Binary bits '0' and '1' can be mapped to 'j' and '1' to '1', respectively, for Baseband Binary Frequency Shift Keying (BFSK). Signals are in phase here. These bits can be mapped into baseband representation for a number of uses, including power spectral density (PSD) calculations. For passband BFSK transmission, we can modulate signal 'j' with a lower carrier frequency and signal '1' with a higher carrier frequency while transmitting over a wireless channel.
Let's assume we are transmitting carrier signal fc1 for the transmission of binary bit '1' and carrier signal fc2 for the transmission of binary bit '0'.
Simulator for 2-FSK Constellation Diagram
Simulator for 2-FSK Constellation Diagram
Energy per bit (Eb):
For transmission of binary ‘1’
Starting from the passband signal \( s_1(t)=A_c\cos(2\pi f_1 t) \) over \(0\le t\le T_b\):
\[ E_b \;=\; \int_{0}^{T_b}\!\big(A_c\cos 2\pi f_1 t\big)^2\,dt \;=\; \int_{0}^{T_b}\!\frac{A_c^2}{2}\,dt \;+\; \int_{0}^{T_b}\!\frac{A_c^2}{2}\cos(4\pi f_1 t)\,dt \] \[ \;=\; \int_{0}^{T_b}\!\frac{A_c^2}{2}\,dt \;+\; 0 \quad \text{(second term averages to 0 over a full cycle)} \;=\; \frac{A_c^2}{2}\,T_b . \]
For transmission of binary ‘0’
Similarly for \( s_2(t)=A_c\cos(2\pi f_2 t) \):
\[ E_b \;=\; \int_{0}^{T_b}\!\big(A_c\cos 2\pi f_2 t\big)^2\,dt \;=\; \int_{0}^{T_b}\!\frac{A_c^2}{2}\,dt \;+\; \int_{0}^{T_b}\!\frac{A_c^2}{2}\cos(4\pi f_2 t)\,dt \] \[ \;=\; \int_{0}^{T_b}\!\frac{A_c^2}{2}\,dt \;+\; 0 \;=\; \frac{A_c^2}{2}\,T_b . \]
Amplitude in terms of \(E_b\)
\[ A_c \;=\; \sqrt{\frac{2E_b}{T_b}} \;. \]
Constellation Diagram of FSK
In Binary FSK (BFSK), two orthogonal signals represent binary symbols:
\( s_1(t) = \sqrt{\frac{2E_b}{T_b}} \cos(2\pi f_1 t), \quad 0 \leq t \leq T_b \)
\( s_2(t) = \sqrt{\frac{2E_b}{T_b}} \cos(2\pi f_2 t), \quad 0 \leq t \leq T_b \)
The points lie on orthogonal axes because \( s_1(t) \) and \( s_2(t) \) are orthogonal signals.
Fig 1: Constellation Diagram of FSK
In the above figure values are in terms of the normalized functions. √(2/Tb).cos2Пf1t and √(2/Tb).cos2Пf2t are orthogonal functions in the interval (0, Tb). And the distance between signaling points, d12 = √(2Eb)
By interpreting these functions as vectors, the phase angle between the resulting vectors will be 90 degrees.
Using more frequency shifts to display multiple symbols or bits of digital data is known as high-order frequency shift keying (FSK). Every frequency in FSK corresponds to a distinct symbol or collection of bits. Higher data rates are possible with high-order FSK schemes, but they may also be more vulnerable to channel impairments and noise.
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