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.
Also read about
- Constellation Diagram of ASK in detail
- Constellation Diagram of PSK in detail
- Baseband ASK, FSK, and PSK
- Theoretical BER vs SNR for binary ASK, FSK, and PSK
BER vs SNR from Constellation Diagram of FSK
The Bit Error Rate (BER) versus Signal-to-Noise Ratio (SNR) can be derived from the Euclidean distance between constellation points. Once the minimum distance between the symbols is known, the probability of making an incorrect decision in an Additive White Gaussian Noise (AWGN) channel can be computed using the Q-function.
The Q-function gives the probability that Gaussian noise exceeds the decision threshold, causing the received symbol to cross into the neighboring decision region and resulting in an incorrect bit detection.
Example: Coherent Orthogonal FSK
For coherent orthogonal Frequency Shift Keying (FSK), the theoretical BER is
The Euclidean distance between the two orthogonal signal points is
Substituting this distance into the general AWGN error probability formula gives
= Q(√(2Eb) / √(2N0))
= Q(√(Eb / N0))
Why is BER approximately 0.15 at 0 dB?
At an SNR of 0 dB,
Therefore,
= Q(1)
Using the standard Gaussian Q-function table,
Thus, at an SNR of 0 dB, the theoretical BER for coherent orthogonal FSK is approximately
This matches the theoretical BER curve for coherent orthogonal FSK in an AWGN channel.
Q(√(2Eb/N0)), the BER at 0 dB becomes
Q(√2) = Q(1.4142) ≈ 0.0786, which corresponds to
coherent BPSK (and QPSK), not coherent orthogonal FSK.