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## Transient behavior of a pure sine wave

DSP, Plug-in and Host development discussion.
KVRAF

3920 posts since 16 Feb, 2005
Let's say you start (a sinusoid signal) by calculating sine with the necessary parameters. There is an obvious discontinuity at the start of it, in any phase, because of the amplitude increasing to unity immediately. Now try to relate this to Quality Factor and say this sine wave has a Q of 1/infinity maybe? It is maximally broadband because it allows a "phase anomaly" to pollute the signal. That phase anomaly does not exist in the steady state signal.

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
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KVRAF

3920 posts since 16 Feb, 2005
What do you know the transient of an absolutely perfect sine wave to look like?
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
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?
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KVRAF

8822 posts since 7 Dec, 2004, from Vancouver, Canada
Short explanation:

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
I took on the idea because I want to generate a fundamental tone without distortion. A totally sinusoid input or "impulse" into the equation. The problem is absolutely time-domain. I do see the relationship to windowing in Fourier type transforms, and it's also quite funny how windowing has its own problems and strengths.

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
An infinite signal doesn't exist. Sine doesn't start or stop.

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
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.
DOLPH WILL PWNZ0R J00r LAWZ!!!!
KVRAF

3927 posts since 11 Feb, 2006, from Helsinki, Finland
How about a cubic "smooth-step" envelope? Very cheap and C1 continuous.
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KVRist

110 posts since 10 Nov, 2013, from Germany
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
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.
DOLPH WILL PWNZ0R J00r LAWZ!!!!
KVRAF

3920 posts since 16 Feb, 2005
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
Any thoughts on the matter?

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
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.
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KVRAF

3920 posts since 16 Feb, 2005
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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