Understanding the Concept:
The bit error probability ($P_e$) of various binary digital signaling configurations operating under an Additive White Gaussian Noise (AWGN) channel context is fundamentally governed by the Euclidean distance separating the signaling points in their respective signal constellation space diagrams.
The bit error probability expressions for coherent digital systems are given below:
• Coherent Binary Phase Shift Keying (BPSK):
$$P_{e,\text{BPSK}} = Q\left(\sqrt{\frac{2E_b}{N_0}}\right)$$
• Coherent Binary Frequency Shift Keying (BFSK):
$$P_{e,\text{BFSK}} = Q\left(\sqrt{\frac{E_b}{N_0}}\right)$$
• Coherent Binary Amplitude Shift Keying (BASK ASK):
$$P_{e,\text{ASK}} = Q\left(\sqrt{\frac{E_b}{2N_0}}\right)$$
• Differential Phase Shift Keying (DPSK - Non-coherent):
$$P_{e,\text{DPSK}} = \frac{1}{2}\exp\left(-\frac{E_b}{N_0}\right)$$
Step-by-step Evaluation:
• The $Q(x)$ function is a monotonically decreasing function as its argument $x$ increases. Therefore, the scheme that maximizes the interior argument inside the radical will yield the lowest numeric value for the probability of error.
• Let us evaluate the scalar multipliers of $\frac{E_b}{N_0}$ inside the respective square roots:
• For PSK (BPSK): Multiplier is $2$.
• For FSK: Multiplier is $1$.
• For ASK: Multiplier is $0.5$.
• Comparing the terms, we observe that:
$$2 > 1 > 0.5$$
$$\Rightarrow \sqrt{\frac{2E_b}{N_0}} > \sqrt{\frac{E_b}{N_0}} > \sqrt{\frac{E_b}{2N_0}}$$
• Since the argument for PSK is the largest, its corresponding $Q$-value output is the lowest, giving it the highest noise immunity. Non-coherent DPSK performs worse than coherent PSK by roughly $3\text{ dB}$ at typical high SNRs because of its differential phase reference noise tracking.