Differentiable Sawtooth.

[swestrup]

I'm wondering if anyone here has a clue as to where to look for a relatively simple function to calculate an approximate sawtooth curve. I need something that is continuous and differentiable or I'd just use something standard like the floor function. I've looked at fourier approximations and they need a lot more terms than I like to approximate a reasonable sawtooth, PLUS they are awfully bumpy.

So far all attempts to look up periodic asymmetrical functions on the net have failed me.

Comments

[sps.livejournal.com]

Welllll, if you can't get your differentiability by filter theory, and you only need to be differentiable once, what about using splines? Continuous and everywhere differentiable, small number of terms and non-bumpy, very easy to tweak, at the cost of having to do the modular reduction yourself.

[cpirate.livejournal.com]

I saw this the other day, which doesn't directly answer your question, but may give you some inspiration: http://www.johndcook.com/blog/2010/01/13/soft-maximum/

[swestrup]

Yeah, I'm looking into splines. After all, a sawtooth is mostly linear, so it won't take many segments. Something I realized earlier today though is that I want something invertible within a period. I'm not sure how to do that with splines.

That starts to make an fft look better, but I still don't like the idea of the work involved just to get a reasonable approximation of a sawtooth.

[swestrup]

Hmm. It is nice, but I was already looking into similar exponential functions without luck. Still, I'm glad you pointed it out, as it looks useful.

[sps.livejournal.com]

I don't think you get invertible within a period together with continuous. The steep side can't be completely vertical... or perhaps I misunderstood.

[swestrup]

I don't want the steep side to be completely vertical. That's why I'm not using a plain sawtooth. (Or actually, I currently AM and don't want to).

[swestrup]

Actually, what would be ideal is some sort of skewed sine wave, but I can't figure out how to construct such a thing.

[sps.livejournal.com]

So you want to jitter the x axis periodically. Something like sin(x + k sin(x))?

[sps.livejournal.com]

I understand. But so long invertibility. In each cycle, each y value has to be passed twice.

[swestrup]

Yeah, but I've had no luck fitting something like that to my data. I need a curve that goes through the points -3, -2, -1, 0, 1, 2, 3, 4, 5 and back to -3 at a constant increment. What a the curve does between any two values doesn't matter, so long as it stays between those values.

[sps.livejournal.com]

Spline.