Chapter 8Frequency Analysis

This chapter describes the features for examining signals such as vibration by frequency: FFT view, which displays the graph by frequency; Filter, applied per series; and STFT peak, Wavelet, and Freq. anomaly on the Analysis tab.

The examples in this chapter use the practice sample data sample_vibration_en.csv. It is data from a fictitious motor vibration test (not a record of a real test).

8.1What frequency analysis can do

All analyses on the Analysis tab are used in the order Target (choose the Analysis target and the Range) → Run → Results. The results appear as a table in Analysis results at the bottom of the screen. This flow and how to read the results are explained in Chapter 9. This chapter describes only the settings and results specific to frequency analysis.

8.2Frequency units

Frequency values are derived from the unit of the X axis. The unit used by each feature is as follows.

8.3FFT view

The FFT view applies a Fourier transform to series and displays graphs (spectra) with frequency on the horizontal axis.

1. On the Series tab, check the series you want to examine to display them.
2. Click FFT on the toolbar. A spectrum is shown in its own tier for each displayed series. In the sidebar, FFT view opens on the Analysis tab.
3. To return to the time-series view, click Overlay or Stacked.

1. 1FFT: Displays the graph by frequency
2. 2Tier name: Shows the frequency of the largest component (Peak) after the series name. When there is a caution about the calculation, Warning is added; hover to see the details
3. 3Target, Quantity: The time range used for the calculation, and the quantity shown on the vertical axis
4. 4X-axis display, Log tick labels, Minor ticks: Whether the horizontal axis is linear or logarithmic, and how log ticks are shown
5. 5Window function, Frequency unit: The window function used for the calculation, and the frequency unit
6. 6Remove DC, Show peaks: Whether to subtract the mean before calculating, and whether to show Peak in the tier name

In the sample data, peaks appear at 50 Hz (about 0.96), 120 Hz (about 0.50), and 230 Hz (about 0.27). The 230 Hz component has an amplitude of 1.5, but it is present for only 4 of the 18 seconds, so it appears small when calculated over the whole data. To know its size during that time only, set Target to Current display range, display 12–16 seconds in the time-series view, and then return to FFT. This way, 230 Hz reads about 1.50.

List of settings

Turn the wheel over a spectrum to zoom the frequency axis in or out, and drag to move it. Only the display range changes; the spectrum is not recalculated.

Display amount

p-p and RMS are correct values for sine-wave components. To compare the size of noise spread over a wide frequency range, use PSD.

Window functions

A window function is a weighting that smoothly brings both ends of the calculated interval close to 0. The choice slightly changes the height and width of peaks. The values read for the sample data's 50 Hz component (amplitude 1.0) with Amplitude 0-p are included.

With Hann, when a component's frequency falls between the calculation step (resolution) ticks, the amplitude can read up to about 15% low. When you use amplitude values in a report, choose Flat top (amplitude accuracy).

PSD and the Welch method

When you set Quantity to PSD, you can choose the PSD method.

• Segment length is specified in seconds (seconds even for data with an X axis in milliseconds). If set to 0, one eighth of the number of data points is used as the interval.
• Overlap is the proportion by which adjacent intervals overlap (0%, 50%, 75%).
• The frequency resolution is 1 ÷ (the number of points in the interval rounded up to a power of 2 × the step). With the sample data and a segment length of 0.5 seconds, 500 points are rounded up to 512 points, and the resolution is about 1.95 Hz. Peak frequencies are also read on this grid, so the 50 Hz component is shown as 50.8 Hz.
• If two or more intervals cannot be obtained, the calculation uses Periodogram (single segment), and Warning is added to the tier name (Too few segments; computed as a periodogram. Use a shorter segment length.).

Calculation rules

• The values used for the calculation are the values plotted on the graph (after Unit conversion / rename and filters).
• When there are more than 65,536 points, an averaging that suppresses aliasing (high frequencies appearing as low frequencies) is applied, and the points are reduced before calculating. High-frequency components become smaller. Warning is added to the tier name.
• When there are fewer than 8 points, nothing is calculated.
• When X goes backward or the same X repeats somewhere, nothing is calculated (The data is not uniformly sampled, so the FFT is not computed is shown).
• When the step deviates from the median by more than 5% somewhere, the data is treated as evenly spaced for the calculation, and Warning is added (with a note that the FFT frequencies are approximate because the X spacing varies widely). This caution also appears when there are missing values or time jumps (see the section "Handling missing values and time jumps").

8.4Series filters

A filter removes components at set frequencies from a series' waveform. On the Series tab, click ▾ for the series and set it up in the Filter field.

1. 1Type: Choose from None, Low-pass, High-pass, Band-pass, Band-stop, Notch, Moving average, and Moving median. The fields below change with the type (described later)
2. 2IIR type: Choose from Butterworth, Chebyshev (1dB), and Bessel
3. 3Degree: 1 to 6. The larger the value, the more steeply frequencies outside the cutoff are reduced
4. 4Phase: Choose Normal or Zero-phase
5. 5Cutoff fc: The cutoff frequency. For Band-pass, there are two: Low fc and High fc
6. 6The data step and the highest frequency that can be handled (Nyq = Nyquist frequency). Enter the cutoff frequency in this unit
7. 7Explanations and cautions about the calculation (described later)
8. 8Response graph: shows how much this filter reduces each frequency (the amplitude ratio in dB) (described later)
9. 9Expand…: Opens the larger view

The cutoff frequency is kept within the range from one trillionth of the Nyquist frequency to just below the Nyquist frequency.

Filtered values are used for the graph, the FFT view, and analyses (when Data is Filtered (as displayed)). When Unit conversion / rename is used, the filter is applied to the converted values. The loaded data itself does not change.

What the cutoff frequency (fc) means

For all three characteristics, fc is "the frequency at which a single pass gives −3 dB". For Chebyshev (1dB) too, fc is the −3 dB point, not the edge of the passband ripple.

These are the results of filtering a sine wave at the same frequency as fc and measuring the amplitude ratio (1000 Hz data, fc 50 Hz; orders 1–6 and all three characteristics gave the same values).

Differences between characteristics

These are the results of measuring the amplitude ratio (dB) at each frequency and the overshoot in response to a step input, with Low-pass, 4th order, Normal, and fc 50 Hz.

• Butterworth: Passes frequencies below fc almost flat. Choose this if you are unsure.
• Chebyshev (1dB): Reduces frequencies above fc most steeply. In exchange, the amplitude ripples by up to about 1 dB below fc, and it rings strongly after a step.
• Bessel: The attenuation is gradual, but it hardly distorts the waveform shape. Choose it when you want to see the shape of steps and pulses.

Normal phase and zero phase

How the band-pass is built

Band-pass is a high-pass at Low fc followed by a low-pass at High fc. Degree is used for each of the high-pass and the low-pass. When the low and high limits are close, the attenuation of the two filters overlaps, and frequencies in between are reduced too.

Band-stop and notch

Band-stop removes the band from Low fc to High fc and passes the frequencies outside it (the opposite of a band-pass). It is built by transforming the low-pass prototype of the IIR type into a band-stop filter. The attenuation is −3 dB at the low and high limits (−6 dB with Zero-phase).

Notch removes only a very narrow band around Center f0. Use it to remove 50/60 Hz mains noise or the first order of a rotation.

The amplitude ratios (dB) below were computed for 1000 Hz data.

Moving average and moving median

These smooth the series over time by combining neighboring points. Both compute over Window (pts) points centered on each point, so there is no time delay. Near the ends of the series, only the available points are used.

For 1000 Hz data, the −3 dB frequency of a Moving average is about 90 Hz with 5 points and about 40 Hz with 11 points.

Response graph

Below the Filter field in the series details, a response graph for the current settings appears. The horizontal axis is frequency (log), and the vertical axis is the amplitude ratio (dB). Shading marks the passband; dotted lines mark −3 dB and fc. The curve is computed from the very coefficients that are applied. With Zero-phase, it shows the attenuation of the two passes. A Moving median has no frequency response, so an explanation appears instead of the graph.

Click Expand… to open a larger view.

1. 1A summary of the settings and the data step (sampling frequency and Nyquist frequency)
2. 2Overlay the spectrum of this series: Overlays the spectrum of the series before (gray) and after (orange) filtering
3. 3Amplitude ratio (dB). Hover over the graph to see the attenuation (dB and gain) at that frequency
4. 4Phase shift. With Zero-phase, there is none
5. 5The frequencies where the attenuation reaches −3 dB, −20 dB, and −40 dB, and the attenuation at 2× and 10× fc

To close it, click Close or press Esc.

Explanations and cautions shown

The following text is shown below the Filter field in the series details.

Missing values and time jumps

The filter splits a series into segments at missing values (non-numeric values such as blanks) and time jumps (intervals of more than 1.5 times the median step), and applies the filter to each segment. Values are never connected across segments. Missing positions remain empty after filtering. The step used is the median of the intervals between adjacent points. Missing values and jumps do not cause the step to be overestimated.

8.5STFT peak

STFT peak divides the data into short time windows, calculates a spectrum for each window, and finds the strongest frequency in each window. Use it to track changes in rotational speed or components that appear partway.

1. On the Analysis tab, choose the series to examine in Analysis target.
2. Open STFT peak and enter Window and Step.
3. Click Compute peak trend.

1. 1Window, Step, Results shown: The window length, the interval by which the window shifts, and the number of rows in the table
2. 2Compute peak trend: Calculates
3. 3Results table: The time at the center of the window, the strongest frequency, its strength, and Assessment. Sorted by strength in descending order
4. 4Add peak series: Adds the peak frequency of each window to the graph as a new series

In the sample data, the peak frequency changes from 49.8 Hz to 230.5 Hz only between 12 and 16 seconds. The added series (its name includes STFT peak, and its unit is Hz) shows when the change occurred.

• The frequency resolution is 1 ÷ (the number of points in the window rounded up to a power of 2 × the step). With a 1-second window (1000 points → 1024 points), it is about 0.98 Hz. That is why 50 Hz is shown as 49.8 Hz and 230 Hz as 230.5 Hz.
• The window function and frequency unit are those of Window function and Frequency unit in FFT view. Strength is amplitude (0-p). When Quantity in FFT view is Power, it is power.
• Assessment is the change compared with the peak frequency of the previous window. More than 8% higher is Rising, more than 8% lower is Falling, and anything else is Flat.

8.6Wavelet analysis

Wavelet analysis displays component strength on both time and frequency axes as a color distribution (scalogram). It is suitable for finding components that appear only briefly. The calculation uses the Morlet wavelet.

1. On the Analysis tab, choose the series to examine in Analysis target.
2. Open Wavelet and, if necessary, enter Min frequency and Max frequency.
3. Click Compute wavelet.

1. 1Min frequency, Max frequency, Frequency bins, Time limit: The frequency range to calculate, and how frequency and time are divided
2. 2Compute wavelet: Calculates
3. 3Scalogram: Time runs horizontally and frequency vertically (low at the bottom, logarithmic scale). Redder means a stronger component. The dark areas at the edges are the Cone of influence (COI)
4. 4Results table: The strongest frequency at each time (the ridge), sorted by strength in descending order

In the sample data, a 50 Hz band is visible throughout, and a 230 Hz band from 12 to 16 seconds.

• The frequency unit follows Frequency unit in FFT view. When Frequency unit is Hz, the automatic lower limit is calculated in seconds regardless of the X-axis unit (when 1/X is chosen, it is calculated in X-axis units).
• The Cone of influence (COI) (the range affected by the boundaries) is the low-frequency part near the ends of the data. Strength in this part comes out small because the calculation is cut off at the ends. In the table, Assessment is Inside COI (indicative only), and these rows are also excluded from the ridge calculation. Other rows are Energy burst if their strength is at least half of the table maximum, and Baseline if less than half.
• With Add peak series, you can add the ridge frequency at each time to the graph as a new series.

8.7Frequency anomaly

Frequency anomaly extracts one metric from the spectrum of each window and lists windows where that metric differs greatly from usual as candidates.

1. On the Analysis tab, choose the series to examine in Analysis target.
2. Open Freq. anomaly and enter Metric, Window, Step, and Threshold.
3. Click Detect frequency anomalies.

1. 1Metric, Window, Step, Threshold: What to compare, the window length and interval, and the size of Z that makes a candidate
2. 2Detect frequency anomalies: Detects
3. 3Results table: The interval, main frequency, change, and verdict of each candidate window. Sorted in time order

In the sample data, 8 windows from 11.8 to 16.2 seconds become candidates. The main frequency is 230.47 Hz (about 4.6 times the baseline of 49.80 Hz), and Assessment is Higher freq..

• Of candidates that are close together (closer than half the window width), the one with the larger Z is kept.
• Dominant freq. (baseline {b}) in the results table is the peak frequency of the candidate window. The baseline is the median of the peak frequencies of all windows. Change is the ratio to the baseline: more than 1.15 times is Higher freq., less than 0.87 times is Lower freq., and anything else is Flat.
• The window function and frequency unit use the FFT view settings.
• With Add score series, you can add Z for each window to the graph as a new series.
• As with STFT peak, make the window shorter than the change you want to see. If you calculate the sample data with the default settings, only 5 windows can be taken, and there are 0 candidates.

8.8Handling missing values and time jumps

Missing values (non-numeric values such as blanks) and time jumps (intervals of more than 1.5 times the median step) are handled by each feature as follows.

• Series in which X goes backward or the same X repeats are not calculated by any feature. The reason is shown on screen.
• When time jumps are too large to arrange the data on an evenly spaced grid, the analysis stops and asks you to select a range.