ylm.lib

A library to compute real N3D¹ Spherical Harmonics at arbitrary degree $l$ and order $m$, according to the following formula:

\[\begin{equation} Y_{l,m}(\theta, \phi) = \left\lbrace \begin{aligned} N_{l,m} P_l^m\left(\sin(\phi)\right) \cos(m \theta) & & \text{for} & & m \geq 0 \\ N_{l,m} P_l^m\left(\sin(\phi)\right) \sin(m \theta) & & \text{for} & & m < 0 \end{aligned} \right., \label{eq:sph} \end{equation}\]

where $P_l^m$ are the associated Legendre polynomial of the first kind of degree $l$ and order $m$, with ${(l,m) \in (\mathbb{N}, \mathbb{Z}) ~|~ |m| \leq l}$. The Condon-Shortley Phase is not included, and the ACN convention² is used. $N_{l,m}$ is the “N3D” normalization factor ensuring the orthonormality of the SHs on the unit sphere³. It is given by: \begin{equation} N_{l,m} = \sqrt{(2- \delta_m) (2 l + 1)\frac{(l-|m|)!}{(l+|m|)!}}, \label{eq:norm} \end{equation} where $\delta_m$ is the Kronecker delta function. Details of implementation are given in⁴ (Note that spherical coordinate system and notations differ in⁴)

Functions

factorial(m)

Computes the factorial $m!$ of integer $m$.

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factorial2(m)

Computes the double factorial $m!!$ of integer $m$.

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legendre(l,x)

Computes the Legendre polynomial $P_{l}(x)$ of degree $l$.

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alegendre(l,x)

Computes the associated Legendre polynomial $P_{l,m}(x)$ of degree $l$ and order $m$.

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chebychev2(m,x)

Computes the Chebyshev polynomial $U_m(x)$ of $2^{\text{nd}}$ kind.

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n3d(l,m)

Computes the “N3D” norm $N_{l,|m|}$ of Eq. \eqref{eq:norm}.

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ylm(l,m,t,p)

Computes the N3D real Spherical Harmonic $Y_{l,m}(\theta,\phi)$ of Eq. \eqref{eq:sph}. –

sylm(l,m,t,p)

Computes the N3D real Spherical Harmonic $Y_{l,m}(\theta,\phi)$ of Eq. \eqref{eq:sph}, with smoothing functions.

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acn(l,m)

Computes the Ambisonic Channel Number (ACN)² from indexes $(l,m)$:

\[\begin{equation} \left\lbrace \begin{aligned} &\text{ACN}(l,m) = i = l^2 + l + m \\ &m = \lfloor \sqrt{i} \rfloor \\ &l = i - m^2 - m \end{aligned} \right. \label{eq:acn} \end{equation}\]

In Eq. \eqref{eq:acn}, $\lfloor \cdot \rfloor$ is the floor function.

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yacn(i,t,p)

Computes the $i^\text{th}$ N3D real Spherical Harmonic $Y_i(\theta,\phi)$ with ACN $i$.

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syacn(i,t,p)

Computes the $i^\text{th}$ N3D real Spherical Harmonic $Y_i(\theta,\phi)$ with ACN $i$ and smoothing function.

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yvec(imax,t,p)

Vector of Spherical Harmonics $\mathbf{y}(\theta,\phi)$. A signal bus of dimensions $i_\text{max}$ is multiplied with the corresponding Spherical Harmonics, i.e., the $i^\text{th}$ signal is multiplied with $Y_i(\theta,\phi)$.

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syvec(imax,t,p)

Vector of Spherical Harmonics $\mathbf{y}(\theta,\phi)$ with smoothing function. A signal bus of dimensions $i_\text{max}$ is multiplied with the corresponding Spherical Harmonics, i.e., the $i^\text{th}$ signal is multiplied with $Y_i(\theta,\phi)$.

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costheta(t1,p1,t2,p2)

Cosine of angle $\Theta$ between directions $(\theta_1,\phi_1)$ and $(\theta_2,\phi_2)$ such that:

\[\cos(\Theta) = \cos(\phi_1)\cos(\phi_2)\cos(\theta_1 - \theta_2) + \sin(\phi_1) \sin(\phi_2)\]

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buswg(c)

Apply the gains defined in the list c to the input bus. Useful for matrix multiplication.

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sup(c)

Returns the maximum of input bus.

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wre(lmax, l1, l)

For a on-axis normalized max-$r_E$ beampattern at degree $L$, the weights at degree $l$, denoted $w_L(l)$ are given by⁵:

\[\begin{equation} w_{\text{max-}r_E,l}(L) = \left\lbrace \begin{aligned} &\frac{P_l\left(\cos\left(\frac{137.9^\circ}{L + 1.51}\right)\right)}{\sum_\limits{i=0}^{L} (2 i + 1) P_i\left(\cos\left(\frac{137.9^\circ}{L + 1.51}\right)\right)} & \text{for } & l \leq L \\ &0 & \text{for } & l > L \end{aligned} \right. \label{eq:acn2} \end{equation}\]

• lmax: maximum degree needed at compilation time: If l1 > lmax, then wre(lmax, l1, l) = 0.
• l1: beampattern degree,
• l: required degree.

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vecwre(lmax,l)

Vector of on-axis normalized max-$r_E$ beampattern weights.

• lmax: maximum degree needed at compilation time: If l1 > lmax, then wre(lmax, l1, l) = 0.
• l1: beampattern degree.

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maxre(lmax,l1)

Vector of on-axis normalized max-$r_E$ beampattern weights. The beampattern degree is l1 and its steering angle $(\theta_0, \phi_0)$

• lmax: maximum degree needed at compilation time: If l1 > lmax, then wre(lmax, l1, l) = 0.
• l1: beampattern degree.

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