is defined as:

N is the total number of the samples and w(k) is window function.

• 互功率谱：cross power spectrum，Gxy To compute the cross-power spectral density Gxy between channel x and channel y:

Step 1, compute the Fourier transform

Step 2,   compute the instantaneous cross power spectral density

Sxy = Sx*   Sy T

Step 2,   average the M frames of Sxx to get averaged PSD Gxx

Gxy’ = Average (Sxy)

Step 3,   Do area correction and double the value for single-sided spectra

Gxy = 2 Gxy’ / AreaCorr

• 传递函数(transfer function)：FRF，Hxy The cross power spectrum method is used for estimating the transfer function between channel x and channel y. The equation is:

where Gxy is the averaged cross-spectrum between the input channel x and output channel y. Gxx is the averaged auto-spectrum of the input. Either power spectrum, power spectral density or energy spectral density can be used here because of the linear relationship between input and output.

This approach will reduce the effect of the noise at the output measurement end, as shown below. The transfer function has a complex data format. Your can view it in real, imaginary, magnitude or phase display format.

• 相关系数: coherence，Coh(x,y) The coherence function is defined as:

where Gxy is the averaged cross-spectrum between the input channel x and output channel y. Gxx and Gyy are the averaged auto-spectrum of the input and output. Either power spectrum, power spectral density or energy spectral density can be used here because of the linear relationship between input and output.

When the averaging number is 1, coherence function has a meaningless result of 1.0 due to the estimation error of the coherence function.

The coherence function is a non-dimensional real function in the frequency domain. You can only view it in the real format.

Note: When describing a system with input x and output y as shown above, some people are used to a notation Hyx instead of Hxy. The most Most DSA products follow the convention used in the reference books listed before. Hxy stands for a transfer function with input x and output y.