Basic Harmonics Question

Not a critique, a question...

This waveform is Odd...

/\ /\ / \ / / \ /

----------------------- / \ / / \ / / \/

This waveform is even.

/\ /\ / \ / / \ / / \ / / \ / / \/

-----------------------

Yet the only difference is a DC component (and some ascii art distortion).

How does the above rule deal with that?

BTW, in Thunderbird I seem to have a problem where copying multiple adjacent spaces one copy as a single space when pasted anywhere. Anyone else see that? Or do I have a defective version (38.2.0)?

BTW, in addition to being polarity reverse, the second half of the waveform has to be mirrored in time.

The waveform that's odd happens to also have all odd harmonics -- not because it's odd, but because it follows the "2nd cycle is the negative of the 1st" rule.

If you put the t=0 axis in the correct place, a sawtooth waveform is odd, but it has both even and odd harmonics.

The waveform that's even does NOT have all odd harmonics, because it can't go negative. The DC component is at frequency zero which in an even harmonic (or at least a non-odd harmonic: I seem to remember seeing someone trying to say that 0 is not an even number; it's certainly a non- odd number, though).

I don't follow the purpose of your observations. I thought your rule was that if the waveform has odd symmetry, it only has odd harmonics. If a waveform has even symmetry it has only even harmonics. The two waveforms given above are the same waveform but one includes a DC component. By the odd rule, the waveform with odd symmetry has a fundamental and only odd harmonics. The other waveform should have the same spectrum plus a DC component, but the even symmetry rule says it should have only even harmonics. What part of the rules am I not getting? Is there a DC exclusion in the rules?

Maybe I've found the issue thanks to wikipedia. The odd/even harmonic relation is to the transfer function, not to the waveform itself. The full wave rectifier (ideal) is an even function which looks like a "V" on the x/y graph. So the output will have only even harmonics of the input, which also excludes the fundamental.

The example given for odd symmetry is symmetrical clipping in a push-pull amplifier with an odd symmetry transfer function, so the harmonics will be all odd and include the fundamental.

So the even/odd symmetry rule has to examine the transfer function, not the waveform.

Yes, but if you read the whole thread you'll see that I was wrong. So I found another rule (which is the posting that you were responding to) that seems to be correct.

The post I replied to is not very clear as to what you are saying. I think I found my answer in the part that you snipped.

DC offset doesn't change evenness or oddness. sin(x) is odd, cos(x) is even, and both are the same shape except for a phase shift.

The only-odd-harmonics property isn't the same as odd symmetry.

Cheers

Phil Hobbs

Misspoke, sorry. Any DC offset destroys odd symmetry, but doesn't affect even symmetry.

Cheers

Phil Hobbs

Exactly. For the harmonics it doesn't matter whether it's odd or even.

I described in an earlier post (maybe it got lost?) that it's the fact that the even harmonics are cancelled out in the full period integration. As one stated already succinctly: they are orthogonal with the fundamental frequency. The fourier series' terms are integrals over the full fundamental period of the product of a harmonic function with fundamental frequency and one with the frequency of the n-th harmonic.

The properties 'odd' and 'even' only facilitate in removing the sine and cosine term respectively from the calculation of a_n and b_n.

joe