## Understanding Filter Characteristics

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.

Figure 1: A Simple Sinewave

Figure 2: A Swept Sinewave

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.

Figure 3: Spectrum of simple sinewave

Figure 4: Spectrum of swept sinewave

So the two types of sinewave are quite similar when viewed in the time domain. If viewing a single cycle it would be hard to distinguish them. However their frequency content is very different. Importantly, the frequency content in the swept sine wave changes uniformly across the time range of the signal.

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.

Figure 5: Characteristics of 500Hz low pass filter

The high pass, shown in Figure 6, would block frequencies below 500Hz, but allow frequencies above 500Hz.

Figure 6: Characteristics of 500Hz high pass filter

The band pass, shown in Figure 7, would block frequencies below 250Hz, allow frequencies between 250Hz and 750Hz, then block frequencies above 750Hz.

Figure 7: Characteristics of 250Hz to 750Hz band pass filter

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.

Figure 8: Characteristics of 250Hz to 750Hz band stop filter

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 9: Swept sinewave before filtering

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 10: Swept sinewave with low pass filter applied

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 11: Swept sinewave with high pass filter applied

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 12: Swept sinewave with band stop filter applied

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.

Figure 13: Swept sinewave with band pass filter applied

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…

Audio Equalisation Filter & Parametric Filtering

High Pass Filtering And Tacho Signals

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#### James Wren

Solutions Engineer and Sales & Marketing Manager at Prosig
James Wren is a Solutions Engineer and the Sales & Marketing Manager for Prosig Ltd. James graduated from Portsmouth University in 2001, with a Masters degree in Electronic Engineering. He is a Chartered Engineer and a registered Eur Ing. He has been involved with motorsport from a very early age with special interest in data acquisition. James is a founder member of the Dalmeny Racing team.

## 12 thoughts on “Understanding Filter Characteristics”

1. Jon Wilson

I teach and consult on shock and vibration testing and measurement. I would like to use some of this material in my seminars, if it is available. I will provide proper attribution.
May I have permission to use it? Can you email it to me in a form that I can adapt to PowerPoint slides?

Jon Wilson
Jon S. Wilson Consulting

2. James Wren Post author

Hello Jon,

Thank you for posting on our blog.

We would be more than happy to support you and provide you with both our permission to use this material and the material itself as long as we are correctly attributed.

I will contact you directly to arrange to supply you with the material.

3. Kyle

James,
I am working on a project for my analog electronic class and am having troubles finding a band pass circuit that can create a curve like the one you display. could you perhaps post a link or give me a little insight on how to create a bandpass that has that sharp of drops at the low and high limits?
Thanks,
Kyle

4. James Wren Post author

Hello Kyle,

Thank you for asking a question on our blog.

I should explain about the differences between DSP (Digital Signal Processing) and Analogue Electronics.

The signals processed by a DSP software package, when filters are applied, have a number of passes.

The number of passes is the number of times the filter is applied to the signal, each pass sharpens the roll off, 1 pass would look more like a pyramid, 10 or 12 passes more like a tall sky scraper.

In the example above, band pass type filter, the number of passes was 8, so that means the signal was filtered 8 times, in order to repeat it you would need to build 8 filters in series.

5. Ronald Fonseka

Dear Mr. James Wren
I am experimenting on an Ultrasonic Monkey Repeller giving a square wavw output
at 17~25 Khz. When the unit is switched off, I get an ear piercing low frequency
sound for a few seconds.
Could you please give me the values of a RC filter that will attenuate the low frequencies.
Thanks & Best Regards,
Ronald

6. James Wren Post author

Hello Ronald,

Thank you for asking a question on our blog.

If you build a simple first order high pass filter, from a simple Resistor and Capacitor, you will need to apply the following formula to set the filter frequency,

$F_c = \frac{1}{(2\pi}RC)$

Where Fc is in Hertz and R is in Ohms and C is in Farads.

The frequency Fc will be the cut off frequency and about which point the output power will be half the input power.

So it should simply be a case of select the cut off frequency that suits your requirements, I would suggest 5,000Hz cut off should cut out the low frequency’s your having a problem with.

1. James Wren Post author

Hi Ankonmahmud,

Thank you for asking a question on our blog.

The article details the low pass filter, I have copied the following section for you.

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.

I believe this covers the low pass filter very well, what sort of characteristics would like to consider further?

Please feel free to post back any further points you’d like to raise.

7. samir

Hi! Please, can you explain some more about amplitude correction of high-pass filtered signal? Ouput signal (after RC filter) always have lower amplitude from cut-off frequency to -3db frequency. Might we correct this amplitude if we interested in this frequency range? Thanks.

1. James Wren Post author

Hello Samir,

Thank you for posting a question on our blog.

I would like to understand why you think amplitude correction is necessary. If the high pass filter is correctly implemented, then there will no requirement for any corrections in the pass band.

Please keep in mind that the filter cut off frequency is the 3dB point. That means that the signal will have been attenuated by 3dB at the cut of frequency.

8. riduan

Greeting James,

I wan to ask some basic question which i still not understand:

Would you explain to me how to design slope steepness in design of low pass, high pass, band pass and band stop graph?

In frequency domain graph, what is the meaning of spectrum? What is the peak value mean in that graph? Why the graph resemble like mountain? Please give me clear explanation in this thing.

What is the meaning with db point? How to intepreting it in time domain signal when frequency weighting applied to the signal. Some graph db point vs frequency in iso 2631 still confusing me until now.

Thank you.

1. James Wren Post author

Hi Riduan,

Thank you for posting on our blog.

It sounds like you may need more assistance than we could provide on our blog.

Have you considered a signal processing training course?
We can provide these locally or at our head offices.

Please let us know if that is something you would be interested in discussing?

I can guide you on one or two issues however.
The 3dB point is the X axis point at which a filter has cut in and attenuated the signal by 3dB.

The roll off or roll on rate of a filter is effected by the type of filter and the number of times the filter is applied or ‘passed’.

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