## Transient behavior of a pure sine wave

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- KVRAF
- 3920 posts since 16 Feb, 2005

by **camsr**; Fri Jan 10, 2014 10:04 pm
Transient behavior of a pure sine wave

What I have done to mitigate this behavior of sine, is applied a first order integrator filter to a stairstep signal which acts as an envelope for the sinewave signal. The filter is minimum phase and tuned with the f3 at the frequency of the sine. A first order filter performed well when the starting phase was 0º, but had the same phase artifacts as the signal without the integrator when the envelope was released.

So I kept adding integrators until I reached an order of 8. Using 8 integrators to smooth the sinewave essentially made the phase anomalies disappear in all phases. I guess I can say it has a Q factor of 8.

To simulate a slewed phase angle based on some real apparatus, I also adjusted the phase of the sinewave by the envelope to a max of 90º. I think this limits the slew of any first order system, no matter what the bandwidth.

So the integrator order is the only confusing thing here, because it slews the time properties of the signal against the phase properties which remain constant.

- KVRAF
- 3927 posts since 11 Feb, 2006, from Helsinki, Finland

by **mystran**; Sat Jan 11, 2014 12:42 am
Re: Transient behavior of a pure sine wave

- KVRAF
- 3920 posts since 16 Feb, 2005

by **camsr**; Sat Jan 11, 2014 2:34 pm
Re: Transient behavior of a pure sine wave

I haven't seen one

I have a Reaktor ensemble doing what I described above, but figured it's simple enough to explain.

- KVRAF
- 3927 posts since 11 Feb, 2006, from Helsinki, Finland

by **mystran**; Sat Jan 11, 2014 4:27 pm
Re: Transient behavior of a pure sine wave

camsr wrote:What do you know the transient of an absolutely perfect sine wave to look like?

It's not a perfect sine-wave if it has a transient: a sine-wave is a periodic infinite signal in both directions.

To treat sine-waves that "start at some point" you need to (time-domain) multiply it with a step-function, which as an 1/f response (and the time-domain multiplication amounts to convolution in the spectral domain). It is quite easy to see that the "purity" of the result can be improved by low-pass filtering the step-function (so the spectral expansion from the convolution is reduced).

Also, I'm not quite sure how your definition of Q factors is supposed to map to the normal idea of Q factors.. are you trying to use a decaying sinusoid as a substitute for resonance or what?

- KVRAF
- 8822 posts since 7 Dec, 2004, from Vancouver, Canada

by **aciddose**; Sat Jan 11, 2014 6:30 pm
Re: Transient behavior of a pure sine wave

It does not make sense to consider the issue in time-domain.

Longer explanation:

You're looking at harmonics generated by modulating the sine with another signal, the amplitude envelope. If your envelope is rectangular you're going to be multiplying an impulse with 1/n harmonics when you introduce the sine.

It is easy to produce various envelopes and their spectral properties are very well understood. So it becomes a simple matter of modulation where you want to minimize the side-bands. See windowing with respect to Fourier transforms for example.

- KVRAF
- 3920 posts since 16 Feb, 2005

by **camsr**; Sun Jan 12, 2014 2:08 am
Re: Transient behavior of a pure sine wave

Okay, so for every order increase of the envelope filters, sideband rejection increases, as well as the time it takes to reach steady state. I just call that time "Q" because it seems similar in practice.

I am just confused by the idea of an "infinite" signal starting and stopping without sidebands.

Not many natural "signals" happens this way.

- KVRAF
- 8822 posts since 7 Dec, 2004, from Vancouver, Canada

by **aciddose**; Sun Jan 12, 2014 2:26 am
Re: Transient behavior of a pure sine wave

Pure frequencies do not occur in nature.

Look at the 100s of years put into heterodyne research and fourier transforms.

- KVRAF
- 3425 posts since 19 Jun, 2002

by **tony tony chopper**; Sun Jan 12, 2014 2:34 am
Re: Transient behavior of a pure sine wave

I took on the idea because I want to generate a fundamental tone without distortion.

the transient itself is distortion, what you're looking for is making it "inaudible", or to make the tone start like it starts the fastest possible, but without "clicking". I'd just to a (half-sine window or whatever) ramp over a period that's directly related to the freq of the tone.

- KVRAF
- 3927 posts since 11 Feb, 2006, from Helsinki, Finland

by **mystran**; Sun Jan 12, 2014 2:45 am
Re: Transient behavior of a pure sine wave

- KVRist
- 110 posts since 10 Nov, 2013, from Germany

by **Chris-S**; Sun Jan 12, 2014 4:57 am
Re: Transient behavior of a pure sine wave

tony tony chopper wrote: what you're looking for is making it "inaudible", or to make the tone start like it starts the fastest possible, but without "clicking". I'd just to a (half-sine window or whatever) ramp over a period that's directly related to the freq of the tone.

Doesn't it depend of the phase? I don't think a sin-wave starting at phase 0 is clicking.

- KVRAF
- 3425 posts since 19 Jun, 2002

by **tony tony chopper**; Sun Jan 12, 2014 5:11 am
Re: Transient behavior of a pure sine wave

Chris-S wrote:tony tony chopper wrote: I don't think a sin-wave starting at phase 0 is clicking.

of course it does click, it's an abrupt discontinuity (& that "cut at zero-crossings to avoid clicking" is just widespread bs. It's better than nothing, but it's still bad).

It's on a spectrogram that you should look at this.

I think that a sinewave starting at 0 requires the same ramp, it's just that the same ramp will probably work a bit better on it, as it will have less impact at the beginning.

- KVRAF
- 3920 posts since 16 Feb, 2005

by **camsr**; Sun Jan 12, 2014 3:27 pm
Re: Transient behavior of a pure sine wave

tony tony chopper wrote:I took on the idea because I want to generate a fundamental tone without distortion.

the transient itself is distortion, what you're looking for is making it "inaudible", or to make the tone start like it starts the fastest possible, but without "clicking". I'd just to a (half-sine window or whatever) ramp over a period that's directly related to the freq of the tone.

By trying to make it inaudible, it should also conserve energy. The real goal I am after is applying some kind of "conservation" to the transient impulse with relation to F=M*A. Mass law is somewhat a first order equation, and if I shift the phase of the sine with the same amplitude envelope, I think this acts to mimic the slew rate of the "mass" making the soundwave. I guess I am trying to insure a constant phase angle or something, I am not very familiar with this part of physics

- KVRAF
- 3920 posts since 16 Feb, 2005

by **camsr**; Sun Jan 19, 2014 1:04 pm
Re: Transient behavior of a pure sine wave

Smooth envelopes are very clean but they lack "transient", IDK how to describe it

Turns out my phase shifting was only shifting sidebands from below the fundamental to above it. IT sounds more natural but still not what I am after.

I could apply a bandpass to a sine and let the Q define the envelope, but it sounds slow just like the smooth envelopes.

I suppose I am looking for the most "natural" way to develop a sine with the least sideband and fastest response.

- KVRAF
- 3927 posts since 11 Feb, 2006, from Helsinki, Finland

by **mystran**; Sun Jan 19, 2014 2:17 pm
Re: Transient behavior of a pure sine wave

camsr wrote:I suppose I am looking for the most "natural" way to develop a sine with the least sideband and fastest response.

You can't have it, because the sidebands and the fast response are the exact same thing just viewed in two different domains.

- KVRAF
- 3920 posts since 16 Feb, 2005

by **camsr**; Sun Jan 19, 2014 3:38 pm
Re: Transient behavior of a pure sine wave

mystran wrote:camsr wrote:I suppose I am looking for the most "natural" way to develop a sine with the least sideband and fastest response.

You can't have it, because the sidebands and the fast response are the exact same thing just viewed in two different domains.

So a sine wave is only a resultant behavior of some other kind of energy input then.

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