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Asked by: Jaswanth Karri
Created at: 2025-08-01 10:39:10
Asked by: Jaswanth Karri
Created at: 2025-08-01 10:39:10
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### Given:
- **OOK modulator** detected by ideal matched filter receiver.
- Non-zero symbol at matched filter input: **Rectangular pulse**
- **Amplitude** = 100 mV = 0.1 V
- **Duration** = 10 ms = 0.01 s
- **Noise**: White & Gaussian, **RMS value** = 140 mV = 0.14 V (measured in 10 kHz bandwidth)
---
## **3.5.1 Energy per non-zero symbol**
Energy of a rectangular pulse:
\[
E_s = (\text{Amplitude})^2 \times \text{Duration}
\]
\[
E_s = (0.1~V)^2 \times 0.01~s = 0.01~V^2 \times 0.01~s = 0.0001~V^2s
\]
\[
\boxed{E_s = 1 \times 10^{-4}~\text{J (per non-zero symbol)}}
\]
---
## **3.5.2 Average energy per symbol**
For OOK, if the probability of "1" is 0.5 and "0" is 0.5 (assuming equiprobable symbols):
\[
E_{avg} = P(1) \times E_s + P(0) \times 0 = 0.5 \times 1 \times 10^{-4} + 0.5 \times 0 = 0.5 \times 10^{-4}~\text{J}
\]
\[
\boxed{E_{avg} = 5 \times 10^{-5}~\text{J}}
\]
---
## **3.5.3 Noise Power Spectral Density**
RMS noise voltage (\( V_{rms} \)) over bandwidth (\( B \)) relates to \( N_0 \) as:
\[
V_{rms}^2 = N_0 B
\]
\[
(0.14)^2 = N_0 \times 10,000
\]
\[
0.0196 = N_0 \times 10,000
\]
\[
N_0 = \frac{0.0196}{10,000} = 1.96 \times 10^{-6}~\text{V}^2/\text{Hz}
\]
\[
\boxed{N_0 = 1.96 \times 10^{-6}~\text{V}^2/\text{Hz}}
\]
---
## **3.5.4 Probability of Symbol Error (\(P_e\))**
For OOK with matched filter and white Gaussian noise:
\[
P_e = Q \left( \sqrt{\frac{E_s}{2N_0}} \right)
\]
Where \( Q(x) = \frac{1}{\sqrt{2\pi}} \int_x^\infty e^{-t^2/2} dt \)
Let's calculate the argument:
\[
\frac{E_s}{2N_0} = \frac{1 \times 10^{-4}}{2 \times 1.96 \times 10^{-6}} = \frac{1 \times 10^{-4}}{3.92 \times 10^{-6}} \approx 25.51
\]
\[
\sqrt{\frac{E_s}{2N_0}} \approx \sqrt{25.51} \approx 5.05
\]
For large arguments, \( Q(5.05) \) is extremely small:
\[
P_e \approx Q(5.05) \approx 2.2 \times 10^{-7}
\]
(From Q-function tables)
\[
\boxed{P_e \approx 2.2 \times 10^{-7}}
\]
---
## **Summary Table**
| Part | Answer |
|-----------|------------------------------------------------|
| 3.5.1 | \(1 \times 10^{-4}\) J per non-zero symbol |
| 3.5.2 | \(5 \times 10^{-5}\) J average energy/symbol |
| 3.5.3 | \(1.96 \times 10^{-6}\) V\(^2\)/Hz |
| 3.5.4 | \(2.2 \times 10^{-7}\) (Probability of error) |
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