Creating odd harmonics without the fundamental

Question

I'm interested in creating a series of odd only harmonics, without the fundamental, for audio use.

As an example the input could be a sine wave at specific frequency $f$ and output is $3f$, $5f$, $7f$, etc. - but without the original $1f$.

Since the $1f$ (in a real audio system) is unknown and the input is multi-tone, just filtering the $1f$ out is not an option.

Any idea for replacing an input of $1f$ with an output of $3f$, $5f$, $7f$, etc. ?

Edit: Additional information from the deleted post

Example: Let's assume the fundamental to be anything possible between 100 and 4000 Hz. Whats more, what if the input is multitone (as most music is...). 50 and 425 Hz input simulatenously ??

Im looking for a mathematical function which can recreate any fundamentals odd harmonics, in a multitone input...

Answer 1

Im looking for a mathematical function which can recreate any fundamentals odd harmonics, in a multitone input...

(emphasis by me)

What you ask for is not possible. If you demand multiple tones and complex tones as they appear in music, then your problem leads to a conflict. A complex tone comes with overtones. Even if you have a process that adds odd harmonics to each sinusoidal component, then the even overtones of the fundamental frequency will contribute even harmonics. That is because an odd number multiplied with an even number is again even.

So circumvent that, you would have to identify the fundamental frequency of complex tones first, leading you directly to the problem of polyphonic pitch detection which remains mostly unsolved.

Maybe you should explain what you're really trying to accomplish, as in what the application is that you have in mind. With that information it's quite likely that we can come up with a relaxed set of requirements that can actually be implemented.

Answer 2

• $y(t) = x(t)^3 - \frac 3 4 x(t)$ (3rd harmonic)
• $y(t) = x(t)^5 - \frac {10} {16} x(t)$ (harmonics 3, 5)
• $y(t) = x(t)^7 - \frac {35} {64} x(t)$ (harmonics 3, 5, 7)
• etc.

Presumably this is a sampled system, so you could choose N based on your Nyquist frequency?

There's a pattern to the coefficients that cancel out the fundamental but I don't know what it is. :)

Answer 3

(Note that this answer was written before the question was edited in a way which made clear that the input is not sinusoidal, and that filtering out the fundamental frequency is also no option. Anyway, for the time being I'll leave it as is.)

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One easy way to achieve this is to symmetrically clip the sinusoid, which will create all odd multiples of the fundamental frequency, and then use a high-pass filter to filter out the fundamental frequency.

Answer 4

The function should be something like this:

SR=sampling rate
T=duration in second of the output
t=0:(1/SR):T
x=your sin wave input
X=fft(x)
freq=index of max(mod(X)) from 0 to SR/2 (in [Hz])
ai=[1 1 0.5 ... ] %harmonics amplitude
y=zeros(length(t))
for i=1:n %n harmonics, n=floor(SR/2/(freq*2))-1 if you want all the harmonics
   y=y+ai(i)*sin(2*pi*(2*i+1)*freq*t)
end

EDIT:

If x is a sin and you don't want to use FFT you can find the freq quite well in this way $O(n)$:

sgn=sign(x);
i=1;
periods=0;
Psmp=0;
while i<length(t)
    if (sgn(i)~=sgn(i+1) && sgn(i+1)~=0) || sgn(i)==0
       for j=(i+1):length(t)-1
          if (sgn(j)~=sgn(j+1) && sgn(j+1)~=0) || sgn(j)==0
            Psmp=Psmp+(j+j+(sgn(j)~=0))/2-(i+i+(sgn(i)~=0))/2;
            periods=periods+1;
            break;
          end
       end
       i=j;
    end
    i=i+1;
end

if periods>0
    Psmp=Psmp/periods/2;
    P=Psmp/SR;
    freq=1/P;
end

It makes a linear interpolation of each couples of samples about 0 and finds the average of the periods.