1. Starting with PWM
In PWM (Pulse Width Modulation):
- You have a carrier signal with a fixed period (e.g., 2 ms) and a duty cycle that can be adjusted.
- For example, if you start with a 50% duty cycle, the pulse will stay "on" for 1 ms in a 2 ms period.
- The modulating signal (often a sine wave or other waveform) influences the duty cycle. If the modulating signal has an amplitude of \( t = 0 \), this could result in a pulse width of 1 ms.
2. Transitioning to PPM
In PPM (Pulse Position Modulation):
- The pulse position changes instead of the pulse width. The idea is that instead of adjusting the width of the pulse, you adjust the position of the pulse within the carrier period.
- The modulating signal (amplitude \( x(t_i) \)) at each sample point determines where the pulse will occur within the period of the carrier.
3. How the Equation Works
The equation:
\( t_{\text{pulse}}(t_i) = t_i + \Delta t \cdot x(t_i) \)
This calculates the position of the pulse for each sample point:
t_iis the base time (the time for the start of the period, e.g., 0 ms, 2 ms, 4 ms, etc.).Îtis the sampling interval, or the time step between samples (e.g., 1 ms).x(t_i)is the amplitude of the modulating signal at timet_i.
The pulse position is shifted by an amount proportional to the amplitude of the modulating signal.
Example Breakdown
At \( t_i = 0 \) (start of the carrier period):
- Assume the modulating signal has an amplitude of \( x(0) = 0.5 \).
- The pulse position is calculated as:
\( t_{\text{pulse}}(0) = 0 + 1 \, \text{ms} \cdot 0.5 = 0.5 \, \text{ms} \)
- So, the pulse occurs at 0.5 ms within the 2 ms period.
At \( t_i = 2 \, \text{ms} \) (next carrier period):
- Assume the modulating signal is still \( x(2) = 0.5 \).
- The pulse position is:
\( t_{\text{pulse}}(2) = 2 \, \text{ms} + 1 \, \text{ms} \cdot 0.5 = 2.5 \, \text{ms} \)
- So, the pulse occurs at 2.5 ms within the next 2 ms period.
At \( t_i = 4 \, \text{ms} \) (next cycle):
- The modulating signal might still be \( x(4) = 0.5 \).
- The pulse position is:
\( t_{\text{pulse}}(4) = 4 \, \text{ms} + 1 \, \text{ms} \cdot 0.5 = 4.5 \, \text{ms} \)
- So, the pulse occurs at 4.5 ms.
Recap of Key Points
- In PWM, the width of the pulse (its duty cycle) changes based on the modulating signal. For example, if the modulating signal has a value of 0.5, the pulse width might be 50% of the carrier period.
- In PPM, the position of the pulse shifts based on the modulating signal. At each sample point, the pulse position is calculated using the formula:
\( t_{\text{pulse}}(t_i) = t_i + \Delta t \cdot x(t_i) \)
- In your case, the pulse position is shifting according to the modulating signal’s amplitude, so the pulse occurs at times like 0.5 ms, 2.5 ms, 4.5 ms, and so on, within each 2 ms carrier period.
Final Summary
Your interpretation is correct. In PPM, the pulse position is adjusted based on the modulating signal's amplitude at each sample point, as determined by the equation:
\( t_{\text{pulse}}(t_i) = t_i + \Delta t \cdot x(t_i) \)