Frequency Shift Keying (FSK)
Theoretical Foundations:
Frequency Shift Keying (FSK) is a discrete frequency modulation scheme wherein the digital information is encoded via instantaneous shifts in the carrier signal's frequency. The fundamental implementation is Binary FSK (BFSK), which maps binary data onto two distinct, discrete spectral states.
A binary '1' (the "mark" state) is represented by a carrier frequency \( f_1 \), while a binary '0' (the "space" state) corresponds to frequency \( f_2 \). Each symbol is sustained for a bit interval denoted by \( T_b \).
FSK Transmitter Characterization:
The mathematical model for the modulated BFSK output \( s(t) \) is defined as:
\[ s(t) = \begin{cases} A_c \cos(2\pi f_1 t), & \text{for } m = 1 \\ A_c \cos(2\pi f_2 t), & \text{for } m = 0 \end{cases} \]
In a Binary Frequency Shift Keying (BFSK) architecture, the baseband bitstream modulates a Voltage-Controlled Oscillator (VCO). The VCO functions as a frequency synthesizer where the instantaneous oscillation frequency is a function of the input voltage level.
Coherent FSK Demodulation
The coherent (synchronous) BFSK receiver functions by exploiting the orthogonality of the two signaling frequencies. The process assumes precise phase and frequency alignment through locally synthesized carrier references. The received waveform \( y(t) \) is processed via a dual-branch architecture, where each branch is tuned to one of the possible transmitted tones.
Detailed Step-by-Step Breakdown:
- Product Modulation (Correlation): The incoming signal is multiplied by two local carriers. If the incoming frequency matches the local carrier (e.g., \( f_1 \)), the multiplication results in a DC component (representing signal energy) and a high-frequency component (\( 2f_1 \)). If they do not match, the result consists only of high-frequency "sum and difference" tones.
- Integration and Filtering: The Low-Pass Filter (or Integrator) acts as an energy detector. It effectively averages the signal over the bit period \( T_b \). It suppresses the high-frequency products while allowing the DC component (if present) to pass through as a "sufficient statistic."
- Comparison and Decision: The outputs of the two branches are compared. The branch whose local frequency matches the transmitted tone will produce a high voltage level, while the "mismatched" branch produces a level near zero. A comparator chooses the maximum of the two to reconstruct the original bit.
Analytical State Analysis
The transmitted signal states are formally represented as: \( S_{1}(t) = A \cos(2\pi f_1 t) \) (Bit '1') and \( S_{2}(t) = A \cos(2\pi f_2 t) \) (Bit '0').
Case 1: Bit '1' is Received
- Branch 1 Output: \( \int [A \cos(2\pi f_1 t) \cdot \cos(2\pi f_1 t)] dt \rightarrow \) High Value (\( A/2 \))
- Branch 2 Output: \( \int [A \cos(2\pi f_1 t) \cdot \cos(2\pi f_2 t)] dt \rightarrow \) Zero (due to orthogonality)
- Decision: Branch 1 > Branch 2 → Logic '1'
Case 2: Bit '0' is Received
- Branch 1 Output: \( \int [A \cos(2\pi f_2 t) \cdot \cos(2\pi f_1 t)] dt \rightarrow \) Zero
- Branch 2 Output: \( \int [A \cos(2\pi f_2 t) \cdot \cos(2\pi f_2 t)] dt \rightarrow \) High Value (\( A/2 \))
- Decision: Branch 2 > Branch 1 → Logic '0'
Signal-Space (Constellation) Representation
BFSK signals are mapped onto a 2D signal space using orthonormal basis functions, reflecting the dual-frequency nature of the modulation:
\[ \phi_1(t) = \sqrt{\frac{2}{T_b}} \cos(2\pi f_1 t), \quad \phi_2(t) = \sqrt{\frac{2}{T_b}} \cos(2\pi f_2 t) \]
The constellation points are located on perpendicular axes. The minimum Euclidean distance between states is: \[ d_{12} = \sqrt{2E_b} \]