Alias Rejection and Alias-free Bandwidth

There are two steps for converting an analog signal into a digital signal – quantization in time (sampling) and quantization of amplitude. Any signal that goes through the sampling process is subject to the effect of aliasing. The common use of the word ‘alias’ is used to describe a false name or identity, which is similar to how the term is used in signal processing.

A signal that is sampled in time has a frequency spectrum that is periodic, with identical copies of the spectrum occurring at multiples of the Nyquist frequency (half of the sampling rate). When a signal is sampled at a particular sampling rate, the sampled signal’s highest frequency is the Nyquist frequency. If frequencies above the Nyquist frequency were present in the signal before sampling, they will show up as false frequencies (aliases) lower than the Nyquist frequency.

For a numerical example, if a signal only contains a sine wave at 1050 Hz but the Nyquist frequency is 1000 Hz, the 1050 Hz will show up at 950 Hz, an ‘alias’ that did not exist in the original signal. Aliases are not able to be removed through filtering after the signal has been sampled, so it is essential for a DAQ to remove frequencies above the Nyquist rate before sampling the signal.

With a delta-sigma ADC, it is possible to use a digital decimation filter that is built into the architecture of the ADC. This allows for the analog anti-aliasing filter requirement to be relaxed (which helps to minimize channel-to-channel phase mismatch), as the digital decimation filter does most of the work to remove frequencies from the input signal that are above the Nyquist frequency.

Digital decimation filters are often designed to have a certain amount of attenuation at the Nyquist frequency, which results in a filter that has attenuation starting around 0.42∙fs, like Filter1 (black curve) shown in the following figure.

With Filter1, there will be some attenuation before the Nyquist frequency, which means that a small range of frequencies in the passband will be attenuated. In some applications this design choice may be acceptable, but not in applications where it is desirable to have the entire passband as flat as possible. For a numerical example, if the sample rate of a DAQ is equal to 4 kHz, attenuation from the digital decimation filter will be significant between 1.7 kHz and 2 kHz, which is 15% of the total bandwidth.

An alternative design option is to increase the cutoff to a higher frequency, represented by Filter2 (red curve) in the figure. This results in the desired stopband attenuation occurring at 0.55∙fs instead of 0.5∙fs. The reason this works is because the usable bandwidth can be limited to 0.45∙fs, which is where 0.55∙fs gets reflected back to when it is aliased. The tradeoff for using Filter2 is a bandwidth reduction of 10% for the benefit of a passband that has no attenuation. 

In conclusion, the Crystal Instruments DAQs provide an alias rejection of 110 dB and an alias-free bandwidth of 0.45∙fs.