Zero phase filter.pdf
- 1. How-to, version: 3.1, Date: 04.05.2016 DEWESoft d.o.o. Gabrsko 11a, 1420 Trbovlje, Slovenia www.dewesoft.com support@dewesoft.com ANTI-ALIASING FILTERS WITH ZERO PHASE DISTORTION
- 2. 1 www.dewesoft.com Dewesoft has a possibility to set different sample rates on different channels in Channel setup. On each signal, which has sample rate less than acquisition rate, decimation is performed. With lower sample rate, we have to take care of signal aliasing. For dynamic acquisition sample rate there is analog filter on hardware, to filter out frequencies, higher than Nyquist frequency, to avoid aliasing. Nyquist frequency is half of the sampling rate of a signal. For lower sample rates, Dewesoft uses digital anti- aliasing filters. It is possible to choose between Anti-aliasing filter (IIR) and AAF (zero-phase distortion). IIR filters have phase delay depending on cutoff frequency. In applications, where there is phase information needed, e.g. torsional vibration, phase delay would corrupt results (AC signals). Here comes in zero phase distortion filters, which have attenuation slope symmetrical on both sides of ideal cutoff frequency and as the name implies, they have linear phase delay. Comparing to IIR filters, which have −3 𝑑𝐵 point at cutoff frequency, here signal is attenuated for 6 𝑑𝐵 at cutoff frequency (−6 𝑑𝐵 point). In other words, magnitude is attenuated by half at −6 𝑑𝐵 point (cutoff frequency). For applications described before, zero phase distortion filters are suitable as they do not cause any phase shifts and have also sharper attenuation. Figure 1: Channels with different sample rates. Figure 2: Anti-aliasing filter options.
- 3. 2 www.dewesoft.com Anti-aliasing IIR filters have cutoff frequency defined as 𝑠𝑎𝑚𝑝𝑙𝑒 𝑟𝑎𝑡𝑒 2.56 ⁄ , since also analog filters are defined that way. For example, channel with sample rate 1000 𝐻𝑧, has −3 𝑑𝐵 point at approximately 400 𝐻𝑧. At zero phase distortion anti-aliasing filters, cutoff frequency is set to 𝑠𝑎𝑚𝑝𝑙𝑒 𝑟𝑎𝑡𝑒 2 ⁄ , so attenuation of signal doesn't start too early. In terms of previous example, cutoff frequency would be at 500 𝐻𝑧. Figure 3 shows that this is optimal setting for cutoff frequency of zero phase distortion filters. Normalised frequency is denoted with 𝜔, 0 ≤ 𝜔 ≤ 1, cutoff frequency is denoted with 𝜔𝑐 and Nyquist frequency with 𝜔𝑁𝑦𝑞. Aliasing starts at Nyquist frequency. Blue triangle represents region with aliases when cutoff frequency, 𝜔𝑐, equals Nyquist frequency. This is now the case in Dewesoft. Blue region is present in the area, where attenuation has already started. But if cutoff frequency would be set any higher, aliases would be present in signal where attenuation hasn't started yet. This implies that current settings in Dewesoft are optimal. Transition band width is an important filter property that defines sharpness of roll-off. Noise floor begins at 𝜔𝑠 and roll-off start frequency is denoted with 𝜔𝑎, according to Figure 3. These two frequencies are symmetrical with respect to 𝜔𝑐. Specifically, in Dewesoft, noise floor is defined at −70 𝑑𝐵 and inside optimally flat pass band, that's before the start of roll-off, there is ±1% of amplitude error. In Advanced settings's Hardware tab, under Digital filters section there are two options for AAF (zero- phase distortion) transition width; Auto adjust and User defined (Figure 4). Auto adjust By default, Auto adjust option is selected. Dewesoft's zero phase distortion anti-aliasing filter's edge of transition band or in other words, beginning of noise floor, is set to 𝜔𝑠 = 0.66 ∙ 𝑠𝑎𝑚𝑝𝑙𝑒 𝑟𝑎𝑡𝑒, hence transition band width is equal to 16% of 𝑠𝑎𝑚𝑝𝑙𝑒 𝑟𝑎𝑡𝑒. Symmetrically, attenuation strats at 𝜔𝑎 = 0.34 ∙ 𝑠𝑎𝑚𝑝𝑙𝑒 𝑟𝑎𝑡𝑒. Because of limited computing power, auto adjust option is recommended, as it has set upper limit for number of filter's coefficients. Above described settings for transition band width (16% Figure 3: Magnitude response of zero phase distortion filter. Blue triangle shows aliases if 𝝎𝒄 = 𝝎𝑵𝒚𝒒.
- 4. 3 www.dewesoft.com of 𝑠𝑎𝑚𝑝𝑙𝑒 𝑟𝑎𝑡𝑒) are valid up to the predefined ratio 𝑎𝑐𝑞𝑢𝑖𝑠𝑖𝑡𝑖𝑜𝑛 𝑠𝑎𝑚𝑝𝑙𝑒 𝑟𝑎𝑡𝑒 ∶ 𝑠𝑎𝑚𝑝𝑙𝑒 𝑟𝑎𝑡𝑒 = 100: 1. What happens, when larger ratio is used on a channel as 100? Upper limit for number of coefficients of zero phase distortion filters is set to 1800 coefficients. When this limit is exceeded at filter design (larger ratio than 100 was used), wanted transition band width can not be reached anymore. Transition band width starts to expand according to equation 𝑇𝐵𝑊 = 11.13 ∙ 𝑆𝑅𝑅𝑎𝑡𝑖𝑜 4 ∙ 𝑁 , where 𝑇𝐵𝑊 represents normalised transition band width, 𝑆𝑅𝑅𝑎𝑡𝑖𝑜 is sample rate ratio, 𝑁 is number of filter coefficients and scale factor 11.13 derives from some particular filter design method. Lets use upper equation; we take default 𝑆𝑅𝑅𝑎𝑡𝑖𝑜 = 100 and for 16% of 𝑠𝑎𝑚𝑝𝑙𝑒 𝑟𝑎𝑡𝑒 transition band width we have 𝑇𝐵𝑊 = 0.16. This results in 1739 filter coefficients. Now consider the case, where upper limit is exceeded. Take larger ratio than 100, let's say 𝑆𝑅𝑅𝑎𝑡𝑖𝑜 = 125 and the same 𝑇𝐵𝑊. As result, we get 2174 filter coefficients, which is over the limit, so 1800 coefficients will be taken instead. In this case, normalised transition bandwidth is expanded to 𝑇𝐵𝑊 = (11.13 ∙ 125) (4 ∙ 1800) ≈ 0.19 ⁄ , i.e. 19% of 𝑠𝑎𝑚𝑝𝑙𝑒 𝑟𝑎𝑡𝑒 in previously used terms. For lower acquisition sample rate dividers, we get less coefficients than limit value. Figure 4: AAF (zero-phase distortion) 𝒂𝒄𝒒𝒖𝒊𝒔𝒊𝒕𝒊𝒐𝒏 𝒔𝒂𝒎𝒑𝒍𝒆 𝒓𝒂𝒕𝒆: 𝒔𝒂𝒎𝒑𝒍𝒆 𝒓𝒂𝒕𝒆 ratio settings.
- 5. 4 www.dewesoft.com For example, if acquisition sample rate is 20 𝑘𝐻𝑧, then transition band width equals 16% of 𝑠𝑎𝑚𝑝𝑙𝑒 𝑟𝑎𝑡𝑒 up to sample rate ratio of 100, i.e. 200 𝐻𝑧 sample rate. Any lower sample rate (less than 200 𝐻𝑧), will result in expanded transition band width. User defined With this option it is possible to set arbitrary transition band width in percents of sample rate. Auto adjust option has this value always set to 16%. Here there is no upper limit for number of coefficients, hence transition band width will always be exact. But at higher sample rate ratios (more than 100), calculation can be too slow because of large amount of filter coefficients. Both of upper options are limited with maximal allowable calculation delay, which means that number of coefficients can never exceed 20% 𝑜𝑓 𝑎𝑐𝑞𝑢𝑖𝑠𝑖𝑡𝑖𝑜𝑛 𝑠𝑎𝑚𝑝𝑙𝑒 𝑟𝑎𝑡𝑒. For example, if Auto adjust option is selected and acquisition sample rate is 1000 𝐻𝑧, upper limit for number of filter's coefficient will be already at 200 and not 1800. Below there is comparison between attenuation slope of IIR and zero phase distortion filters. For first example, dynamic acquisition rate was set to 10 𝑘𝐻𝑧 and sine sweep from 10 𝐻𝑧 to 10 𝑘𝐻𝑧 with amplitude ± 5 𝑉 and duration of 180 seconds was used on all channels. On first two channels we have 1000 𝐻𝑧 sample rate, on first there is anti-aliasing IIR filter used and on second there is zero phase distortion anti-aliasing filter set. Last, third channel has sample rate equal to acquisition rate, 10 𝑘𝐻𝑧, which means that analog filter from hardware will be used, no matter the anti-aliasing filter setting. Figure 5: User defined AAF (zero-phase distortion) transition band width Figure 6: First example's setup.
- 6. 5 www.dewesoft.com Green signal was filtered with IIR filter and blue signal has zero phase distortion anti-aliasing filter. Attenuation starts later at second one (blue), at approximately 340 𝐻𝑧. At first one attenuation starts immediately. On Figure 7 are marked −3 𝑑𝐵 and −6 𝑑𝐵 points. Sample rate of signal on Figure 8 is equal to acquisition rate, therefore no decimation is needed and digital anti-aliasing filter is not used (no matter what your anti-aliasing filter setting is, in this example it is set on OFF). Here is used analog filter, −3 𝑑𝐵 point is at approximately 5000 𝐻𝑧, attenuation starts at around 4600 𝐻𝑧. For second example, dynamic acquisition rate was set to 20 𝑘𝐻𝑧 and sine sweep from 100 𝐻𝑧 to 20 𝑘𝐻𝑧 with amplitude ± 5 𝑉 and duration of 180 seconds was used on all channels. Figure 7: Comparison between IIR and zero phase distortion anti-aliasing filter at 𝟏𝟎𝟎𝟎 𝑯𝒛 sample rate. Figure 8: Signal with 10 kHz sample rate. Figure 9: Second example's setup.
- 7. 6 www.dewesoft.com Figure 12: Phase comparison example. Similar to first example, first two channels have 10000 𝐻𝑧 sample rate, first has anti-aliasing IIR filter used and second has zero phase distortion anti-aliasing filter. Last, third channel has sample rate equal to acquisition rate, 20 𝑘𝐻𝑧, so analog filter from hardware will be used. Green signal was filtered with IIR filter and blue signal has zero phase distortion anti-aliasing filter. Attenuation starts later at second one (blue), at approximately 3400 𝐻𝑧. At first one, attenuation starts a lot earlier (start is cropped out of Figure 10). On Figure 10 are marked −3 𝑑𝐵 and −6 𝑑𝐵 points. Similar as before, sample rate of signal on Figure 11 is equal to acquisition rate. Analog filter is used, −3 𝑑𝐵 point is at approximately 10000 𝐻𝑧, attenuation starts at around 8900 𝐻𝑧. With next example, phase shift of anti-aliasing IIR filters will be pointed out. Dynamic acquisition rate was set to 10 𝑘𝐻𝑧 and sine sweep from 10 𝐻𝑧 to 10 𝑘𝐻𝑧 with amplitude ± 5 𝑉 and duration of 180 seconds was used on all channels. Figure 11: Signal with 𝟐𝟎 𝒌𝑯𝒛 sample rate. Figure 10: Comparison between IIR and zero phase distortion anti-aliasing filter at 𝟏𝟎𝟎𝟎𝟎 𝑯𝒛 sample rate.
- 8. 7 www.dewesoft.com In order to compare phase shifts between channels with different sample rates, for each type of filter we have two channels, one with 5000 𝐻𝑧 sample rate and another with 1000 𝐻𝑧 sample rate. For reference there is also channel with sample rate equal to acquisition rate, 10 𝑘𝐻𝑧, so analog filter from hardware will be used. Figure 13: Phase comparison az 100 Hz. Fragment of approximate two periods on Figure 13 is taken around sweep's frequency of 100 𝐻𝑧. On upper graph there are channels with zero phase distortion anti-aliasing filter applied, and on bottom are channels with IIR anti-aliasing filter. On both graphs there is also channel without software filter (red) for reference. There is quite noticable difference in phase shift between channels with different types of filters used. Both channels with zero phase distortion anti-aliasing filter (cyan and pink) are completely aligned, between each other and with the reference channel. We can't say that for channels which have anti- aliasing IIR filter used. Channel with 5000 Hz sample rate and IIR anti-aliasing filter (green) is already shifted from reference channel (red). With 1000 Hz channel (blue) this shift only expands. As the name suggests, zero phase distortion anti-aliasing filters don't introduce any phase shifts between signals with different sample rates as that happens with anti-aliasing IIR filters. Sweep signal that was used in previous example on all channels had some higher frequency orders added (50th and 100th order). On Figure 14, there is one period shown at frequency around 16 𝐻𝑧.
- 9. 8 www.dewesoft.com Figure 14: Filtered out higher frequencies. Red signal on graph is reference signal (without filter). Other two both have sample rate set to 1000 𝐻𝑧. Pink channel has zero phase distortion anti-aliasing filter and blue channel has IIR anti- aliasing filter used. At that moment, captured on Figure 14, previously mentioned higher frequencies are at around 800 𝐻𝑧 and 1600 𝐻𝑧. Zero phase distortion anti-aliasing filter nicely filters out these higher orders and outcome is a smooth sine curve, aligned with reference channel. Because of IIR filter's slower attenuation, we can still se some bumps on blue signal. They are effect form 50th frequency order, which is not yet completely filtered out. Zero phase distortion anti-aliasing filters have steeper cutoff slope, hence there is no more effect seen from 50th order.