# Anti-aliasing Filter Design and Applications in Sampling

January 13, 2020

Aliasing is a well-known phenomenon in digital photography, but it also arises in analog-to-digital conversion.

Whenever I watched car commercials where the camera pans along the driving vehicle, there was always one particular speed where the spokes in the wheels appeared to remain stationary or even move backwards. When I was younger, I racked my brain trying to figure out how this was possible and why viewing the wheels on camera made a difference. I later learned about sampling rates with digital systems and how this leads to a phenomenon called aliasing.

When aliasing occurs, the signal you’re trying to sample appears to have a constant level, i.e., you cannot observe an oscillation. This happens when the sampling period is equal to the oscillation period, and the sampling device will only reliably measure a fixed signal. This problem can be overcome in a few ways, depending on the harmonic content of your analog signal and the speed of your system.

## Anti-aliasing Filter Design Points

An anti-aliasing filter is effectively a low-pass filter that is placed on the input of an analog-digital converter (ADC). In theory, any type of active low-pass filter with unity gain can be used as an anti-aliasing filter. Any anti-aliasing filter design intends to remove high frequency content from the signal you want to sample with the goal of preventing aliasing. In this way, the goal in anti-aliasing filter design is to set the filter’s -3 dB cutoff frequency equal to half the sampling frequency.

When sampling an analog signal with a specific center frequency, aliasing will occur if the sampling frequency is less than double the center frequency. In other words, if the signal’s center frequency is more than half the sampling frequency, then aliasing will occur, and the ADC will detect a low alias frequency signal in error, rather than detecting the desired center frequency of the input high frequency signal. In the case when aliasing occurs, the ADC will output a series of digital pulses that quantifies a signal with frequency equal to the difference between the sampling rate and the center frequency. This particular sampling frequency at which the system can prevent aliasing is called the Nyquist frequency.

The relationship between accuracy in the detected signal and the sampling rate is shown in the image below. As long as your sampling rate is high enough (more than double the signal frequency), you can always detect the signal with high accuracy; quantization becomes more accurate as long as you do not increase the noise level. The tradeoff here is that the ADC uses more power, so it generates more heat.

Accuracy in signal sampling as the sampling rate increases.

## Sinusoidal vs. Arbitrary Analog Signals

If you’re trying to sample a perfect sinusoid, the only job of an anti-aliasing filter is to remove noise. You are effectively trying to sample a single sinusoidal frequency, and you’ll only need to set the sampling rate to at least double the signal frequency. Your primary consideration at this point should be to use a device with the correct resolution and remove noise in order to prevent quantization errors.

If you’re working with an arbitrary analog signal, such as an RF pulse for radio-over-fiber or a voltage impulse for a laser driver circuit, your application might require sampling the signal as part of a feedback loop or signal conditioning loop. Arbitrary signals contain harmonic components that are spread out over a broad spectrum, and the frequency content may exceed the Nyquist frequency. This means any frequency content you try to sample above the Nyquist frequency will cause aliasing. When the input signal is quantified, the digital output will reflect a distorted version of the original input.

This issue with sampling and aliasing is illustrated in the figure below. The graph shows a spectrum for an arbitrary signal (red curve). Note that the sampling rate (fs) is set higher than the end of the spectrum. However, the Nyquist frequency (half the sampling rate) falls into the middle of the spectrum. Any frequency components below the Nyquist frequency can be accurately sampled, while all higher order frequencies will be aliased and will be interpreted incorrectly as low frequency components by the ADC.

Schematic showing aliasing and distortion with an arbitrary analog signal

In this example situation, you have two options to prevent the digitized output from failing to accurately reflect the input analog signal:

1. Increase the sampling rate so that the Nyquist frequency is larger than the end of the frequency spectrum. As arbitrary signals have frequency content that extends out to infinity, you cannot increase the sampling rate to infinity. The other option is to choose a suitable maximum frequency that you need to sample. This brings us to the second point...

2. You should use an anti-aliasing filter to remove all frequency content greater than the Nyquist frequency.

This second point should illustrate the advantage of an anti-aliasing filter. An anti-aliasing filter is just a low pass filter with the cutoff frequency (i.e., the -3 dB frequency) set to the Nyquist frequency. This filter cuts out any higher order frequency content in the input signal as any frequencies higher than the Nyquist frequency would be aliased. With these frequencies removed from the signal, the ADC can now sample the remaining harmonic content without creating false low-frequency errors.

## Designing an Anti-aliasing Filter For Use With an ADC

Anti-aliasing filters are typically designed as higher order active filters using a low-noise op-amp. The goal is to design the filter with unity gain across the pass band and to set the -3 dB cutoff frequency to be set precisely equal to the Nyquist frequency, which in turn is half your intended sampling rate. If you are using an adjustable ADC, always set the -3 dB cutoff to correspond to the Nyquist frequency for the smallest sampling frequency you intend to use in your system.

Anti-aliasing filter design is all about engineering the transfer function for an active filter. This can be as simple as selecting a wideband op-amp and wiring a low pass RC filter to the non-inverting input. A slightly more advanced design is to use higher order filtering as this will provide stronger rolloff beyond the -3 dB point in a Bode plot. An example is shown below.

Example second order active low-pass filter that can be used as an anti-aliasing filter.

In the case your signal chain has high noise floor, you can always increase the sampling rate to spread noise out over a larger bandwidth, and then pass the signal through an anti-aliasing filter with the desired cutoff frequency. This is one way to work with noisy signals with an ADC and an anti-aliasing filter.

This also allows you to use an anti-aliasing filter with a lower cutoff frequency alongside an ADC with adjustable sampling rate. In the event that the noise floor becomes too high, you can increase the sampling rate while keeping the cutoff frequency at the original value. This spreading of noise over a broader bandwidth reduces the level of noise that propagates to the ADC output, which reduces quantization error and allows higher resolution to be used.

Creating an anti-aliasing filter and bringing it into your application is much easier when you the right PCB design and analysis software. OrCAD PSpice Simulator and Cadence’s full suite of analysis tools allow you to simulate the behavior of your signal chain and how your filter will sample analog signals. You’ll also have access to manufacturer part search tools as you select the components you need for an anti-aliasing filter and gain the ability to quickly associate a PSpice model to these filters if you do not already find a simulation model for them.