Many technical articles advise using antialiasing filters in digital sampling systems to avoid acquiring bad data. These low-pass filters are required because you never know what frequencies—other than those of interest—are on the signal lines.
Some sources of extraneous noise include power supplies, computer monitors, radiated RF from a variety of sources, or the cell phone in the car that just drove by. Whether the noise generators are transient or repetitive, the critical issues are how this noise affects measurements and how to eliminate it. A good antialiasing filter is vital to ensuring that only the correct data is recorded.
The first step in choosing an antialiasing filter is to understand the specific application and the report generation it requires. For instance, if your application is vibration or acoustics, amplitude accuracy vs frequency is important. If you are measuring shock pulses, amplitude vs time becomes critical. Next, you need to understand three basic parameters—frequency, amplitude, and phase—and how their interaction affects the accuracy of the measurements.
A waveform typically is represented as a series of amplitudes vs time (time domain). The French mathematician Fourier discovered that a waveform can be represented by adding together a series of sinusoidal waveforms. There is an advantage to dismantling a waveform into a series of sine waves via the Fourier transform. Only three parameters determine a sinusoidal waveform: frequency, amplitude, and phase (Figure 1).
Once the waveform is transformed, plotting the frequency on the X-axis and amplitude or phase on the Y-axis will tell the relationship among the three parameters in the frequency domain. Phase can be expressed in degrees or as delay in units of time.
For example, a 1-kHz sine wave repeats itself periodically in 1-ms intervals. As a result, 360° is a complete circle that represents 1 ms, and 90° is a quarter-circle representing 0.25 ms. The phase can be represented in degrees or as a delay in time units.
When you request technical specifications from filter manufacturers, three plots generally are provided to determine the effect of the filter on your signal. These plots are frequency response, phase, and delay (group delay) vs frequency (Figure 2). The frequency-response plot provides the amount of amplitude attenuation of individual frequencies. The amplitude typically is defined in decibels: 20 Log (Vout/Vin).
If the amplitude of the filtered waveform is the same as the waveform input (Vout = Vin), then it is 0 dB in log scale [20Log(1)]. If the amplitude is attenuated by half, it is
-6 dB; that is, 20Log(0.5).
Phase vs frequency and delay provide information on the amount of phase shifting at a particular frequency. The phase plot is displayed in degrees; delay is displayed in time units. The frequency response and delay of various filter types often are normalized in terms of the filter corner frequency (fc), as shown in Figure 3 for Butterworth and Bessel filters.
Table 1 shows the values of amplitude, phase, and delay vs frequency normalized to the fc of Bessel, Butterworth, elliptical, and linear phase filters. The Bessel, Butterworth, and elliptical are 8-pole analog antialiasing filters.
Linear phase filters can be either analog or digital. They are used more and more in today’s data acquisition systems, such as Delta-Sigma analog-to-digital converters (ADCs), or in combination analog and digital filter techniques because of their sharp rolloff. The linear phase filter listed in Table 1 is a combination analog and digital design.
Choosing the best filter for your application comes down to understanding how frequency response and delay plots affect the time-domain signal. Let’s compare the differences.
Amplitude Comparison
Table 1 tells an interesting passband story: the Bessel filter has the worst amplitude matching of all the filters. The table indicates that amplitude attenuates in an increasing linear fashion as frequency increases. Butterworth is excellent up to one-half of the corner frequency, but it is down -3 dB at the fc.
Elliptical seems to give a very good match with little fluctuation at the fc (only 0.035 dB). The linear phase filter seems to be the best because it maintains a true amplitude representation of the signal in the passband.
Phase/Delay Comparison
When you look at the delay vs frequency plots in Table 1, notice that all filters have fairly uniform phase shift in the passband. However, when the signal reaches the fc, the Butterworth, elliptical, and linear phase filters exhibit some nonlinearity.
How does this nonlinearity manifest itself in the time domain? The easiest way to see the manifestation is through a square wave. A square wave is used because it contains many odd, high-frequency harmonics to assist us in understanding the effects of frequency shifts.
To test how the filters affected the data, R.C. Electronics’ RC-AAF PC plug-in, programmable, 8-pole antialiasing filters and a DATAMAX® internal linear phase filter (a combination analog/digital filter design) were used. A function generator produced a 20-Hz square wave and a DATAMAX 16-bit instrumentation data acquisition system recorded the effects of the filter. Figure 4 shows the effect each of the four filter types has on a square wave.
As you can see, all filters except the Bessel cause the square wave to ring at the rising and falling edges. The linear-phase filter has a much smaller ringing amplitude compared to Butterworth or elliptical, but rings on all four corners. The ringing is caused by the phase shift from high-frequency components.
When you combine the sine waves back into square waves, these phase shifts cause the high-frequency waveform to reconstruct out of sync with each other and produce the ringing effect. The Bessel filter is the only one that does not have any ringing because the phase shows no drastic shift in frequency at or after the fc.
Once you understand how the frequency response and delay plot affect the data in the time domain, you can choose the best filter for your application. If an application requires precise measurements in amplitude vs frequency and phase is not important, such as in vibration and acoustic testing, then elliptical or linear phase filters work best because the amplitude in the passband has the least variation.
In low-frequency applications or if the frequency of interest is not close to the fc, such as stress, pressure, or temperature measurements, Butterworth, elliptical, or linear phase filters are good candidates. Butterworth will be the best choice if the frequency of interest is less than half of the fc.
Elliptical or linear-phase filters are best if you need to read the amplitude accurately all the way out to the fc. The Bessel only would be used to record a square wave or a shock pulse to eliminate the ringing effect in which the high-frequency content would extend beyond the fc.
If you have an existing filter that may not be the best for your application, does that mean you cannot collect valid data? As long as you collect nonaliased data, it is possible to find out the filter’s transfer function and reverse-engineer your waveform. This will require some math, but you can replicate the original information.
Summary
We have gone over the filter specifications regarding frequency response, phase, and delay plots of the four most popular filters. If you understand what kind of measurements are important to you and know what type of input to expect from the test, then you can select an appropriate filter for the job.
Linear phase or elliptical is the best general-purpose filter as long your waveform does not have many high-frequency components above the fc. Bessel is the best if your signal is a sharp pulse or square wave. Butterworth, a combination of both designs, is best when most frequencies of your signal are less than half of your filter fc.
Reference
Pang, C., “Antialiasing Filters and Data Validity—The Inside Story,” Sound & Vibration, April, 1998.
About the Author
Chao-Sun Pang is the vice president of marketing at R.C. Electronics. He graduated from the University of California at Santa Barbara with a B.S. in physics and an M.S. in scientific instrumentation. R.C. Electronics, 6464 Hollister Ave., Santa Barbara, CA 93117, (805) 685-7770, e-mail: [email protected].
Sidebar
What Sampling Ratio Should You Use With Your Filter?
So you know that the filter you are using is the correct one for your application. That means your data is valid, right? Simply having a filter in your system does not ensure that your data will be valid. You also must sample your data at the proper frequency.
You can find the appropriate sampling rate by applying this formula:
SR = S/fd = 1+fr/fd
where: S = sample rate
fd = maximum desired data frequency
fr = frequency at the required attenuation
The fd typically specifies the fc of a filter when it reaches attenuation at the -3-dB point. The fr is related to the attenuation required to maintain the desired precision of the data. For a 0.01% (12-bit) system, it is -76 dB; for a 0.0015% (16-bit) system, it is -96 dB. Table 2 shows some of the typical sampling ratios for 2-, 4-, 6-, and 8-pole filters.
Bessel* Butterworth* Elliptical* Linear Phase*
f/fc
Amp
Phase
Delay
Amp
Phase
Delay
A Amp
Phase
Delay
Amp
Phase
Delay
(dB)
(deg)
(µs)
(dB)
(deg)
(µs)
(dB)
(deg)
(µs)
(dB)
(deg)
(µs)
0
0
0
0.506
0
0
0.816
0
0
0.713
0
0
51.5
0.1
-0.029
-18.2
0.506
0
-29.4
0.819
0.004
-25.7
0.716
0
-206
51.5
0.2
-0.117
-36.4
0.506
0
-59
0.828
-0.014
-51.6
0.724
0
-411
51.5
0.3
-0.264
-54.7
0.506
0
-89.1
0.843
-0.024
-77.9
0.74
0
-618
51.5
0.4
-0.47
-72.9
0.506
0
-120
0.867
-0.02
-105
0.767
0
-824
51.5
0.5
0.737
-91.1
0.506
0
-152
0.903
0.007
-133
0.811
0
-1029
51.5
0.6
-1.06
-109
0.506
-0.0011
-185
0.956
0.033
-163
0.872
0
-1235
51.5
0.7
-1.45
-128
0.506
-0.014
-221
1.04
0.014
-196
0.946
0
-1442
51.5
0.8
-1.91
-146
0.506
-0.121
-261
1.19
-0.041
-232
1.04
0
-1649
51.5
0.9
-2.42
-164
0.506
-0.738
-307
1.4
-0.016
-272
1.23
-.002
-1855
51.5
1
-3.01
-182
0.506
-3.01
-360
1.46
-0.035
-323
1.65
-.3
-2060
51.5
1.5
-7.08
-273
0.504
-28.2
-511
0.448
-40.1
-578
0.517
-100
-2738
51.5
2
-13.7
-362
0.468
-48.2
-568
0.226
-87.8
-450
0.168
-95.1
-2320
51.5
3
-33.4
-489
0.241
-76.3
-621
0.094
-99.9
-305
0.057
-101
-2920
51.5
4
-51.8
-552
0.126
-96.3
-646
0.052
-83.1
-140
0.03
-94.5
-2320
51.5
5
-66.8
-587
0.007
-112
-661
0.033
-82.1
-148
0.018
6
-79.2
-610
0.052
-125
-671
0.023
-83.1
-154
0.013
7
-89.8
-626
0.038
-135
-678
0.017
-84.6
-157
0.009
8
-99
-638
0.029
-144
-683
0.013
-86.2
-160
0.007
9
-107
-647
0.023
-153
-687
0.01
-87.8
-163
0.005
10
-114
-655
0.018
-160
-691
0.008
-89.3
-164
0.004
Table 1
Table 2
Filter
-60 dB
(0.1%)
-80 dB
(0.01%)
-96 dB
(0.0015%)
2-pole Bessel
41
128
321
2-pole Butterworth
33
101
252
4-pole Bessel
10
16
26
4-pole Butterworth
6.6
11
17
6-pole Bessel
6.5
9
12
6-pole Butterworth
4
5.6
7
8-pole Bessel
5.5
7
8.5
8-pole Butterworth
3.4
4.2
5
8-pole Cauer (elliptical)
2.7
3
3.5
Linear Phase
2.4
2.45
2.5
Copyright 1999 Nelson Publishing Inc.
May 1999
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