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Odd+even harmonic wave

Discussion in 'Electronic Design' started by Ethan Border, Aug 3, 2007.

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  1. Ethan Border

    Ethan Border Guest

    What is the characteristic wave shape that contains both odd and even
    harmonics in equal amplitudes?

    I have tried mixing a sawtooth with 1) squarewave and 2) triangle wave
    of the same frequency all of the same voltage. This works, to a
    degree, but I don't know if it is the best option.

    Is it necessary to play around with duty cycles or something else. to
    get an optimal result?

    Thank you for any advice.

    Ethan B.
     
  2. D from BC

    D from BC Guest

    I got curious and did some spice....

    V1 N003 N004 SINE(0 10 3khz)
    V2 N004 N005 SINE(0 10 4khz)
    V3 N005 N006 SINE(0 10 5khz)
    V4 N006 N007 SINE(0 10 6khz)
    V5 N007 0 SINE(0 10 7khz)
    V6 N002 N003 SINE(0 10 2khz)
    V7 N001 N002 SINE(0 10 1khz)
    R1 N001 0 10k

    7 harmonics..all equal amplitude..all waves starting at 0 degree.
    The resulting waveform kinda looks like this |\|\|\|\|\ but bumpy.
    I get the impression that with more harmonics it might turn into ramp
    wave...

    D from BC
     
  3. Guest

    The Fourier transform of a repeating spike contains all the harmonics
    of the repetition frequency up to a limit set by the width of the
    spike. The amplitudes of all the harmonics are equal.

    Search on the Dirac spike, which has a finite area, but infinite
    height and zero width.

    http://en.wikipedia.org/wiki/Dirac_delta_function

    Check out the Comb function while you are at it

    http://en.wikipedia.org/wiki/Dirac_comb

    When I was getting to grips with this, I found it helpful to note that
    if you differentiate a square wave you get two interleaved comb
    functions, one with positive-going spikes and one with negative-going
    spikes.

    Differentiating the harmonics of a square wave gives all the odd
    harmonics at equal amplitudes - you can see what has happened to the
    even harmonics ...
     
  4. me

    me Guest

    for sin () you get a series of negative and positive impulses.

    | | |
    ---------------- sorta like this.
    | | |



    for cos() you get a series of positive impulses.

    | | |
    ---------------- hope this helps....
     
  5. LVMarc

    LVMarc Guest

    excellent!
     
  6. Tom Bruhns

    Tom Bruhns Guest

    Others have pointed out that repeating impulses yield a spectrum of
    equal amplitude harmonics; they all equal the fundamental in
    amplitude. But that's not the only waveform that results in equal
    amplitude harmonics. In fact, there is an infinite set of them--maybe
    an infinity of infinities, because the phase of each of an infinite
    number of harmonics may take on any of an infinite set of values, and
    each set of phases will result in a different wave shape.

    An example of a second waveform that has equal amplitude harmonics is
    a clip of a white noise that happens to have equal starting and ending
    values, so it may be repeated without introducing a discontinuity
    (!). (Note that a random has an infinitely small but not zero
    probability of having a discontinuity...) So long as you repeat the
    same clip over and over to infinity, the result will have only the
    fundamental at frequency 1/(clip length) and its harmonics. Since the
    clip is limited in length, it won't have exactly equal harmonic
    amplitudes. Consider that the clip can have _any_ shape (so long as
    the end points are the same value), including an impulse, but also
    including one cycle of sine, or one cycle of cosine, and be a valid
    clip from a random waveform. But statistically, a clip with harmonics
    with greatly different amplitudes will be very rare.

    Hope this isn't too confusing...

    Using a (pseudo)random clip is much more practical than using an
    impulse in many cases, since the peak amplitude is not so outrageously
    higher than the RMS. In theory, a random clip _could_ have very high
    peak to RMS amplitude, but in practice it's not a problem; it's
    statistically exceptionally unlikely, and in any event the methods of
    generation guarantee peaks of some maximum value.

    One reasonably quick and easy way to play with this and see what
    various phases of the harmonics gives you is to use Matlab or Scilab
    (free...). You can either build a time domain wave from time domain
    sinusoids, or just build a frequency domain signal with equal
    amplitude and related or random phases, and do an inverse FFT on it.

    Cheers,
    Tom
     
  7. Guest

    I'd one on file with 20 generators. Just cleans up on the general
    shape you're seeing now. Curiously, if you put the first few sources
    180degs out of phase, the wave looks identical to one of those heart
    beat Alpha waves (or whatever they're called).
     

  8. While your waveform was fun to plot is is stunningly and mind bogglingly
    useless.

    It consists of an infinite positive spike just beyond zero phase and an
    infinite negative spike just before 360 degree phase.

    Truncating to fewer harmonics gives you a narrow positive spike just
    beyond zero phase, a bunch of intermediate low level ripple, and a
    narrow negative spike just before 360 phase.

    The harmonics correllate only near zero and 180 degrees; they
    decorrelate otherwise.

    There are twice as many zero crossings as the highest harmonic used.
    The waveform is overwhelmingly positive from 0 to 180 degrees and
    overwhelmingly negative beyond.

    "Real" waveforms tend to have diminishing harmonic values.

    Lots of fun to do with PostScript.

    More useful waveforms appear at http://www.tinaja.com/magsn01.asp
    More on PostScript plotting at http://www.tinaja.com/post01.asp

    Here is the half cycle code using my Gonzo utilities...


    %!PDF

    (C:\\Documents and Settings\\don\\Desktop\\gonzo\\gonzo.ps) run % use
    internal gonzo


    50 50 10 setgrid
    40 20 showgrid


    0 10 mt

    0 0.1 180 {/priang exch store


    priang 20 mul 90 div % scale for one cycle

    priang sin
    priang 2 mul sin add
    priang 3 mul sin add
    priang 4 mul sin add
    priang 5 mul sin add
    priang 6 mul sin add
    priang 7 mul sin add
    priang 8 mul sin add

    priang 9 mul sin add
    priang 10 mul sin add
    priang 11 mul sin add
    priang 12 mul sin add
    priang 13 mul sin add
    priang 14 mul sin add

    priang 15 mul sin add
    priang 16 mul sin add
    priang 17 mul sin add
    priang 18 mul sin add
    priang 19 mul sin add
    priang 20 mul sin add

    2 mul 10 add lineto % should be 20 mul -- scaled in interest of sanity

    } for

    line1 stroke

    showpage


    --
    Many thanks,

    Don Lancaster voice phone: (928)428-4073
    Synergetics 3860 West First Street Box 809 Thatcher, AZ 85552
    rss: http://www.tinaja.com/whtnu.xml email:

    Please visit my GURU's LAIR web site at http://www.tinaja.com
     
  9. Guest

    <snip>

    I'm not so sure about that - if you take *all* the frequencies
    starting from

    1/age of the universe

    You get a single spike of infinite energy at the Big Bang.

    Robin
     
  10. "Ethan Border" ...
    A pure sine wave. All harmonics are zero. :)

    Arie de Muijnck
     
  11. LVMarc

    LVMarc Guest

    I you take the harmonics and adjust the phase so that eacs sucedding
    harmonic number has a phase advance, or quad time delay, you get a
    chirp! so the chrip and the impulse are closely related in a fourier
    sense..whereas the chirp has precise phase advance for each component
    and the impulse is just all of then starting at t=0...

    the paek t average is way different and thi discussion and technique
    moves inot the realm of optimizing waveforms,to produce dsirebale
    perfomance. th chirp has some nice rardar propoerites, and a impulse is
    "simple" and the core for UWB devices..

    Happy waveform generation! BTW, making the chirp is a tricky process!
     
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