====== UVC Hue rotation (YCbCr 4:4:4) ======

This block applies UVC Hue adjustment by rotating the chroma vector \((C_b, C_r)\)
around the neutral point, while leaving luma \(Y\) unchanged.

This block does **not** clamp the output values. Chroma values outside the nominal
12-bit full-range are allowed and are expected to be handled later in the pipeline.

This block takes as input 12-bit per-channel YCbCr 4:4:4 of the following format:
\[
 Y12 \in [0..4095]\\
 C_b12 \in [0..4095]\text{, neutral at }2048\\
 C_r12 \in [0..4095]\text{, neutral at }2048
\]

===== Control input =====

Hue control value is a signed integer \(H\) in degrees \(\times 100\):
\[
H \in [-18000..+18000]\text{, default }0
\]

Rotation angle (in degrees):
\[
\theta = H / 100
\]

===== Step 1: calculate chroma deltas =====

Convert chroma channels into signed deltas around neutral:
\[
\Delta C_b = C_b12 - 2048\\
\Delta C_r = C_r12 - 2048
\]

===== Step 2: rotate chroma vector =====

Apply 2D rotation:
\[
\Delta C_b' = \Delta C_b \cdot \cos(\theta) - \Delta C_r \cdot \sin(\theta)\\
\Delta C_r' = \Delta C_b \cdot \sin(\theta) + \Delta C_r \cdot \cos(\theta)
\]

===== Step 3: re-center chroma =====

Re-apply neutral offset:
\[
C_b^{out} = 2048 + \Delta C_b'\\
C_r^{out} = 2048 + \Delta C_r'\\
Y^{out} = Y12
\]

No clamping or range limiting is performed in this block.

===== Fixed-point implementation =====

==== Trigonometric coefficients (MCU) ====

The microcontroller computes \(\sin(\theta)\) and \(\cos(\theta)\) and writes them
as signed Q18 fixed-point values:
\[
\sin_q = \mathrm{round}(\sin(\theta) \cdot 2^{18})\\
\cos_q = \mathrm{round}(\cos(\theta) \cdot 2^{18})
\]

==== FPGA math (signed Q18) ====

Chroma deltas are treated as signed values:
\[
\Delta C_b, \Delta C_r \in [-2048..+2047]
\]

Multiply-accumulate:
\[
T_b = \Delta C_b \cdot \cos_q - \Delta C_r \cdot \sin_q\\
T_r = \Delta C_b \cdot \sin_q + \Delta C_r \cdot \cos_q
\]

Convert back from Q18 using rounding-to-nearest:
\[
\Delta C_b' = (T_b + 131072) >> 18\\
\Delta C_r' = (T_r + 131072) >> 18
\]

All operations use **signed arithmetic with arithmetic right shift**.

===== Bit width after hue rotation =====

Input deltas:
\[
\Delta C_b, \Delta C_r \in [-2048..+2047]
\]

Maximum chroma vector magnitude:
\[
r_{max} = \sqrt{2048^2 + 2048^2} \approx 2896
\]

Maximum rotated deltas:
\[
\Delta C_b', \Delta C_r' \in [-2896..+2896]
\]

After re-centering:
\[
C_b^{out}, C_r^{out} \in [-848..+4944]
\]

13-bit signed storage is insufficient.

Minimum safe width is **14-bit signed**.
Recommended internal chroma width is **15–16 bits signed** to allow margin for
rounding and subsequent chroma-domain adjustments.