[DAGA] SH HRTF interpolation notebook

[hoyer-a]

Spherical harmonic interpolation of head related transfer functions

[hoyer-a]

Some typos, some suggestions. Overall great notebook!

[hoyer-a]

Suggested change

	"We can apply the spherical harmonic transform - also referred to as spherical Fourier transform - to $\\mathbf{f}$ to get the spherical harmonic spectrum\n",
	"We can apply the spherical harmonic transform - also referred to as spherical Fourier transform - to $\\mathbf{h}$ to get the spherical harmonic spectrum\n",

[hoyer-a]

Suggested change

	"As you can see, $\\mathbf{Y}$ it is of shape $Q$ $\\times$ $(N+1)^2$, and basis functions depend on the colatitude $\\theta$ and the azimuth $\\phi$ of the source positions. "
	"As you can see, $\\mathbf{Y}$ is of shape $Q$ $\\times$ $(N+1)^2$, and the basis functions depend on the colatitude $\\theta$ and the azimuth $\\phi$ of the source positions. "

[hoyer-a]

Suggested change

	"in this case using\n",
	"\n",
	"$$\\mathrm{Y} = [Y_0^0(\\hat{\\theta},\\hat{\\phi}), Y_1^{-1}(\\hat{\\theta},\\hat{\\phi}), Y_1^0(\\hat{\\theta},\\hat{\\phi}), Y_1^1(\\hat{\\theta},\\hat{\\phi}), ..., Y_N^N(\\hat{\\theta},\\hat{\\phi})].$$\n",
	"in this case using the target source positions $\hat{\theta}$ and $\hat{\phi}$ for the spherical harmonic basis function matrix\n",
	"\n",
	"$$\\mathrm{Y} = [Y_0^0(\\hat{\\theta},\\hat{\\phi}), Y_1^{-1}(\\hat{\\theta},\\hat{\\phi}), Y_1^0(\\hat{\\theta},\\hat{\\phi}), Y_1^1(\\hat{\\theta},\\hat{\\phi}), ..., Y_N^N(\\hat{\\theta},\\hat{\\phi})].$$\n",

Or something similar to make it a little bit clearer that Y now represents different directions than before

[hoyer-a]

Suggested change

	"The sampling grid supports spherical harmonic processing up to an order of approximately $N=32$ and for simplicity, we use the default spherical harmonic definition of spharpy.\n",
	"This specific sampling grid supports spherical harmonic processing up to an order of approximately $N=32$ and for simplicity, we use the real valued default spherical harmonic definition of spharpy.\n",

[hoyer-a]

Suggested change

	"and its pseudo inverse"
	"and its pseudo inverse $\mathbf{Y}^\dagger$"

[hoyer-a]

Suggested change

	"We now perform the inverse transform introduced above\n",
	"We now perform the inverse transform introduced above as\n",

[hoyer-a]

Suggested change

	"1. If $N$ is sufficently large, we get $\\hat{\\mathbf{h}} = \\mathbf{h}$ after applying the inverse transform. HRTFs require $N>32$, which means that you will see differences between $\\hat{\\mathbf{h}}$ and $\\mathbf{h}$ \n",
	"1. If $N$ is sufficently large, we get $\\hat{\\mathbf{h}} = \\mathbf{h}$ after applying the inverse transform. For HRTFs this requires $N>32$, which means that you will see differences between $\\hat{\\mathbf{h}}$ and $\\mathbf{h}$ \n",

[hoyer-a]

Suggested change

	"Note that we manually stored the data in a pyfar Signal object after the inverse transform. Starting in spharpy v1.1, there will be a function that performs the transform and directly returns a Signal object."
	"Note that we manually stored the data in a pyfar Signal object after the inverse transform. In spharpy v1.1, a function will be introduced that performs the transform and directly returns a Signal object."

[hoyer-a]

Maybe mention that where these errors occur depends on the maximum order N (hope i remember this correctly)

[hoyer-a]

should we explain why the right ear signal is flipped in sign?

[mberz]

@hoyer-a @f-brinkmann
I've updated a couple of things, condensed the notebook in some parts. Please have a look.