Calculating PSD from a Time History File
June 25, 2019
The Vibration Research software uses Welch’s method for PSD estimation. This method applies the fast Fourier transform (FFT) algorithm to the estimation of power spectra. In regards to his method, Peter D. Welch said, “[the] principal advantages of this method are a reduction in the number of computations and in required core storage, and convenient application in nonstationarity tests” .
The process begins with Gaussian-distributed time-domain input data or, more simply, a time history file. The following is a summary of the steps for calculating PSD from a time history file:
- Data is partitioned into frames of equal length in time; each frame is transformed into the frequency domain using the FFT
- Frequency-domain data is converted to power by taking the squared-magnitude (power value) of each frequency point; the squared-magnitudes for each frame are averaged
- The average is divided by the sample rate to normalize to a single Hertz
A few additional factors—windowing and overlapping specifically—influence the final PSD. Let’s look at each step in more detail.
Moving to the Frequency Domain
Figure 2.8 shows five seconds of Gaussian-distributed random vibration data in a time history file. It is difficult to extract any meaningful information from the graph other than the peak acceleration that appears to be around -30G.
To extract useful information from the vibration data, they must be viewed in the frequency domain. Moving to the frequency domain requires two calculations: the FFT and the PSD.
Fast Fourier Transform (FFT)
The FFT transforms data into acceleration versus frequency. No windowing, averaging, or normalizing functions are used to create an FFT graph.
An FFT graph is often used to monitor the frequency spectrum. Engineers can focus on changes in the frequency spectrum while viewing live data or analyzing a time history file. However, to view energy distribution across the frequency spectrum, the PSD must be calculated from the FFT.
Calculating the PSD from a Time History File
The steps to calculating PSD are as follows:
1. Divide the time history file into frames of equal time length
First, the time history file should be divided into frames of equal time length. The lines of resolution and sample rate can be used to determine the width of each frame. There are two samples per analysis line.
In Figure 2.9, the recorded time history was sampled at 8,192Hz and 4,096 lines of resolution. This resulted in a frame width of one second. If the lines of resolution were changed to 1,024, the frame width would be 0.25 seconds. This is an important step to remember when selecting an initial sample rate. Using a sample rate that is an exponential of 2 (2n) usually results in a PSD with lines spaced at a convenient interval.
2. Calculate the FTT for each frame after applying a window function
The FFT assumes data is an infinite series. Therefore, the starting and ending points of each frame are interpreted as if they are next to each other. Normally, random data is not an infinite series, so a window function needs to be applied before calculating the FTT.
Without a window function, the starting and ending points may be different and result in a transient spike between the two points. The transient spike would show up in the FFT as high-frequency energy.
The sharp transition between starting and ending points is a discontinuity. This discontinuity is called a spectral leakage and is reflected in the FFT calculation. Applying a window function removes the emphasis on the discontinuities and reduces the spectral leakage. Ideally, the real data would have identical starting and ending points in every frame of data. As this is not the case, we must minimize any discontinuity by using a window function (Figure 2.10).
There is more than one window function, so we must determine which function is most suitable for the current application. A window function is evaluated by two components: the side lobe and the main lobe. The Hanning window function is used in Figure 2.10, where the grey waveform is the original waveform and the orange waveform is the windowed data. The Hanning function has a very high and wide main lobe and low side lobes at practically zero. There is little to no discontinuity between the starting and ending points, which results in accurate frequency measurement.
The most commonly used window functions in vibration testing are the Hanning or Blackman functions. Both functions have a good frequency resolution, minimal discontinuity, and, therefore, minimal leakage. Other window functions may be appropriate for other applications. For a comprehensive list of commonly (and not-so-commonly) used window functions, visit the course on Window Functions for Signal Processing.
The windowed data is then used to calculate the FFT for each frame, thereby transforming the signal from the time domain into the frequency domain (see Figure 2.11.) This linear transformation gives us the ability to observe the frequency content of the time history waveform.
FFT is widely used in signal processing and analysis. Most importantly, it shows the frequencies in a waveform and their proportions. This information can be used to determine many things, including what frequencies are being excited during a section of time, the peak acceleration of each frequency inside of a windowed frame of data, the distribution of peaks, harmonic content, and more.
Certain weighting factors can also be applied to an FFT. If our goal was to simply generate an FFT, we might apply a weighting factor to ensure 1Gpk of the time domain data is equal to 1Gpk in the FFT. When the goal is to generate a PSD, the window function is normalized to preserve the input power.
3. Square the individual FFTs for each frame and find an average
Next, square the individual FFTs for each frame and then find the average squared-amplitude (Figure 2.12).
The PSD shows the average energy at a single frequency over a period of time. Initially, it will have a variance or “hashiness.” As more frames of similar data are included in the average, the overall variance will decrease, the accuracy will increase, and the PSD will appear much smoother.
The total amount of time included in a PSD is related to the averaging parameter called degrees-of-freedom (DOF). The more frames of data that are averaged together, the higher the DOF. Simply put, more FFTs will result in a better PSD. However, large numbers of FFTs require more time to collect and calculate.
4. Normalize the calculation to a single Hertz
Finally, take the average squared-FFT and divide it by the bandwidth of the analysis. By doing so, the data is normalized to a single Hertz and a power spectral density is determined (Figure 2.13). For acceleration, the resulting unit is G2/Hz.
Using the PSD, the response of the device under test is clear. As more frames of data are added to the PSD, the variance will continue to decrease and the PSD will become smoother.
For a more in-depth look at variance and methods of PSD smoothing, watch the webinar on Instant Degrees of Freedom, a patented feature from Vibration Research that quickly and effectively reduces the variance and creates a smooth PSD. This is the only mathematically-justified method for displaying a smooth PSD trace in a short period of time.
Additional Parameters to Consider
There is one additional technique that is often used, though not required, during PSD creation: overlapping. Overlapping is used to include more original data in the PSD and to generate more DOF within a period of time.
With a 0% overlap, each frame of data is completely separated. When frames are overlapped, some data in each frame are not accounted for due to the applied window functions. Additionally, for each frame of data included in the PSD, two degrees-of-freedom are calculated for the total average.
For example, if we create a PSD with 0% overlap, 120 DOF, an 8,192Hz sample rate, 4,096 lines of resolution, and a Hanning window function (resulting in 1-second frames), we will need to average 60 seconds worth of data to achieve 120 degrees of freedom.
With a 50% overlap, there would be 0.5 seconds between the start of each frame (each frame would still be 1 second in length.) Frames do not result in 2 DOF per FFT when they are overlapped. A 50% overlap would result in around 1.85 DOF per FFT; a 75% overlap would result in around 1.2 DOF per FFT. Therefore, for a 50% overlap, 120 DOF, 8,192Hz, 4,096 lines of resolution, Hanning window PSD, we can achieve the desired PSD in 64.8 frames in 32.4 seconds.
In the original example with 0% overlap, there were five frames of data, resulting in 5 FFTs. With a 50% overlap, as shown in Figure 2.14, the same section of data results in 9 FFTs.
Lines of Resolution
The last parameter to consider in the PSD calculation is the lines of resolution. Along with the sample rate, the lines of resolution value determines how far apart each analysis point is spaced on the PSD. A higher value will result in a more accurate PSD but requires a larger number of samples.
Many test standards require a specific lines of resolution value to be included inside a resonance to properly display the peak. If too few lines are used in a PSD, the result is similar to under-sampling a waveform: the distance between analysis points will be too great and the gap between will not be appropriately accounted for. A minimum requirement of three or more lines of resolution is needed to properly resolve a resonance.
Welch, P. D. (1967). The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms. IEEE Transactions on Audio and Electroacoustics, AU-15(2), 70-73.