Even Order harmonic pedal

[slideman82]

Harmonic Percolator IIRC

I love this bug!

[Eb7+9]

I would really like to hear these arguments.
As I understand it every signal is made up from sines (Fourier) and therefore multiplication of a signal with itself will produce for every harmonic in the original signal an octave of that harmonic.

this is how I see things ... Fourier analysis strictly applies to periodic signals first of all so the idea of a signal made up of harmonics is a touchy concept as it applies to string derived signals - one should be talking of harmonic bursting, the question is can we really be talking about an infinite sum of sine waves in the context of rapidly changing harmonic bursts ?? ...that's where the discussion almost gets stupid and many people opt for a simplified understanding of "harmonics" ... the modified Fourier theory as it applies to non-periodic signals gets really messy as well and only yields similar results "in the limit" ...

in music signals only the decay portion comes close to being quasi-periodic - so Fourier analysis doesn't apply "idealy" the same way that periodic theory predicts when applied to plucked string signals, esp. near the front end where the periodicity can be conceived as "quite" non-existant depending on how one wants to look at things ... ignoring that part a non-pure signal (ie carrying harmonics) will produce sums and differences of frequencies when going through any non-linear transfer in addition to (self-product) doubled terms ... amongst other things this is why/how you get this one frequency term heading towards DC when bending one note into another (as we do on the guitar) when going through certain non-linear processing blocks like octavers ...

[SeanCostello]

I would really like to hear these arguments.
As I understand it every signal is made up from sines (Fourier) and therefore multiplication of a signal with itself will produce for every harmonic in the original signal an octave of that harmonic.

It's algebra. Imagine that you have an input signal consisting of two sine waves, a and b, with two different frequencies, freq(a) and freq(b). The signal can be represented as

a+b

Multiplying the signal by itself will yield

(a+b)(a+b) = a^2 + 2ab + b^2

a^2 will result in a sine wave of freq(a)+freq(a), which is 2*freq(a), which is an octave above freq(a). It will also create a frequency at freq(a)-freq(a), which is 0 Hz, or DC.
b^2 will result in a sine wave of freq(b)+freq(b), which is 2*freq(ab), which is an octave above freq(b). It will also create a frequency at freq(b)-freq(b), which is 0 Hz, or DC.
2ab will result in a sine wave at freq(a)+freq(b), as well as a sine wave at freq(a)-freq(b). These will not be simple octaves of a or b.

So, it is true that multiplication of a signal by itself will result in an octave of every harmonic in that signal. But, you get a whole mess of other frequencies, as every frequency in the signal is multiplied by every other frequency in the signal. This is why a squaring circuit is not a pitch shifter.

In addition, the amplitude envelope is squared by itself. Imagine that you just have a single sine wave, of amplitude A. When the amplitude is 1.0, the squared amplitude is also 1.0. If the input amplitude is 0.5, the squared output amplitude is 0.25. If the input amplitude is -0.5, assuming a bipolar signal, the output amplitude will be 0.25. This ends up resulting in transients that decay away much quicker, as well as a unipolar output signal (i.e. >0 for all input signals). Such behaviour is useful in envelope detection, and is the S in RMS (root mean squared) which is used in compressors, VU meters, and other places.

Sean Costello

[dirk]

Thanks Sean Costello, that makes perfect sense.

Of topic: As I understand Fourier, it will apply to every real world signal. Because every real world signal is bandlimited, you have a finite amount of sines that would make up the signal. But I'm not a mathematician, so I could be wrong.

[SeanCostello]

Thanks Sean Costello, that makes perfect sense.

Thanks! I am not that good at math, but I was able to dig out the old high-school algebra for that one.

Of topic: As I understand Fourier, it will apply to every real world signal. Because every real world signal is bandlimited, you have a finite amount of sines that would make up the signal. But I'm not a mathematician, so I could be wrong.

It is actually the opposite: real world signals are NOT bandlimited. In many of the signals, the higher order harmonics end up having insignificant amplitudes, but they are still there.  In many cases, the signals above our high frequency thresholds (20+ KHz for younguns, 17 Khz and lower as you reach adulthood) have significant energy. That is why A/D convertors have brick-wall lowpass filters, so the frequencies outside of the audible range do not alias down into the area where we can here them.

Digital signals are bandlimited - they have to be - so they can be represented by a finite number of sines. Psychoacoustic data compression works by throwing out those frequencies that will not be perceived, due to masking and other perceptual pheonomena.

With regards to multiplying signals by themselves, if a harmonic has insignificant amplitude, its products will also be insignificant. It is also worth noting that multiplying a signal by itself will double the bandwidth of the output (i.e. the highest frequency of the output will be twice the highest frequency of the input). No big deal in the analog domain, but a big problem in the digital domain, as this will cause aliasing.

Sean Costello

[Brian Marshall]

That sounds interesting but wouldn't it only generate octaves?

Thats what even order harmonics are.
Of cause this circuit produces for every harmonic in the signal a series of even harmonic overtones.

The exciter process  is much broader in its spectrum and works like a shelving eq across the harmonic series(somehow filtering the dissonant odd-order tones).

With the circuit mentioned you can dial in the level of the second harmonic and the level of the 4th, 6th, 8th etc... More ringmodulators and mixers would allow you to mix the 8th harmonic higher in level than the 6th for example.

I was looking for a full bandwidth solution which would dial out the original signal leaving the distortion (which would have to be very sweet!)

Just turn down the level of the original signal.

I'll see if I can make an example of this circuit with my Clavia micro modular.

I'm pretty sure that only perfect squares are octaves.  6th harmonic is an octave of the 3rd.

edit- nevermind, someone already caught this.

[dirk]

Of topic: As I understand Fourier, it will apply to every real world signal. Because every real world signal is bandlimited, you have a finite amount of sines that would make up the signal. But I'm not a mathematician, so I could be wrong.

It is actually the opposite: real world signals are NOT bandlimited. In many of the signals, the higher order harmonics end up having insignificant amplitudes, but they are still there. 

I don't agree with this.
Every real world signal is bandlimited and has a signal to noise ratio, I was not complete before. If real world signals are not bandlimited, then they would have infinite energy (an endless amount of frequencies with insignificant amplitudes, still adds up to infinity). That is simply not possible.
And therefore Fourier applies to every real world signal.
Here you can find a table that shows what type of Fourier transform applies to what type of signal.
http://en.wikipedia.org/wiki/Fourier_analysis#Discrete_Fourier_transform_.28DFT.29

[R.G.]

That sounds interesting but wouldnt it only generate octaves? The exciter process  is much broader in its spectrum and works like a shelving eq across the harmonic series(somehow filtering the dissonant odd-order tones). I was looking for a full bandwidth solution which would dial out the original signal leaving the distortion (which would have to be very sweet!)

Actually, there are three of those. They are the JFET Doubler, the MOS Doubler and the Mu Doubler, both at GEO. And they work exactly as you describe. They take a signal, use a phase inverter to generate a second, out of phase signal, then run the two split signals into a differential amplifier.

But the differential amplifier also adds the two amplified, out of phase signals back together at the output by tying the drains of a diffamp pair of FETs together. This cancels out the original signal and leaves only even order harmonics, which all reinforce. The original signal and odd-order harmonics cancel. It worked the first time I tried it. JFETs and MOSFETs have a square-law nature, so the result is primarily second harmonic only.

So why don't you see this more? Because the linear characteristics of JFETs and MOSFETs are quirky, and it's difficult to get off-the-shelf devices to work. There is nearly always some tinkering needed to get it to cancel properly and have enough output level. I've been working with a guy on one of these recently.

You need a well-matched pair of FETs for the diffamp, which is where the magic happens. That is what generates the second-order distortion. And you need a "dirty" pair, ones that generate a lot of square-law. Nicely behaved linear FETs don't distort much, so they generate only a tiny amount of distortion to use, and then you have to really work at amplifying it back up.

The closest to repeatable is the MOS Doubler, which uses a CD4007 logic chip for its accessible MOSFET devices. They are monolithic (and so well matched inherently) and logic chips (so nobody slaved over a hot display trying to make them linear). There's even enough devices there to make a post amplifier to get more output and a little grit. The complaints I've received, other than it being quirky to set up, are largely that it's either too quiet or not distorted enough, and a little post-clipping enhances things.

So if you want to tinker with an embodiment of your idea, check out the Doublers at GEO.

[Joe Kramer]

I was wondering if anyone knows of a fuzz or distortion pedal that that produces only EVEN ORDER HARMONICS. Like the process used on Aphex Exciters but carried even further?IF not then any ideas on its development would be cool!

The Ampeg Scrambler does a pretty good job at this IMHO.  With careful adjustment of the BLEND control, you can get a shimmery, Fender-ampish even-order distortion.  Match the differential xstrs and use three germanium diodes in series instead of the stock silicons for a softer sound and better touch sensitivity.

Regards,
Joe

PS: Just found this:

http://electronicdesign.com/Articles/Index.cfm?AD=1&ArticleID=6379

Might work for a FWR-type distortion.  Haven't tried it yet, but looks promising!