Why does a quick pulse of a pure tone have overtones?

[richard31416]

TL;DR Summary

Why does a short burst of a pure tone sound like a hollow tock, not like a tone? What causes the overtone spectrum we hear as hollow?

I'm new to the forum and not sure I've chosen the right section. And while not an engineer or physicist, I'm pretty scienc-y.

I've been wondering this since I was a kid listening to shortwave radio, and decided in my 60s to try once more to understand. The time broadcasts from NIST stations WWV and WWVH include seconds markers that are 5 ms of pure sine-wave tones (1000 Hz or 1200 Hz). But we hear them as hollow tocks, not tones (it's easy to reproduce this with simple tone generators). How come? I’m aware that it results from the overtone spectrum, but what causes an overtone spectrum from 5 ms of a pure tone? An engineer I asked seemed to say that short pulses of sound give out their own spectrum of overtones over and above the underlying sound. (His exact words were: “When you send a tone modulated by a pulse -- in this case a pulse of 5ms -- the spectrum is the result of convolving the spectrum of the pulse with that of the pure tone. “) Is that about right? Is there any way to explain why that is so? Much indebted if anyone can do some explaining or lead me in the right direction.

--Richard

 

[phinds]

It's complicated. Google "Fourier Analysis".

 

[anorlunda]

Elaborating on what @phinds said, here are the results of the Fourier spectrum for a burst of tone.

Whereas below are the Fourier spectrum of a pure tone (continuous, not burst)

Forget the numbers on the axes, just look at the shapes.

 

[BrucePerens]

Ham Radio operators have known for about a century that if you allow the rise and fall times to be too sharp, you get "key clicks" in your radiotelegraph transmission. For the reasons given above. - Bruce K6BP

 

[Averagesupernova]

You cannot vary the amplitude of a sine wave without putting energy in another part of the spectrum. Even if just for a short time.

 

[phinds]

You cannot vary the amplitude of a sine wave without putting energy in another part of the spectrum. Even if just for a short time.

Nicely stated.

 

[Baluncore]

Why does a short burst of a pure tone sound like a hollow tock, not like a tone? What causes the overtone spectrum we hear as hollow?

A pulse of pure tone is constructed in the time domain by multiplying a pure tone by a rectangular pulse or window.

A non-linear product, (multiplication), in the time domain, results in sum and difference frequencies in the frequency domain. That is how frequency mixers and modulators work to add and subtract frequency.

A rectangular pulse is made from a fundamental frequency, plus an infinite series of harmonics or overtones. It is those harmonics and their side-bands of pure tone, that you are hearing in the "pulse of pure tone".

 

[phinds]

So what do you think @richard31416 --- any of this make sense to you?

 

[tech99]

A pulse of pure tone is constructed in the time domain by multiplying a pure tone by a rectangular pulse or window.

A non-linear product, (multiplication), in the time domain, results in sum and difference frequencies in the frequency domain. That is how frequency mixers and modulators work to add and subtract frequency.

A rectangular pulse is made from a fundamental frequency, plus an infinite series of harmonics or overtones. It is those harmonics and their side-bands of pure tone, that you are hearing in the "pulse of pure tone".

Further to this comment, notice that the spectrum shown is plotted on a logarithmic frequency scale, which hides the fact that the signal has the same pattern of side frequencies above and below the tone. Actually, the lower side frequencies cannot extend below zero frequency and so are folded over and then increase. This creates a messy and non musical sound.

 

[Rive]

For most people the whole Fourier analysis thing usually comes first as some esoterical mathematics theory.

Differently shaped continuous waves comes with overtunes, and non-continuous pulses as continuous spectrum... Well, one very interesting theory, right?

It takes some time to get that in electronics, Fourier is no theory but the real deal.

 

[richard31416]

I'm so appreciative of you all for taking the time to give me a number of thoughtful replies. Many thanks.

I am even sort of kinda starting to understand it. The different ways of phrasing/viewing help an amateur like me start to grasp. So it's helpful that different folks chimed in with different angles on this.

At the risk of wearing out my welcome, can I ask two follow-ups?

1) A friend is also interested in this and has a digital oscilloscope with an extremely fast sampling rate, but the tracing of pulses of tones look like pure sine waves. I would guess we're missing seeing the overtone spectrum through some problem with settings or sensitivity in the device--sound right? Or maybe we would see it if the device had an FFT setting that would mathematically analyze squiggles we can't really decode with our eyes?

2) The very initial sounds of musical instruments are apparently key in our interpreting sounds as coming from a, say, clarinet. I wonder if this is a little bit related to the overtones resulting from pulses or quick disturbances?

 

[Baluncore]

A friend is also interested in this and has a digital oscilloscope with an extremely fast sampling rate, but the tracing of pulses of tones look like pure sine waves.

Where the sinewave starts and stops is where the energy is in the rectangular pulse. You will probably not recognise that in the time domain. You will need to use an FFT to see the frequency domain.

Alternatively, you could simulate the pulse of pure sinewave using math, or a simulator like LTspice and look at the FFT that way.

 

[hutchphd]

2) The very initial sounds of musical instruments are apparently key in our interpreting sounds as coming from a, say, clarinet. I wonder if this is a little bit related to the overtones resulting from pulses or quick disturbances?

You need to talk (loudly) to an old person. You are in luck (here I am). The typical aging hearing loss affects high frequencies first and the resulting loss has particular affects upon speech perception. Most difficult are the fricative sounds which contain sharp changes in volume. To differentiate between them requires accurate perception of the high requency parts of the audio spectrum. So to differentate "pace","case" , "face", "base" becomes annoyingly difficult.

I don't find this true for musical instruments because their harmonic structure is markedly different even in the first few overtones, which are still relatively low frequencies (middle C is 256 Hz and hearing loss is maybe 5000 Hz and higher). I don't believe our recognition of musical instruments is generally based on the "attack" profile of the note (one exception is the harpsicord vs pianoforte but that is a different story)

 

[Rive]

2) The very initial sounds of musical instruments are apparently key in our interpreting sounds as coming from a, say, clarinet. I wonder if this is a little bit related to the overtones resulting from pulses or quick disturbances?

In music, the ratio between the length of pulse and the frequency of the note is typically not that dramatic as in the previous examples here, so while those change-originated overones will be present, they are not supposed to be dominant.
Even for percussion instruments the actual length of a (single) sound is not that short.

 

[Baluncore]

Here are 10 cycles of 1 kHz.

The centre frequency is 1 kHz. The pulse lasts for 10 ms.
The reciprocal of 10 ms is 100 Hz, so there is a null every 100 Hz.

 

[tech99]

I think it is likely that WWV passes the signal through a bandpass filter, which will alter its sound. What we may be hearing is the ringing caused by the edges of the passband.

 

[Baluncore]

I think it is likely that WWV passes the signal through a bandpass filter, which will alter its sound.

The WWV signal was very clean last time I looked. The only filter would be in the IF and audio of the communications receiver used, which is under the control of the operator.

IIRC, some tone pulses at the end of the minute, (that encode the time), were shorter than others and so sounded hollower. The longest pulse, at the minute mark, sounded cleanest.

There may also still be some slow digital data, phase-encoded as a quadrature signal, with sidebands very close to the carrier, but you will not hear that.

 

[richard31416]

So grateful for more thoughtful and helpful replies. Captivated by the FFT image! Kind of you to take the trouble.

 

[richard31416]

Quick comment re WWV seconds pulses in case of interest: They go to extra trouble for the seconds markers in particular--they are (amplitude) modulated at 100%, whereas their longer pure tones are at 50%, and voice is 75%. That means they are going for best possible broadcast fidelity of the seconds pulses, yes? (Or at least, so this amateur thinks.)

 

[Baluncore]

That means they are going for best possible broadcast fidelity of the seconds pulses, yes?

Maybe not. The energy in a time pip is proportional to the amplitude and the period of the tone. The modulation depth determines the amplitude of the AM detected wave, so short pulses need greater modulation to sound as loud as longer pulses. The tone or sound of the pulses will change if their duration is different, because the nodes and nulls will be spread into different places in the spectrum.

Modulation depth would also be adjusted in the case of WWV, since it was used to phase lock time clocks. Like your brain, a PLL takes longer to detect and lock to short duration, low amplitude tones.

If you want to simulate the spectrum of a burst of n cycles of pure tone, you could use a copy of LTspice, which is a free download. The spectrum can be simulated in 3 lines of code, using a single voltage source.
.param n=20
V1 out 0 SINE(0 {1414.2/n} 1000 {(1000-n)/2000} 0 0 {n})
.tran 0 1 0 1u

 

[sophiecentaur]

a digital oscilloscope with an extremely fast sampling rate,

In order to sample a waveform such that it can be re-constructed again 'perfectly' you need to sample it very fast (at least twice the frequency of the highest frequency in the waveform to be sampled. The sample pulses will have sidebands (as with any modulation) which lie way outside the frequencies in the original waveform. If the re-constructed waveform is Low Pass Filtered, then the extra products are not visible (too fast) and the scope trace is smooth. Try to sample a waveform that varies too quickly and the display will contain 'alias' frequencies which are not really there. Look up Nyquist Criterion if you feel inclined to take this further but it's another layer of complexity. (You can look upon sampling as a form of modulation.)

 

[richard31416]

Thanks again to all of you for the rich replies. Some of it is way out of my league, but I like trying to reach higher and at least have some grasp about the pulse having its own spectrum beyond what it's a pulse of. Much appreciated.

--Richard