Recently when discussing with an engineering student the characteristics of filters, it became clear that some confusion exists around this subject area. This note attempts to explain the differences between types of filter and the effects of the parameters of those filters.
To begin we will cover some basics of signal processing. This article uses swept sinewaves to explain filtering, so first we must understand what they are.
We can seen the simple sine wave is a repeating pattern, but the swept sine wave is increasing in frequency. That is, the time between the peaks is reducing. The simple sine wave has a fundamental frequency of 1Hz, but the swept sine wave has a varying frequency, starting at 1Hz and finishing at 10Hz over the 2 seconds time period.
Figures 3 and 4 show the simple sinewave and swept sinewave in the frequency domain. As we can see the sinewave has one dominant frequency spike, whilst the swept sinewave shows a spread of frequencies representing the range of the sweep from 1Hz to 10Hz.
The rest of the article will discuss the swept sine wave and the effects of certain types of filter on this swept sine wave.
Let’s look at the 4 basic types of filter. Low pass, high pass, band pass and band stop. Each of these filters has different frequency characteristics.
Low pass filters will allow the low frequencies to pass through, but block the high frequencies. The cut off frequency is the frequency that the filter begins to attenuate the content. So a low pass filter set at 100Hz will remove the frequency content above 100Hz, but not below 100Hz. It follows that a sinewave with a fundamental frequency of 10Hz would not be affected by a 100Hz low pass filter. But a sinewave of 200Hz would be heavily affected by a low pass 100Hz filter as the frequency content above 100Hz would be removed.
High pass filters are the opposite to low pass filters. They remove the frequency content below the cut off frequency.
Band pass filters will have a low and high cut off and will pass frequencies that fall between these two limits.
Band stop filters will block the frequency content between the lower cut off and the higher cut off.
We call the rate at which the filter attenuates the frequency content, the roll off rate. The filter cut off point for a low pass filter of 100Hz does not mean that the filter begins to work at 100Hz. This means that the filter will have attenuated the signals amplitude by about 30% at that point. This is known as the filter 3dB point, where the energy or power of the signal has reduced by 50% (and the amplitude reduced by a factor of 0.7071). The ‘rate’ of the roll off is measured in attenuation per frequency (dB per octave). This is the number of dB being attenuated per frequency octave, where an octave is a doubling of frequency.
Figure 5 shows the characteristics of a low pass filter, this example would allow the low frequencies to pass but block frequencies above 500Hz.
The high pass, shown in Figure 6, would block frequencies below 500Hz, but allow frequencies above 500Hz.
The band pass, shown in Figure 7, would block frequencies below 250Hz, allow frequencies between 250Hz and 750Hz, then block frequencies above 750Hz.
The band stop filter, shown in Figure 8, would allow frequencies up to 250 Hz, block frequencies between 250Hz and 750Hz, but allow frequencies above 750Hz.
Which poses the next question – how would a swept sine wave be affected by these different filters?
Figure 9 shows the first 5 seconds of the swept sine wave before we have applied any filtering. This swept sinewave starts at 1Hz at t=0 seconds and increases to 1000Hz (or 1kHz) at t = 5 seconds.
Figure 10 shows the full swept sine wave after we have applied the Low pass filter. We can see how the signal is unaltered initially, but as the frequency approaches, and passes, the 500Hz cut off we attenuate more and more of the signal.
Figure 11 shows the full swept sine wave after we have applied the high pass filter. Here we see how the signal is attenuated at lower frequencies, but as it passes the 500Hz cut off more of the signal passes through the filter.
Figure 12 shows the full swept sine wave after we have applied the band stop filter. Clearly we can see how the filter attenuates the signal as the frequency of the swept sinewave passes through the 250Hz to 750Hz region.
Figure 13 shows the full swept sine wave after we have applied the band pass filter. Here, we see the opposite effect, where the filter only passes frequencies lying between the two cut offs.
Several properties of a filter can affect the precise form of the output. There are, for instance, many different types of filter (Butterworth, Chebyshev etc.). Also, we should consider the number of passes. This is simply the number of times we apply the filter algorithm to the signal. The more times it is applied the sharper the roll off rate. However, as well as changing the amplitude, passing data through a filter causes phase changes or delays in the output signal. The real change is frequency sensitive and depends on the number of passes, the cut off frequency and the filter type. To find out more about this and how you can use phaseless techniques to filter data, see the earlier article Removing Phase Delay Using Phaseless Filters
If this has whetted your appetite and you would like to read more about filtering techniques, you may also be interested in…
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