Constellation Diagram of PSK in Detail

• 📘 Overview
• 📘 Simulator for Constellation of BPSK
• 📘 Energy Per Bit Calculation for BPSK Constellation
• 🧮 MATLAB Code
• 📚 Further Reading

 

 

 

 

 

 

Fig 1: Constellation Diagram of PSK

 

 In the above figure, the binary bit '1' is represented by S1(t) and the binary bit '0' by S2(t), respectively.

So, energy of S1(t) = (√(Eb))2 = Eb
So, energy of S2(t) = (-√(Eb))2 = Eb

Distance between the signaling points, d12 = 2(√(Eb))

 

Energy per bit for binary '1' and binary '0' (Tb = bit duration)

** For Transmission of binary ‘1’:
$E_b = \int_{0}^{T_b} (A_c \cos(2\Pi f_c t))^2 dt$
$= \int_{0}^{T_b} \frac{(A_c)^2}{2} dt + \int_{0}^{T_b} \frac{(A_c)^2 \cos(4\Pi f_c t)}{2} dt$
$= \int_{0}^{T_b} \frac{(A_c)^2}{2} dt + 0$ (The integral of $\cos(4\Pi f_c t)$ over a complete cycle is zero)
$= \frac{(A_c)^2}{2} \cdot T_b$
From this, $A_c = \sqrt{\frac{2E_b}{T_b}}$

** For Transmission of binary ‘0’:
$E_b = \int_{0}^{T_b} (-A_c \cos(2\Pi f_c t))^2 dt$
$= \int_{0}^{T_b} \frac{(A_c)^2}{2} dt + \int_{0}^{T_b} \frac{(A_c)^2 \cos(4\Pi f_c t)}{2} dt$
$= \int_{0}^{T_b} \frac{(A_c)^2}{2} dt + 0$ (The integral of $\cos(4\Pi f_c t)$ over a complete cycle is zero)
$= \frac{(A_c)^2}{2} \cdot T_b$
From this, $A_c = \sqrt{\frac{2E_b}{T_b}}$

** Constellation Diagram Representation:
For binary '1': $S_1(t) = A_c \cos(2\Pi f_c t) = \sqrt{\frac{2E_b}{T_b}} \cos(2\Pi f_c t) = \sqrt{E_b} \cdot \sqrt{\frac{2}{T_b}} \cos(2\Pi f_c t)$
For binary '0': $S_2(t) = -A_c \cos(2\Pi f_c t) = -\sqrt{\frac{2E_b}{T_b}} \cos(2\Pi f_c t) = -\sqrt{E_b} \cdot \sqrt{\frac{2}{T_b}} \cos(2\Pi f_c t)$

 

 High-order PSK (e.g., 8 PSK, 16 PSK) can transmit more bits per symbol but is more sensitive to noise. Low-order PSK (e.g., BPSK, QPSK) is less susceptible to noise.
PSK modulation can be visualized using a constellation diagram, where each point represents a symbol. In the presence of noise, points may be away from the original positions, making them harder to distinguish. 

Further Reading

1. Constellation Digrams of ASK, FSK, and PSK together
2. Constellation Digram of m-ary PSK
3. Constellation Digram of m-ary QAM