Understanding FFTs

The Fast Fourier Transform is the modern tool used to extract frequency information out of a digitized waveform.

The process has limitations based on windowing, sample rate, and the number of samples collected.

The highest frequency that can be measured in a signal requires that at least twice the number of samples be collected. This is described as the Nyquist frequency.

HGL uses 1024 sample FFTs (1k) for most processing. Other processes in the background use larger FFTs and there are options to customize the size of the displayed FFT. However, HGL has found that using a 1k FFT allows for a good balance in response time and signal accuracy.

Sample Rate Explanation

The blue line in the image above represents a waveform that is being measured. The red lines show that the data is being sampled at regular intervals and the circle is the measured value.

The measured value is actually a digitized value called an ADC Count. The accuracy of the measured amplitude is based on the gain and the resolution capability of the Analog to Digital Converter (ADC). Because of this, the exact amplitude of an analog signal will never be perfectly measured, but it can be very close. For instance, the V4 Dragonfly system has a base resolution of ±10,000 mV and uses a 24-bit ADC. At a gain of 1, this results in a minimum resolution of 0.00119 mV signal that can be detected if there was no noise in the signal. In the picture below, the green trace represents the ADC quantizing of the blue analog signal.

When configuring a test in the HGL Software, users specify the acquisition BANDWIDTH. This represents the highest frequency that the system can reliably measure. The default sample rate is 2.5 times faster than the bandwidth. So, data sampled at 25,000 Hz will give a useable measurement range of 10,000 Hz. 

Real Time FFT

The Hawkeye Client is where users can view the frequency content of a measured signal. The real time FFTs are always reported as a 1024 FFT. This means that it takes 1024 samples to create a single FFT. For example, given a sample rate of 25,000 Hz, a 1k FFT will be created every 0.04096 seconds. 

In the Real Time displays, HGL offers two different FFTs that users will primarily interact with:

  • Real Time 1k (1024 point) FFT.
  • Safety FFT (typically an 8k (8192 points) FFT).

The difference in FFT sizes results in some key differences:

  • The Real Time FFT has a faster response time and will measure higher amplitudes for signals that quickly change.
  • The Safety FFT has finer frequency resolution making it better at distinguishing between two peaks that are close together.

When the two different FFTs are show together in terms of sample size, it becomes readily apparent why there is a difference between the two.

An important technical term is Bin Size, or the effective frequency resolution of an FFT. As described above, an 8k FFT has finer resolution than a 1k FFT, but take more time to create. To calculate the bin size, divide the sample rate by the size of the FFT. At a sample rate of 25000 Hz, the bin size of a 1024 (1k) FFT is 24.4 Hz and the bin size of an 8192 (8k) FFT is 3.1 Hz. The graphic below shows the relationship between the sample rate, the displayed FFT, and a zoom in window showing the individual 1k FFT bins.

Note: For the Real Time displays, FFTs will be generated at a default rate of 20 Hz - or every 0.05 seconds. This can slow down, but currently 20 Hz is the fastest at which the live FFTs are made. There is a slight difference to the timing of the FFT then what is shown... at the described sample rate, the Safety FFTs will actually be overlapping by about 80% and the display FFT will have a slight gap in data. However, the key concepts remain.

Frequency Analysis in Aurora

When post processing data in Aurora, HGL takes a slightly different approach in how the data is created. There are two types of files available:

  1. Frequency (ZMOD) File
  2. Time History File

The time history file is fairly self-explanatory in that any size FFT can be displayed for any point in the time history file. 

The ZMOD file is different because of it's fixed size nature. Here are the key details of a ZMOD file:

  • Every ZMOD file has 360 FFTs that are evenly distributed across the analysis time window.
  • Every FFT is a 1024 (1k) FFT.
  • There are multiple processing methods available that the analysis software uses when finding FFTs to put into the ZMOD file.
Standard ZMOD

When looking at the raw analog signal, the standard ZMOD can be represented like this:

The Standard ZMOD is a gapped analysis. All of the Data in the processing window is not guaranteed to analyzed, so users need to be consider the time period being analyzed to know if it is appropriate. The Standard ZMOD guarantees a few key details:

  • It will provide a consistently fast processing time because only 360 FFTs (per channel) need to be created.
  • All of the data between the channels will be time aligned. This is critical for doing phase analysis.
Spectrum ZMOD

The Spectrum ZMOD is different from the standard ZMOD because:

  • All data is within the analysis period is processed using a 50% overlap of 1k FFTs.
  • Finds the highest responding FFT within each partial segment (spectra) of the analysis file.
  • The results between channels are NOT time aligned. Phase analysis is not valid for Spectrum ZMODs.
  • The time required to create the Spectrum ZMOD file depends on the duration of the file. The longer the analysis period, the more FFTs that are created and need to be compared.

Each FFT in the ZMOD file represents the highest amplitude (whether highest ENERGY or highest PEAK) FFT for the time segment that it represents. This makes the Spectrum ZMOD very useful for automatically reporting the highest response within a test event. The picture above shows that the FFT with the highest response can come from any FFT within the partial spectra analysis segment that it represents. Each spectra analysis segment represents, sequentially, 1/360th of the total analysis time period.

Average ZMOD

The Average ZMOD analysis method also processes:

  • All the data within the analysis period using a 50% FFT overlap.
  • All of the FFTs within each partial analysis segment (spectra) are averaged to create a single FFT for that spectra.
  • The data is not time aligned because it represents all of the signals over a period of time. Since phase would only be appropriate for STATIC measurements, it is not recommended.

Analysis Key Points

All FFTs in the ZMOD are a 1024 (1k) FFT

The chosen FFT in the ZMOD file will vary depending on the specified analysis method.

All FFTs represent an evenly spaced spectrum of data covering the analysis period.

Only Standard Zmod should be used for Phase Analysis.

Carefully choose Analysis Bandwidth and time period so that the data (FFTs) don't have excessive smearing.

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