How do I warp a 1 dimensional signal in frequency domain?

Question

I want to implement VTLN (Vocal Tract Length Normalization).

It is equivalent to warping the signal in the frequency domain according to the following graph.

I can't wrap my head around how one implements such a transformation.

Suppose y = fft(x).

What should be y2 = f(y)?

What do I do after getting the warped frequency?

suppose x = [1, 2, 3, 4, 5]

y = f(X) = [ 10, 20, 30, 40, 50]

x_warped = [ 1, 1, 2, 4, 5].

What should y_warped[] be?

Do we delete some values and repeat some values in the y sequence?

is y_warped = [10, 10, 20, 40, 50]?

Image taken from

Pitz, Michael et al. “Vocal tract normalization equals linear transformation in cepstral space.” IEEE Transactions on Speech and Audio Processing 13 (2001): 930-944.

Answer 1

The mapping in this specific figure is basically a "lookup table" that maps $\omega$ to $\tilde{\omega}$ and the process in detail is outlined in section 5 of the paper that you link.

But on the point of applying that to a series, a function that effects such a transformation that is controlled by a parameter $\alpha$ is $f(x) = x^\alpha, x\in \left[0..1 \right]$ and $x \in \mathbb{R}$.

For example (Using Octave but easily extendable to other platforms):

function y=f(x,a) y=x.^a;end;    
x = 0:0.001:1;
plot(x,f(x,1),x,f(x,2),x,f(x,1./2));
xlabel('x');ylabel('f(x)');
legend("x","x^2","x^{0.5}");
grid on;

Would give you:

Notice here that this is slightly different than the transform that you have with respect to the behaviour of the function with $\alpha$. I am using this for illustrative purposes here.

Notice also that because $x \in \left[0..1 \right]$, we can transpose that function to whatever limits we like. In your particular example it is $\pi$ or the number of bins of your Frequency Transform. Let's call that $N_{FT}$.

To apply the transform, you can either discretise the function or evaluate it on the fly. For example:

x_lookup = round(N_FT .* f(x/N_FT,alpha));

In this case, your input frequency transform bin would be x and that would have to now appear in bin x_lookup.

Notice here that x/N_FT to make sure that x stays within the range $0..1$ (assuming of course its lowest value is 0 here).

Again, section 5 contains all the details for specific f(x) functions you will need to effect this kind of mapping for a specific application.

Hope this helps.