Differential Phase-Shift Keying (DPSK)

This example demonstrates differential modulation schemes in Kaira, specifically DBPSK and DQPSK. Differential modulation encodes information in the phase differences between consecutive symbols, making it robust against phase ambiguity.

import matplotlib.pyplot as plt

Imports and Setup

import numpy as np
import torch

from kaira.channels import AWGNChannel, FlatFadingChannel
from kaira.modulations import DBPSKDemodulator, DBPSKModulator, DQPSKDemodulator, DQPSKModulator
from kaira.modulations.utils import plot_constellation
from kaira.utils import snr_to_noise_power

# Set random seed for reproducibility
torch.manual_seed(42)
np.random.seed(42)

Generate Test Data and Create Modulators

n_symbols = 1000

# Generate random bits
bits_dbpsk = torch.randint(0, 2, (1, n_symbols))
bits_dqpsk = torch.randint(0, 2, (1, 2 * n_symbols))  # DQPSK uses 2 bits per symbol

# Create modulators and demodulators
dbpsk_mod = DBPSKModulator()
dbpsk_demod = DBPSKDemodulator()
dqpsk_mod = DQPSKModulator()
dqpsk_demod = DQPSKDemodulator()

# Modulate the data
dbpsk_symbols = dbpsk_mod(bits_dbpsk)
dqpsk_symbols = dqpsk_mod(bits_dqpsk)

Visualize Phase Transitions

fig, axs = plt.subplots(1, 3, figsize=(15, 5))

# DBPSK phase transitions
axs[0].plot(dbpsk_symbols.real.numpy().flatten()[:-1], dbpsk_symbols.real.numpy().flatten()[1:], "b.", alpha=0.5, label="Transitions")
axs[0].grid(True)
axs[0].set_xlabel("Current Symbol (Real)")
axs[0].set_ylabel("Next Symbol (Real)")
axs[0].set_title("DBPSK Phase Transitions")
axs[0].set_aspect("equal")

# DQPSK constellation
plot_constellation(dqpsk_symbols.flatten(), title="DQPSK Constellation", marker=".", ax=axs[1])

# DQPSK phase transitions
angles = torch.angle(dqpsk_symbols)
axs[2].plot(angles[0, :-1].numpy(), angles[0, 1:].numpy(), "r.", alpha=0.5)
axs[2].set_xlabel("Current Phase (rad)")
axs[2].set_ylabel("Next Phase (rad)")
axs[2].set_title("DQPSK Phase Transitions")
axs[2].grid(True)

plt.tight_layout()
plt.show()

Compare Performance in AWGN and Fading Channels

snr_db_range = np.arange(0, 21, 2)
n_trials = 100  # Number of trials for each SNR point

# Initialize arrays for storing BER results
ber_dbpsk_awgn = []
ber_dqpsk_awgn = []
ber_dbpsk_fading = []
ber_dqpsk_fading = []

for snr_db in snr_db_range:
    # Setup channels
    # Convert numpy value to torch tensor before passing to snr_to_noise_power
    noise_power = snr_to_noise_power(1.0, torch.tensor(snr_db))
    awgn_channel = AWGNChannel(avg_noise_power=noise_power)
    fading_channel = FlatFadingChannel(fading_type="rayleigh", coherence_time=10, avg_noise_power=noise_power)  # Use Rayleigh fading  # Fading remains constant for 10 symbols

    # Arrays for storing BER for current SNR
    ber_dbpsk_awgn_trials = []
    ber_dqpsk_awgn_trials = []
    ber_dbpsk_fading_trials = []
    ber_dqpsk_fading_trials = []

    for _ in range(n_trials):
        # AWGN channel transmission
        received_dbpsk_awgn = awgn_channel(dbpsk_symbols)
        received_dqpsk_awgn = awgn_channel(dqpsk_symbols)

        demod_dbpsk_awgn = dbpsk_demod(received_dbpsk_awgn)
        demod_dqpsk_awgn = dqpsk_demod(received_dqpsk_awgn)

        # Note: Differential demodulation produces one less symbol, so we skip the first bit
        ber_dbpsk_awgn_trials.append(torch.mean((demod_dbpsk_awgn != bits_dbpsk[:, 1:]).float()).item())
        ber_dqpsk_awgn_trials.append(torch.mean((demod_dqpsk_awgn != bits_dqpsk[:, 2:]).float()).item())

        # Fading channel transmission
        received_dbpsk_fading = fading_channel(dbpsk_symbols)
        received_dqpsk_fading = fading_channel(dqpsk_symbols)

        demod_dbpsk_fading = dbpsk_demod(received_dbpsk_fading)
        demod_dqpsk_fading = dqpsk_demod(received_dqpsk_fading)

        ber_dbpsk_fading_trials.append(torch.mean((demod_dbpsk_fading != bits_dbpsk[:, 1:]).float()).item())
        ber_dqpsk_fading_trials.append(torch.mean((demod_dqpsk_fading != bits_dqpsk[:, 2:]).float()).item())

    # Store average BER for current SNR
    ber_dbpsk_awgn.append(np.mean(ber_dbpsk_awgn_trials))
    ber_dqpsk_awgn.append(np.mean(ber_dqpsk_awgn_trials))
    ber_dbpsk_fading.append(np.mean(ber_dbpsk_fading_trials))
    ber_dqpsk_fading.append(np.mean(ber_dqpsk_fading_trials))

Plot BER Performance Comparison

plt.figure(figsize=(12, 5))

# AWGN channel performance
plt.subplot(121)
plt.semilogy(snr_db_range, ber_dbpsk_awgn, "bo-", label="DBPSK")
plt.semilogy(snr_db_range, ber_dqpsk_awgn, "ro-", label="DQPSK")
plt.grid(True)
plt.xlabel("SNR (dB)")
plt.ylabel("Bit Error Rate (BER)")
plt.title("Performance in AWGN Channel")
plt.legend()

# Fading channel performance
plt.subplot(122)
plt.semilogy(snr_db_range, ber_dbpsk_fading, "bo-", label="DBPSK")
plt.semilogy(snr_db_range, ber_dqpsk_fading, "ro-", label="DQPSK")
plt.grid(True)
plt.xlabel("SNR (dB)")
plt.ylabel("Bit Error Rate (BER)")
plt.title("Performance in Flat Fading Channel")
plt.legend()

plt.tight_layout()
plt.show()

Visualize Phase Recovery

n_test = 100
test_bits = torch.randint(0, 2, (1, n_test))
test_symbols = dbpsk_mod(test_bits)

# Add phase rotation
phase_rotation = np.pi / 3  # 60 degrees rotation
rotated_symbols = test_symbols * torch.exp(1j * torch.tensor(phase_rotation))

# Demodulate rotated symbols
demod_bits = dbpsk_demod(rotated_symbols)

# Plot original and rotated symbols
fig, axs = plt.subplots(1, 3, figsize=(15, 5))

# Original constellation
plot_constellation(test_symbols.flatten(), title="Original DBPSK Symbols", marker=".", ax=axs[0])

# Rotated constellation
plot_constellation(rotated_symbols.flatten(), title=f"Rotated Symbols ({phase_rotation:.1f} rad)", marker=".", ax=axs[1])

# Phase differences
phase_diff = torch.angle(rotated_symbols[0, 1:] * torch.conj(rotated_symbols[0, :-1]))
axs[2].hist(phase_diff.numpy(), bins=50)
axs[2].set_title("Phase Differences Distribution")
axs[2].set_xlabel("Phase Difference (rad)")
axs[2].set_ylabel("Count")
axs[2].grid(True)

plt.tight_layout()
plt.show()

# Print demodulation accuracy despite rotation - compare with bits[1:] since differential demodulation loses first symbol
accuracy = torch.mean((demod_bits == test_bits[:, 1:]).float()).item()
print(f"Demodulation accuracy with {phase_rotation:.1f} rad rotation: {accuracy:.2%}")

Demodulation accuracy with 1.0 rad rotation: 100.00%

Conclusion

This example demonstrated:

1. Implementation of DBPSK and DQPSK modulation using Kaira

2. Visualization of phase transitions and constellation diagrams

3. Performance comparison in AWGN and fading channels

4. Phase ambiguity tolerance of differential modulation

Key observations:

• Differential modulation maintains performance under phase rotation

• DBPSK offers more robust performance than DQPSK

• Performance degradation is more severe in fading channels

• Phase differences remain consistent despite absolute phase rotation

• DQPSK provides higher spectral efficiency at the cost of BER performance