This article provides some tips on how to fine-tune the characteristics of a low-pass filter.

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In my experience, the most common filter design tasks—i.e., when you actually have to do some math, choose component values, and consider different topologies—involve the low-pass response. Electrical engineers often find themselves dealing with signals that have low-frequency information and high-frequency noise or interference. Perhaps we need to suppress unpleasant sounds in an audio signal, or remove spurious variations from a sensor signal, or eliminate the undesired spectrum created during a demodulation process. And then there are the anti-aliasing filters that help to maintain the quality of our digitized data, even when we could otherwise ignore the signal’s higher-frequency content.

Choosing the cutoff frequency of a low-pass filter initially seems quite simple, but when you think about it more carefully—such as when a real-life design forces you to think about it more carefully—you realize that there actually are some subtle details and complexities.

 

First Things First: What Is a Cutoff Frequency?

We have to keep in mind that a cutoff frequency is not some sort of precise dividing line between “good” frequencies and “bad” frequencies. Low-pass filters always transition smoothly from the passband to the stopband. Furthermore, there is nothing magical about the “cutoff” frequency, which is more accurately referred to as the –3dB frequency, i.e., the frequency at which the magnitude response is 3 dB lower than the value at 0 Hz.

 

 

It seems to me that this is the primary cause of the complication surrounding the design of low-pass filters—it’s difficult to choose a proper cutoff frequency because this term’s literal meaning refers to something that doesn’t exist in real-life circuit design, namely, a precise point at which the filter “cuts off” the undesired frequencies, leaving the desired frequencies untouched. Nevertheless, the more fundamental portion of the design task is thoroughly understanding the signals that will enter the filter and the signals that should come out of the filter.

 

Inputs and Outputs

To optimize your low-pass filter, you have to know as much as possible about the expected frequency content of your input and the desired frequency content of your output.

If you plan to use a first-order filter, the frequency response will always have the same basic characteristics, and consequently there are only two generic scenarios that occur to me:

 

Focusing on the Passband

In this first situation, there are frequencies toward the end of the passband that cannot experience significant attention. For example, you know that all of your signals will be below 10 kHz, but you have an important sensor output that tends to stay around 7.5 kHz. You may not want a cutoff frequency of 10 kHz because this would apply almost 2 dB of attenuation to the 7.5 kHz signal:

 

 

In this case it would be better to increase the cutoff frequency until 7.5 kHz is closer to the almost-flat section of the passband. For example, if you push the cutoff out to 20 kHz, the attenuation at 7.5 kHz is less than 0.6 dB:

 

 

Further increases in cutoff frequency would produce corresponding reductions in the 7.5 kHz attenuation, but as usual, there are trade-offs involved. By increasing the cutoff, you include more unnecessary (and possibly noisy) frequencies in the passband and you reduce the attenuation of frequencies in the stopband. (In this context “stopband” is a vague term—I’m using it to refer to frequencies that experience major attenuation, say, at least 20 dB.)

 

Focusing on the Stopband

The second scenario is when the priority is to suppress a particular frequency in the stopband, rather than to preserve a particular frequency in the passband. For example, you might have a clock signal or RF transmitter that is contaminating your fragile analog signal. You are relying on a low-pass filter to suppress this interference (you’re not using a notch filter because you also want typical broadband noise reduction).

Within the limitations of a first-order filter, all you can do to increase the attenuation at a particular frequency is move the cutoff closer to 0 Hz. If you can’t thoroughly attenuate a strong interfering signal and adequately preserve the amplitude of the signals in the passband, it’s time to think about a second-order filter.

 

The Second-Order Option

This is an expansive topic, and we’re only going to skim the surface here. The important thing to understand in the context of this article is that second-order filters are more malleable. A second-order filter can be adjusted so as to offer a flatter passband (a Butterworth filter), a steeper roll-off (a Chebyshev filter), or a more-linear phase response (a Bessel filter). The following three plots provide a visual comparison of Butterworth, Chebyshev, and Bessel responses.

 

Generated using Analog Devices’ Analog Filter Wizard.

 

Generated using Analog Devices’ Analog Filter Wizard.

 

Generated using Analog Devices’ Analog Filter Wizard.

 

The relationship between cutoff frequency and the characteristics of second-order filters is the following: Your choice of cutoff frequency might be influenced by the type of filter that you use.

Let’s say you have strict requirements for suppressing a higher-frequency interfering signal. If you use the Chebyshev filter, which has a rapid transition from passband to stopband, you may not have to decrease your cutoff frequency as much as you initially thought. If your primary concern is preserving the amplitude of a frequency somewhere in the passband, a Butterworth filter’s flat passband response gives you more flexibility in locating the cutoff frequency.

 

Summary

Choosing a cutoff frequency begins with having a vague idea of which frequencies should pass through the filter and which should be blocked by the filter. After that, you have to consider the details of the filter’s transition from low attenuation to high attenuation, along with the frequency content of the input waveforms and the signal-processing objectives that the filter is expected to accomplish.

 

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