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Nyquist Frequency

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Re: Nyquist Frequency

#41

Post by InitialLastName »

spacechild1 wrote: post #35
_kb wrote: post #33

Had a great encounter with this recently! In an environment I work there's multichannel audio recordings that are archived. The archival recordings all had a perfect 4kHz tone appearing, seemingly out of nowhere. This was happening on every channel, across every room, but only in one building. Nowhere else. Absolutely nothing of the sort showed up on live monitoring. The systems were all the same and yet this behavio…

> It turns out the archival was downsampling to 24kHz But... why?

In situations where you don't need the archival to be at "perfect reproduction" quality (including things like broadcast archives or recordings of voice comms) you can get by with a 12kHz maximum frequency without losing the essentials (especially clarity of voices). Many adults can't hear much past 12kHz anyway and most music and voice content doesn't have content past 10khz. You don't lose much, but you save half your file size by x2 downsampling.

Re: Nyquist Frequency

#42

Post by gaze »

EarthIsHome wrote: post #3

One misconception that many make regarding the Nyquist frequency is thinking that the sampling rate needs to be twice the highest frequency. Your sampling should really really be twice the bandwidth . e.g. your bandwidth is 100 MHz centered at 1 GHz (it needs to actually be bandlimited to 100 MHz**). You do not need to sample at 2.2 GHz. You sample at 200 MSPS (really, you should sample a little more than that, say 2…

Yeah but you also need the bandwidth of the sampler to exceed the highest frequency of the sample. Most samplers are limited by some kind of RC time and not their sinc envelope. Most.

Re: Nyquist Frequency

#43

Post by ska »

titzer wrote: post #36
PaulDavisThe1st wrote: post #23

Earlier quoted context omitted.

This is incorrect, though subtly, and for several different reasons: 1) It is completely possible to create a sawtooth wave that contains only a single frequency. However, you could also consider the wave to be an (infinite) sum of sinusoids at different frequencies. Both views are "correct", and which is more appropriate depends on the context. 2) Related to (1): natural (acoustic) sounds are almost always best cons…

> It is completely possible to create a sawtooth wave For a loose definition of "wave". All of the math behind information theory and sampling signals assumes waves are sinusoids. It also happens that waves in nature behave like (dampened) sinusoids. It's a completely natural way to model them mathematically when one has no a prior knowledge of the source, which is what the comment above you is pointing out. To recog…

> All of the math behind information theory and sampling signals assumes waves are sinusoids

This isn't really true. The point about the sinusiods is mostly that the form a very convenient complete basis of a useful space of functions, hence the fourier expansion. This doesn't amount to an assumption about how the signals are generated, rather how they are represented. You could pick a different basis and you'd get a different representation, but as functions they are identical. By definition this applies equally to any signal in the class, however you generate it.

Where the shape of the underlying basis vectors does show up is in errors and estimation, e.g. the similar estimation error in fourier vs. Haar will show up as sinusoids or steps.

Re: Nyquist Frequency

#44

Post by PaulDavisThe1st »

jameshart wrote: post #34
PaulDavisThe1st wrote: post #31

Earlier quoted context omitted.

> A sawtooth waveform has infinite harmonics This is only true if you consider the waveform to be a sine series. As I indicated, this is a perfectly legitimate way to think about a sawtooth (and indeed, it appears to be fundamentally how the human ear works too). But a sawtooth waveform is also nothing more than a very sharp rise/drop in air pressure followed by a longer drop/rise, repeated over and over again. If yo…

You can’t physically construct a speaker that makes a sawtooth wave. Its cone would need to change velocities from -n to +n or vice-versa instantaneously in order to generate the ‘teeth’ of your wave. The air particles you are moving would likewise need to instantly accelerate. That is a physical impossibility - these things have mass, accelerating them requires force, infinite acceleration requires infinite force. T…

> You can’t physically construct a speaker that makes a sawtooth wave.

This was my point (3), though you've added an additional set of reasons why it is particularly hard for shapes like a sawtooth.

Re: Nyquist Frequency

#45

Post by abstrakraft »

Sesse__ wrote: post #27
abstrakraft wrote: post #20

Earlier quoted context omitted.

From an information theoretic perspective (which is the perspective Nyquist was originally coming from, though it didn't yet have that name), you don't need to mix the signal down. Assuming it is truly band-limited, you can sample the signal directly at RF, and reproduce it from those samples. Additionally, you will need to modulate the reproduced signal into the original band, which means you need to know where that…

I've worked on ultrasound systems that definitely worked this way, not just in theory but also in practice. Bandpass filter 20–40 kHz, sample directly at 40 kHz (giving 20 kHz bandwidth). No mixer step involved, but your spectrum becomes inverted (e.g. if you do an FFT, a 22 kHz tone will be in the 18 kHz bin, not the 2 kHz bin as you would perhaps expect).

Aliasing makes more sense (to me, anyway) if you think about the spectrum of complex signals, in which signals of real samples are modeled as the sum of positive and negative frequencies.

In the sampling operation, all sinusoids are shifted down to the "natural baseband" by adding or subtracting some multiple of the sampling frequency that places the resulting frequency within +/- half of the sampling frequency. So for your example of 22kHz, that real frequency has two components: +22kHz that gets shifted down to -18kHz=22kHz-40kHz, and -22kHz that gets shifted up to +18kHz=-22kHz+40kHz.

Note that this "natural baseband" is an abstraction of our own invention. You can just as easily think of the spectrum as ranging from 0Hz to the sampling frequency f_s, rather than -f_s/2 to f_s/2. The fact that some prefer one over the other is precisely why fftshift exists.

Re: Nyquist Frequency

#46

Post by Sesse__ »

InitialLastName wrote: post #41
spacechild1 wrote: post #35

Earlier quoted context omitted.

> It turns out the archival was downsampling to 24kHz But... why?

In situations where you don't need the archival to be at "perfect reproduction" quality (including things like broadcast archives or recordings of voice comms) you can get by with a 12kHz maximum frequency without losing the essentials (especially clarity of voices). Many adults can't hear much past 12kHz anyway and most music and voice content doesn't have content past 10khz. You don't lose much, but you save half y…

I'd guess the “why” was “why on earth did they not have an antialiasing filter”, not “why did they downsample”. A good lowpass filter is easy to design, cheap to apply, and protects you from this kind of stuff.

Re: Nyquist Frequency

#47

Post by lumb63 »

Signals and systems was a tough course for me. It was what crushed my 4.0 GPA. Nyquist frequency was a concept I could not wrap my head around. I’ve improved, but it still doesn’t click as I’d like it to.

When I took the course, it made no sense to me that you could sample at twice the frequency of the signal and reconstruct it. Consider a sine wave at 1 Hz. If you sample at 2 Hz, you’d get readings of 0, 1, 0, -1, etc. If you graph that, it’s a perfect triangle wave, not a sine wave! That’s what I couldn’t not get past. I thought you’d need an infinite sampling rate to accurately capture the sine wave.

As I type this out, I’m realizing that a critical component of this that I wasn’t taught (or I didn’t grasp) is the need for the signal to be bandlimited. Returning to my sine example from above, what bothered me was, if I don’t sample more points, how do I know that it’s only a sine wave, and nothing more? That only works if you pretend there are no higher frequencies (or filter them out, though an ideal filter is impossible in practice). If there aren’t higher frequencies, there can’t be anything you “can’t capture” by sampling at the Nyquist frequency.

Re: Nyquist Frequency

#48

Post by InitialLastName »

Sesse__ wrote: post #46
InitialLastName wrote: post #41

Earlier quoted context omitted.

In situations where you don't need the archival to be at "perfect reproduction" quality (including things like broadcast archives or recordings of voice comms) you can get by with a 12kHz maximum frequency without losing the essentials (especially clarity of voices). Many adults can't hear much past 12kHz anyway and most music and voice content doesn't have content past 10khz. You don't lose much, but you save half y…

I'd guess the “why” was “why on earth did they not have an antialiasing filter”, not “why did they downsample”. A good lowpass filter is easy to design, cheap to apply, and protects you from this kind of stuff.

I was working off the quote, but I can see some reasons that someone would decide not to AA filter. Depending on the context it might be reasonable to assume that the signal is band-limited anyway (talk-oriented radio especially is often low-pass filtered) and it's easy to miss that some point in the system can introduce an (inaudible to most humans) artifact. Those assumptions, along with the desire to avoid complexity (every step in the signal path is an opportunity for failure) could easily tip you to "just downsample".

I'd also emphasize how little most of the people involved in these systems care about the quality of the archive. If it's good enough to a) confirm there was signal on the channel and b) understand the voices involved, it's good enough to not worry about further.

Re: Nyquist Frequency

#49

Post by mikepavone »

labcomputer wrote: post #38
mikepavone wrote: post #17

Earlier quoted context omitted.

Is this assuming you have some analog hardware that's demodulating the signal in front of your ADC? How do you demodulate a signal from a 1GHz carrier with 200 MSPS?

As the sibling comment mentioned, you don’t need to demodulate first, because that is actually what the sampling process of your ADC does. You can think of it as multiplying the original signal by a comb (in the time domain) of delta functions, which folds everything (in the frequency domain) back into the nyquist frequency of your ADC. Each delta function corresponds to one sample. If your original signal was truly…

Thanks for the explanation! Between your comment and the Undersampling wiki page diydsp linked to I think I am on the path to enlightenment.

> If your original signal was truly band-limited to 100MHz

In practice, this means you need to band pass before the ADC, right? i.e. "signal" in this case is the entire input to the ADC and not just the particular modulated signal you care about

Re: Nyquist Frequency

#50

Post by tomjakubowski »

lumb63 wrote: post #47

Signals and systems was a tough course for me. It was what crushed my 4.0 GPA. Nyquist frequency was a concept I could not wrap my head around. I’ve improved, but it still doesn’t click as I’d like it to. When I took the course, it made no sense to me that you could sample at twice the frequency of the signal and reconstruct it. Consider a sine wave at 1 Hz. If you sample at 2 Hz, you’d get readings of 0, 1, 0, -1, e…

A triangle wave at 1Hz would have many higher frequency components. If you know a priori that the highest frequency of the signal is 1Hz, sampling at 2Hz is enough to infer 0, 1, 0, -1, ... came from a sine wave.
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