Square + triangle = sine (almost)
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From: Martin Brown on 7 Mar 2010 04:26 Harold Larsen wrote: > If a squarewave contains all odd harmonics of the fundamental > frequency, and a triangle all even, will I get ALL harmonics if I mix > the two waveforms? > > It looks like a cross between a squarewave and sinewave. Not in this world it doesn't. Both contain only the odd harmonics but in varying amounts. You get from square to triangle by integrating it. _ _ _ _| |_| |_| |_ A square wave is sum (-1)^(2n+1).sin((2n+1)wt)/(2n+1) n=0 .. inf When you integrate a square wave you get a triangle wave - usually available off the timing capacitor with a bit of buffering. /\ /\ /\ \/ \/ \/ The expression for the square wave can be integrated to give: A triangle wave is sum sin((2n+1)wt)/(2n+1)^2 n=0 .. inf You could take the linear combination of triangle + square/3 to null out the third harmonic but the waveform would look nothing like a sine wave because of all the other uncancelled higher harmonics. And the zero crossing would be perpendicular which is not right. > > I have not seen any tech references to the practical value of this. > Does it have any? None at all. > > For example, to roughly approximate a sinewave without filtering. A much better way ISTR originally poineered by HP is to take a triangle wave and apply diode shaping to it. First order is to just clip the top off and the next order chamfers the rough edges then a low pass filter. Neater methods by varying gain with amplitude exist too. Although the neatest of all is probably based on log shaping. Almost all of these tricks have been displaced by direct digital synthesis now. Natsemi has an app note that reviews sine generation methods that you might find interesting: http://www.nalanda.nitc.ac.in/industry/appnotes/Natsemi/AN-263.pdf And venerable Intersil ICL8038 part that first embodied square, triangle and a sinewave shaper on one chip is still online at http://www.intersil.com/data/fn/fn2864.pdf Regards, Martin Brown
From: Robert Baer on 7 Mar 2010 04:38 Harold Larsen wrote: > If a squarewave contains all odd harmonics of the fundamental > frequency, and a triangle all even, will I get ALL harmonics if I mix > the two waveforms? > > It looks like a cross between a squarewave and sinewave. > > I have not seen any tech references to the practical value of this. > Does it have any? > > For example, to roughly approximate a sinewave without filtering. > > Harold Larsen Use no filtering and only the triangle waveform: pass thru what 40-50 years ago was called a DFG (diode function generator) at least 16 segments for each polarity; THD result can be quite low (less than 0.5% aka 46dB THD.
From: Robert Baer on 7 Mar 2010 04:41 Harold Larsen wrote: > If a squarewave contains all odd harmonics of the fundamental > frequency, and a triangle all even, will I get ALL harmonics if I mix > the two waveforms? > > It looks like a cross between a squarewave and sinewave. > > I have not seen any tech references to the practical value of this. > Does it have any? > > For example, to roughly approximate a sinewave without filtering. > > Harold Larsen Correction: SIX diodes for each polarity, NOT 16 (remembered incorrectly).
From: Robert Baer on 7 Mar 2010 04:42 Ron Tanner wrote: > On Sun, 7 Mar 2010 14:31:48 +1100, "Phil Allison" <phil_a(a)tpg.com.au> > wrote: > >> "Harold Larsen" >>> If a squarewave contains all odd harmonics of the fundamental >>> frequency, and a triangle all even, >> >> ** Sorry - that is WRONG . >> >> A triangle wave contains only odd harmonics too. >> >> http://en.wikipedia.org/wiki/Triangle_wave >> >> A "sawtooth" wave contains all integer harmonics. >> > > OK thanks for the pull-up, but how about using a triangle-square wave > mix, in place of a filter, to simulate a sinewave . > > I have not seen that method applied or described anywhere, but it > makes a fair approximation, at least to my eye. > > Harold Larsen > > You have a crappy "eye"; no cigar - in fact no tobacco!
From: Robert Baer on 7 Mar 2010 04:44
Richard Henry wrote: > On Mar 6, 8:04 pm, rontan...(a)esterbrook.com (Ron Tanner) wrote: >> On Sun, 7 Mar 2010 14:31:48 +1100, "Phil Allison" <phi...(a)tpg.com.au> >> wrote: >> >> >> >>> "Harold Larsen" >>>> If a squarewave contains all odd harmonics of the fundamental >>>> frequency, and a triangle all even, >>> ** Sorry - that is WRONG . >>> A triangle wave contains only odd harmonics too. >>> http://en.wikipedia.org/wiki/Triangle_wave >>> A "sawtooth" wave contains all integer harmonics. >> OK thanks for the pull-up, but how about using a triangle-square wave >> mix, in place of a filter, to simulate a sinewave . >> >> I have not seen that method applied or described anywhere, but it >> makes a fair approximation, at least to my eye. >> >> Harold Larsen > > Why would you need to simulate a sine wave? It is well characterized > in the literature and there are lots of extra ones lying around > unused. > ....and one does not need a movie star to SINE the autograph! |