Simcenter Testing Solutions Acquisition Filters and Overshoot

Simcenter SCADAS Simcenter Testlab



Acquisition Filters and Overshoot

Applying filters (for example low-pass anti-aliasing filters) to time data can create overshoots.  When a sudden change in amplitude occurs in a time signal, the filter can cause a noticeable variation in amplitude as shown in Figure 1.  This is caused by the removal of frequency content from the original signal.
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Figure 1: Square wave signal exhibiting overshoots at the leading and trailing edges of a square wave.

This overshoot might need to be addressed depending on how the data will be analyzed:
  • Frequency Domain: When data is primarily viewed in the frequency domain, the overshoot is of little consequence as it is spread over a large frequency range and alters the amplitude very little or not at all. Noise and vibration analysis is often done in the frequency domain.
  • Time Domain: If the time domain amplitude is important, the overshoot creates higher levels than in the original signal.  Durability analysis is performed with the time domain data, and overshoot could cause incorrect fatigue life predictions.  
This article details the causes of overshoot and how it can be addressed by selection of the appropriate filter:
1    Time Domain versus Frequency Domain
2.   Set Measurement Bandwidth Correctly
   2.1   Nyquist-Shannon
   2.2   Why Is A Filter Needed?
3.   Choosing The Right Acquisition Filter
   3.1  Why You Should Care About Your Filter
   3.2  Filter Type And Order
   3.3  Phase: As Linear As Possible
   3.4  Magnitude: Ideal For Frequency Might Not Be For Time
   3.5 Cause Of Overshoot
   3.6 Signal Overshoot Is Not Gibbs
   3.7  Sharpness Versus Smoothness
4.  Real World Signals Are Not Square Waves

5. Filters and Overshoot In Image Processing
6.  Conclusion

1. Time Domain versus Frequency Domain

In frequency domain analysis, time data is converted to the frequency domain by Fourier analysis.  The analysis can be performed without much of a worry because the overshoot effect is broad in frequency but causes a very small change (or none depending) in the amplitude of the frequency domain. 

In data acquisition and analysis, the amplitude in the time domain matters.  Examples of time based testing and analysis include shock testing, road load data acquisition, and transient analysis. These time data measurements need to be just right: statistics, peaks, times at level, level crossings, range pairs, rain flows, and logical triggers all depend on accurate amplitude time data (Figure 2).
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Figure 2: Durability data relies on correct amplitude levels in the time domain to calculate rainflows (left) and range pairs versus level (right) from load data.

Just small changes in the amplitude can change fatigue life estimates drastically.  See this knowledge article for more information: What is a SN-Curve?.

2.   Set Measurement Bandwidth Correctly

The information of an analog signal is contained in its bandwidth, aka its frequency content (both in magnitude and phase). Most analog signals have a bandwidth between zero Hertz and some frequency (aka bandwidth of interest). Two situations are possible based on the analog signal frequency content versus the measurement bandwidth setting:

2.1   Nyquist-Shannon

Following Nyquist-Shannon sampling theorem and its hypotheses, if the digital data is acquired with a sample rate higher than the double of the signal bandwidth, all the information of the original analog signal will be correctly digitized/encoded, and even perfectly reconstructed if necessary. In other words, if the analog signal bandwidth is below of half of the sample rate (aka Nyquist frequency) then no artifacts are possible.
2.2   Why Is A Filter Needed?

Unfortunately, analog signals often have signal content greater than the bandwidth used to digitize the data.  This can cause undesirable noise (aliasing) to occur as the higher frequency signals "foldback" into the measurement bandwidth as shown in Figure 3.
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Figure 3: If a signal (green) is not band limited to Nyquist or below, then if will foldback to appear with lower frequency (red signals) based on its frequency content relative to the Nyquist frequency.

Think about the input signal as a combination of actual expected signal and undesired noise. Acquisition low pass filters (from now on, just filters) are then needed, to mold the input signal into a smaller bandwidth, and more importantly, to get rid of noise and alias frequencies.

More about sampling rates, bandwidth, and aliasing in these knowledge articles:

3.   Choosing The Right Acquisition Filter

Both for frequency and time domains, there are two options: either set the measurement bandwidth high enough to capture the entire analog content of the signal, or choose the acquisition filter type appropriately. For the second choice, depending on the filter type used, overshoot in the time data might appear more, less, or not at all.

3.1  Why You Should Care About Your Filter

The measurement bandwidth is defined by the filter passband, or cut-off frequency (Fco), generally at -3dB (Figure 4).
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    Figure 4: A filter is defined  by a pass-band, transition band, and stop band.  The cut-off frequency (Fco) delineates the pass-band and transition band, and is usually defined at the 3dB attenuation point of the pass-band.
    Implementable filters aren’t ideal:
    • Filters won’t totally stop or totally let frequency components pass
    • Different frequency components are delayed in time by different amounts.
    • Space is needed between cut-off frequency (Fco) and stopband (i.e. transition band can’t be 0)
    • Their behavior further deviates from expectation depending on their implementation (analog, digital (FIR, IIR), etc.)
    It’s clear that some information of the original input (noise + signal) might get modified by acquisition filters: The filter becomes both a solution, and if not selected and configured correctly, becomes a threat by creating undesired artifacts in the measurement.

    3.2  Filter Type And Order

    The transfer function of a filter is basically characterized by filter type and order
    • Type defines its shape and phase response in passband, transition band and stopband. There are some well-known classical filter types, most of them originating as analog versions like Butterworth, Bessel, Chebyshev, Elliptic, etc as shown in Figure 5.
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    Figure 5: Comparison of magnitude frequency response of different filter types.
    All of them optimize certain attributes at the price of compromising others.
    • Order (or number of poles) sizes its buffer (i.e. energy storing devices contributing to the process delay, like capacitors for analog filters, or memory vectors for digital filters), and it will account for the roll-off and help to minimize the transition band (Figure 6).
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    Figure 6: Filter roll-off performance versus order. The higher the order (for the same filter), the greater the roll-off.
    Increasing the order increases the roll-off, but compromises filter complexity and process delay.

    More about filters in the knowledge article: Introduction to Filters: FIR versus IIR

    3.3  Phase: As Linear As Possible

    Classical analog and digital IIR filters have non-linear phase responses (Figure 7) because they are intrinsically causal. Each signal frequency component won’t propagate through the filter with the same time delay.
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    Figure 7: Non-overshooting non-linear filter effect.
    Delay is an undesirable artifact of a filter.  Filters have existed for years (especially the analog ones), so their behavior is well known. Furthermore, IIR filters are easy to design: they are cheap, simple, and very performant. However, they exhibit non-linearity in their phase responses. In the time domain, this produces scattering (or asymmetries).

    Phase linearity in the passband means that group delay (proportional to the derivative of phase) is as constant as possible.  This ensures minimal shape distortion and helps distributing overshoot, if any, before and after transients.

    In general, increasing filter order reduces phase linearity (Figure 8).
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    Figure 8: Overshooting non-linear filter effect.
    But there are exceptions where phase linearity improves with filter order, like Bessel and transitional filters like Gaussian.

    Finally, only a subset of synthesized digital Finite Impulse Response (FIR) filters can have exactly linear phase transfer functions. If possible, use a linear phase FIR filter when collecting data. The Simcenter SCADAS hardware has both sharp and smooth linear filters choices.

    3.4   Magnitude: Ideal For Frequency Might Not Be For Time

    If the measurement bandwidth is not high enough relative to the signal frequency content, information will be lost after filtering. The best filtered version for time analysis (smooth in time), is not necessarily the best version for frequency analysis (overshooting in time). Differences between some filter types and their orders are shown in Figure 9.
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    Figure 9: Time domain step response of three different filters for various number of poles. Response is progressively smoother from left to right.

    Good news is that an acquisition filter can be selected to tune how it impacts the measurement. This is especially helpful for the time domain when overshoot comes into play.

    3.5 Cause Of Overshoot

    Overshoot is a product of a sharp filter impulse response magnitude, when there is frequency content falling into the stopband as shown in Figure 10.
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    Figure 10: As signal (green, left) is passed through a filter (middle) it can result in overshoot being added to the signal (right).
    There is an actual need to store stopband energy in the filter itself, and it may be abruptly released and overlapped later into the passband. As a rule of thumb:
    • The sharper the filter, the larger the overshoot amplitude
    • The higher the order, the longer its duration

    3.6 Signal Overshoot Is Not Gibbs

    The Gibbs Phenomenon is commonly confused with data acquisition artifacts, like overshoot. Although the result resembles in both cases a ringing artifact, while overshoot may (or not) be produced after filtering a signal, Gibbs phenomenon always appears after approximating a discontinuous function with a finite set of its Fourier Series (Figure 11).
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    Figure 11: The Gibbs Phenomenon is the result of trying to approximate a discontinuous function (black line right) with a Fourier Series (multi-colored sine waves, right).

    Overshoot is a physical product of a filter, Gibbs is a mathematical consequence of a suboptimal approximation exercise.

    3.7    Sharpness Versus Smoothness

    If the signal frequency content goes beyond the measurement bandwidth, sharpness is desired for frequency domain transition band, allowing more efficient usage of outhe measurement bandwidth given a sample rate.

    On the other hand, smoothness is better for time domain, allowing smoother energy release; thus, overshoot is minimized and eventually avoided. Sharpness depends only on the filter type (Bessel, Butterworth, Cauer, etc). Smooth and sharp filters are shown in Figure 12.
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    Figure 12: Sharp (left) versus smooth (right) filter shapes.
    To achieve better stopband attenuation figures using a smooth anti-aliasing filter, extra care should be taken to use higher sample rates so that Nyquist frequency gets pushed safely above cut-off frequency (Fco).

    The time domain differences and frequency domain differences between sharp and smooth filters are shown in Figure 13.
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    Figure 13: A sharp filter (elliptic, upper left) in the frequency domain results in overshoots in the time domain (upper right).  A gradual, smooth filter (Bessel, lower left) has no overshoot (lower right).
    A sharp filter (elliptic) has a narrow transition band in the frequency domain.  However, the sharp filter results in overshoots in the time domain.  A smooth filter (Bessel) has no overshoot in the time domain, but has a more gradual attenuation performance in the frequency domain relative to the sharp, elliptic filter..

    4.  Real World Signals Are Not Square Waves

    There are many types of signals in the real world, not all of them are square waves.  Consider the real world transient signal shown in Figure 14:
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    Figure 14: Time recording of pyrotechnic transient events sampled at 44.1 kHz.

    The signal has significant frequency content up to 17 kHz as shown in Figure 15:
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    Figure 15: The transient signal has significant frequency content to 17 kHz.

    Assume the signal needs to have high frequency content (above 11 kHz) to be removed. A "sharp" filter might be used. For example a 14th order Butterworth low-pass filter as shown in Figure 16.
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    Figure 16: Original signal frequency content (red) versus low-pass filtered (11 kHz cutoff) frequency content (green).

    It might be expected that removing signal content would result in smaller amplitudes in the filtered signals.  However, due to overshoot created by the Butterworth filter, some of the amplitudes have increased as shown in Figure 17:
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    Figure 17: Some peak amplitudes are higher in the filtered signal (green) than in the original signal (red) due to overshoot created by the sharp Butterworth filter.
    Filtering the transient time history with a smooth filter results in less overshoot as shown in Figure 18:
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    Figure 18: Top - Frequency content of original signal (red) versus the filtered signal (green).  Bottom - No overshoot appears in the filtered time history (green) versus the original time history (red) when using a less sharp Bessel filter.

    Instead of a Butterworth filter, a less sharp 2nd order Bessel filter was used to low-pass filter the same data. With the Bessel, the filtered time history has the same or lower amplitude than the original time history.  The Bessel filtered time history exhibits no overshoot.

    Please note that overshoot can only occur when a filter truncates frequency content in a signal.  If the frequency content of a signal is entirely contained in the pass band of a filter, then no overshoot can occur.
    5.  Filters And Overshoot In Image Processing

    Filtering and overshoot applies to images processing as well! Consider a black and white image. Instead of a two dimensional (2D) time plot, data is now in three dimensional (3D) spatial signals. The spatial co-ordinates behave like time.

    Two dimensional (2D) understanding can be practically applied following this reasoning: there is 2x "time" or spatial axis (x,y axis) making a plane (image). The voltage or amplitude (z axis) would be the intensity of color (white), rendering zero volts to black, and the gray scale all the way up to the maximum range, up to the white color. A sudden color transition, like in the images, represents a transient in the amplitude. The sudden color transition creates high frequency components in the image's Fourier transform.

    Consider the images processed with different filters in Figure 19:
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    Figure 19: Three images - Original (left), processed with sharp filter (middle), processed with smooth filter (right).

    To remove high frequency components (sudden changes in color) from the image on the left, an overshooting filter will ring the colors near the transitions as shown in the middle image, while a non-overshooting filter will smooth them out as shown in the image on the right.
    6.  Conclusion

    Some conclusions:
    • For frequency domain analysis of a signal, always choose a sharp linear FIR filter. For example, use the built-in decimation filter of Simcenter SCADAS Mobile and SCADAS Recorder series, or the Steep FIR of the Simcenter SCADAS RS series. Then set the Nyquist frequency quite close to the measurement bandwidth. Rule of thumb: 25% higher, for an excellent alias rejection.
    • If interested in the time domain and the signal frequency content might exceed the measurement bandwidth: choose a smooth linear FIR filter, like the Gaussian filter in Simcenter SCADAS RS series, or a smooth near-linear phase filter like Bessel both in SCADAS RS and SCADAS Lab/Mobile/Recorder series. Set the Nyquist frequency far enough from the measurement bandwidth (rule of thumb: 1x to 5x times higher, depending on filter order), for good alias rejection.
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