What is anti-alias pre-filter for preventing aliasing after under-sampling?

Question

We know that the under-sampling results in aliasing and frequencies higher than half of the Nyquist rate is not distinguishable. I've a base band signal that I want to use the higher frequencies which are higher than the half of the Nyquist rate (Nyquist frequency) as well as low frequencies(all parts). I've a special process with this path:

$$\textrm{Input}{\longrightarrow}\boxed{\textrm{anti-aliasing pre-filter}}{\longrightarrow}\boxed{\textrm{decimate}}{\longrightarrow}\boxed{\textrm{FFT}}{\longrightarrow}\boxed{\textrm{tune on special part}\\{\textrm{of the signal}}}$$

The low-pass post-filter that people usually use as anti-aliasing filter removes the high frequencies that are of interest for me. What is the digital or analog anti-aliasing pre-filter that I don't lose high frequencies.

Answer 1

I think you're looking for a free lunch that does not exist. Your original question and response to Peter K's answer suggests that you want to sample a signal that has both lowpass and highpass content, with the highpass content extending beyond the Nyquist frequency associated with your target sample rate. That is probably not going to work.

Given a sample rate $f_s$ (and real samples), you can only unambiguously represent frequencies on the interval $[0, \frac{f_s}{2})$. More generally, you can only represent a swath of bandwidth that is up to $\frac{f_s}{2}$ wide. Frequencies above the Nyquist rate alias down such that they appear to be located in this interval after you sample them. If you have a signal of interest that meets this bandwidth constraint, then you can use bandpass sampling techniques; basically you select a sample rate taking into account the center frequency and bandwidth of the desired signal. You allow the signal to alias in a "controlled" manner, so that it appears to be present in a contiguous portion of $[0, \frac{f_s}{2})$ after you sample it (perhaps with the spectrum inverted, but that is easily fixed).

This does not seem to line up well with what you seem to want. Your question indicated that you have lowpass content (i.e. content near zero frequency) that you want to preserve in addition to highpass content above $\frac{f_s}{2}$. In many cases, this is not going to be achievable without the highpass content aliasing down on top of the lowpass signal of interest after you sample. However, under certain conditions, you might be able to make this work. If:

• The lowpass and highpass components are separated in frequency (i.e. there is a gap between the two regions where you don't care about preserving the signal's content),

• You know the center frequency and bandwidth of the highpass portion (so it is more accurately termed "bandpass" instead),

• And you have control over the sample rate,

Then you may be able to make it work. In that relatively special case, you would simply apply the bandpass sampling approach described before, except the sample rate must be selected with caution so that the higher-frequency content does not alias down into the portion of the band that the lowpass signal occupies.

Whether you would actually want to do this in a practical system is still an open problem. It's not clear specifically what you're trying to accomplish, or what the constraints in your application are. An alternate approach would be to separate the two signal components using analog filters (lowpass for one channel, highpass/bandpass for the other), then sample them independently. This could allow you to use a lower sample rate, commensurate with the bandwidths of each component.

Answer 2

Provided you can meet the conditions shown in this answer,

$$\frac{2 f_H}{n} \le f_s \le \frac{2 f_L}{n - 1}$$

your anti-aliasing pre-filter should be a bandpass filter, with $f_L$ the lower band limit and $f_H$ the higher bandlimit, filtering the signal frequencies of interest to you.

Answer 3

Put a notch in the low-pass filter where the image of the high frequency of interest will show up after sampling, and parallel this notched low-pass filter with a narrow-band filter of the high frequency spectrum of interest that is narrower than the notch.

If you can't put a wide-enough notch in the low-pass anti-alias filter so that the 2 spectra don't overlap, you won't be able to unscramble the egg. (...unless something else is going on, like a cleanly separated time-multiplexing of spectral content, etc.)