# Ignoring Nyquist

What happens if I ignore Nyquist? I have a 16.368MHz digital signal coming out of a GPS front-end chip. My microcontroller (which is reading that digital stream of data) operates at a maximum of 32MHz so sampling at ~16MHz seems unlikely. If I undersample my signal with say M = 3, then I have a 5.456MHz signal which I can sample.

The Intermediate Frequency of the digital signal coming from the GPS chip is at 4.092MHz, it has a bandwidth of about 2MHz but there does seem to be noise at all frequencies. So clearly, downsampling shouldn't destroy my signal. When I simulate the downsampling, even values of M seem to completely corrupt the data (frequency domain looks like pure noise) however odd values of M give a recoverable spectrum.

So what if I do undersample my signal without filtering it first? Is there any way to quantify the extra loss this will add to my signal without ever recovering the full 16.368MHz digital signal? Also, are there any tricks I could use here to avoid this problem completely?

UPDATE: Here is a frequency domain representation of the 16.368MHz signal (and yes it's a Real signal, despite me plotting both sides of the spectrum):

• Can you say a little more about where the plot comes from? Is it the 16.368 MHz signal sampled at ~16 MHz? If so, is there any way you can obtain a spectrum sampled closer to 40 MHz? It is hard to tell what I am looking at now, if the aliasing has already taken place. Oct 30, 2013 at 18:24
• The plot is the frequency spectrum of the 16.368MHz digital signal coming out of the ADC, it's already been downconverted in the analog front end. The intermediate frequency is 4.092MHz. The signal being sampled is at 1575.42MHz (GPS L1, C/A). The peaks you see at +/-4.092MHz are due to the IF filter as the GPS signal is well below the noise ( > 20db below noise). We'd expect to see a flat noise spectrum if the noise wasn't filtered. According to the front end chip's datasheet the IF filter's bandwidth is 2.2MHz. Nov 1, 2013 at 0:40
• I think I'm getting closer to understanding the plot. So in the signal you are sampling, the GPS signal is at the IF of 4 MHz? Or is it at baseband? Nov 1, 2013 at 19:00
• Yep, the GPS signal is at the IF of ~4MHz. Nov 3, 2013 at 5:00

You should never ignore Nyquist, but you can successfully sample bandpass signals at a rate lower than Nyquist if you are careful. This is often called undersampling or bandpass sampling.

The main thing to keep in mind when undersampling is that, not only does the signal of interest alias down to baseband, but so does much of the out-of-band noise. The filtering that precedes downsampling is not usually meant to condition the signal itself, it is meant to squash all of the noise that would otherwise alias on top of your downsampled signal. The more you downsample (higher values of M), the more this effect will bite you.

The other thing to remember when undersampling is that a real signal has positive and negative frequency support. Ignoring this fact can cause your negative frequency aliases to land on top of your positive frequency aliases. This might explain what you are seeing with even-parity vs. odd-parity values of M.

• You say that the filtering that precedes downsampling is not meant to condition the signal itself. By this do you mean that it's a band-pass anti-aliasing filter? Designed to reject noise which is not in the signal's narrow bandwidth so that when you downsample the folded noise (outside of the bandwidth of the signal) will never collide with the signal itself. Oct 30, 2013 at 1:15
• Yes... Why do you ask? Oct 30, 2013 at 5:39
• I was looking for a way to quantify the noise. It looks like, assuming the signal is small compared to the noise (as it is in a GPS signal), that I will end up with a M/2 times more noise in my signal, equivalent to a gain of 2/M. Nov 1, 2013 at 0:29
• It seems to me like your noise will increase by a factor of M when downsampling by a factor of M. I have never worked with GPS signals, but when you already have a negative SNR it seems like decreasing your SNR by 5 to 10 dB is a pretty big deal Nov 1, 2013 at 18:55

The Nyquist limit is not in regards to the highest frequency of the signal, but in regards to the total bandwidth of the signal being sampled.

You can under-sample a signal much higher in frequency than Fs/2, without filtering, as long as the contiguous frequency band of the signal does not touch or cross any of the folding frequencies (N * Fs/2 for all integer N). (Or, if the signal spectrum is sparse, as long as there is no overlap of any spectrum frequencies with non-zero energy after folding).

But note that any jitter in the sampling clock creates a lot more sampling noise if the frequency band being sampled is above Fs/2.

• +1, Consideration for sampling clock jitter can be very important in this type of situation. An ADC which performs adequately at sampling signals below Nyquist may be too noisy to perform bandpass sampling. Oct 30, 2013 at 18:09