Timing Mathematics¶

Mathematical Description of Carrier and Packet Timing

Overview¶

REVA operates using two independent timing layers:

Carrier Timing
Packet Timing

The carrier defines pulse repetition rate.

The packet system defines how those pulses are organized into bursts and envelopes.

Carrier Frequency¶

The carrier frequency is generated by Timer1.

For a carrier frequency (f_c), the carrier period is:

$$ T_c = \frac{1}{f_c} $$

Example: 90 kHz¶

$$ T_c = \frac{1}{90000} $$

$$ T_c = 11.11\ \mu s $$

Example: 120 kHz¶

$$ T_c = \frac{1}{120000} $$

$$ T_c = 8.33\ \mu s $$

Timer1 Frequency Equation¶

$$ f_c = \frac{F_{CPU}}{2N(1+OCR1A)} $$

Where:

Symbol	Meaning
(F_{CPU})	CPU clock frequency
(N)	Prescaler
(OCR1A)	Compare register

For REVA:

$$ F_{CPU} = 16\,MHz $$

Packet Structure¶

Packet
 ├─ Segment 1
 ├─ Segment 2
 ├─ Segment 3
 └─ ...

Each segment contains:

Pulse Count
Gap Time

Segment Duration¶

If a segment contains (N) carrier cycles:

$$ T_{pulse} = N \cdot T_c $$

The total segment duration becomes:

$$ T_{segment} = T_{pulse} + T_{gap} $$

Packet Duration¶

$$ T_{packet} = \sum T_{segment} + T_{final} $$

Packet Frequency¶

$$ f_{packet} = \frac{1}{T_{packet}} $$

Packet Density¶

$$ Density = \frac{T_{active}}{T_{packet}} $$

Pulse Scaling¶

$$ N' = k_p \cdot N $$

Gap Scaling¶

$$ T'{gap} = k_g \cdot T{gap} $$

Geometry Preservation¶

Scaling changes size but preserves structure.

Original Geometry
        ↓
Scaling
        ↓
Larger Geometry