4 ms·

A 1Hz sawtooth contains frequencies above 1Hz. It actually has frequency components that go out to infinity, so its impossible to perfectly reconstruct a sawto

• PaulDavisThe1st 4y ago
  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 considered as a sine series. While there are such sounds which are most easily described as a sawtooth, when you consider the physical/mechanical process by which they are formed, the sine series is a more obvious approach. 3) A digital 1Hz sinusoid can trivially contain no harmonics at all. However, the moment you attempt to convert this into an acoustic pressure wave, the nature of the physical world essentially guarantees that the acoustic pressure wave will have a series of harmonics going out far beyond the base frequency. Once you start actually moving things (like magnetic coils, speaker cones and air), it's more or less impossible to avoid generating harmonics. But since the original signal was genuinely a pure sine tone, it becomes a little tricky to decide what the correct way to describe this is.
  • stagger87 4y ago
    At the "textbook"/"theory" level, the person you are replying to is not wrong. A sawtooth waveform has infinite harmonics. If you were going to be nitpicky (which your response was in that spirit), the best thing to have said (IMO) was that the high frequency harmonics are going to drop off and be below any sort of "noise floor" or sensitivity of the system and not matter anyways. Instead you wrote a bunch of stuff about sounds and pressure waves that I don't think had the effect you intended. I think you lost the plot somewhere along the way.
    • PaulDavisThe1st 4y ago
      > 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 you want to synthesize a sawtooth wavefrom with analog equipment, then thinking of it as an (infinite) sine series makes sense, because that's how you will end up approximating the (perfect) sawtooth. However, digital synthesis does not require this sort of conception at all, and can be constructed without any summing of a harmonic series. Also, I find it assuming that in the comments of a post about nyquist, you would write > a bunch of stuff about sounds and pressure waves that I don't think had the effect you intended. I think you lost the plot somewhere along the way. What do you the plot is?
      • jameshart 4y ago
        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. Those physical constraints manifest as limits on the frequency of sinusoidal harmonics it is possible for you to put into the wave; for the medium to carry; and for you to physically detect at the other end. Mathematicians don’t break functions down into sinusoidal harmonics because they like trig functions. They do it because they fundamentally are what’s happening.
        • PaulDavisThe1st 4y ago
          > 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.
      • Gordonjcp 4y ago
  • titzer 4y ago
    > 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 recognize and then reconstruct a sawtooth with no a priori knowledge, you need to sample much higher than the frequency of the sawtooth. You can compress said information quite well if you have not only wavelets, but sawtooths in your encoding. I am no audio expert but I don't think codecs exploit sawtooths (sawteeth?) for compression because they sound unnatural (because they are). Note that even digitally you can't create a perfect sawtooth wave because there is a fundamental quantization of time in digital systems. It's a question of, again, how fast you can alter voltages, i.e. a frequency, so you end up generating a step-like function, inescapably. Yeah, sure, you can switch digital systems at MHz or GHz, but still.
    • ska 4y ago
      > 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.